Subgroups — C₆₇×#22752 #30702003

Order 17152.

888 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2290	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1924	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1923	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1916	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1916	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1870	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1869	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1832	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#1768	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#1753	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#431×C₆₇	—	2	4	no	—
22	#424×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	#388×C₆₇	C₄	1	1	yes	characteristic
40	#388×C₆₇	C₄	1	1	yes	characteristic
41	#387×C₆₇	C₄	1	1	yes	characteristic
42	#387×C₆₇	C₄	1	1	yes	characteristic
43	#386×C₆₇	C₄	1	1	yes	characteristic
44	#386×C₆₇	C₄	1	1	yes	characteristic
45	#385×C₆₇	C₄	1	1	yes	characteristic
46	#385×C₆₇	C₄	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
48	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
49	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
50	#376×C₆₇	C₂²	1	1	yes	characteristic
51	#336×C₆₇	C₂²	1	1	yes	characteristic
52	#336×C₆₇	C₂²	1	1	yes	characteristic
53	#336×C₆₇	C₂²	1	1	yes	characteristic
54	#336×C₆₇	C₂²	1	1	yes	characteristic
55	#336×C₆₇	C₂²	1	1	yes	characteristic
56	#336×C₆₇	C₂²	1	1	yes	characteristic
57	#336×C₆₇	C₂²	1	1	yes	characteristic
58	#336×C₆₇	C₂²	1	1	yes	characteristic
59	#336×C₆₇	C₂²	1	1	yes	characteristic
60	#336×C₆₇	C₂²	1	1	yes	characteristic
61	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₆₇	—	2	2	no	—
67	C₂×M₁₆×C₆₇	—	2	2	no	—
68	C₂×M₁₆×C₆₇	—	4	1	no	—
69	C₂²×C₈×C₆₇	—	2	2	no	—
70	C₂²×C₈×C₆₇	—	2	2	no	—
71	C₂²×C₈×C₆₇	—	4	1	no	—
72	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₄×D₄×C₆₇	—	2	2	no	—
81	C₄×D₄×C₆₇	—	2	2	no	—
82	#117×C₆₇	—	2	2	no	—
83	#117×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	—	2	2	no	—
98	C₂×C₄²×C₆₇	—	2	2	no	—
99	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
106	C₄⋊C₈×C₆₇	—	2	2	no	—
107	C₄⋊C₈×C₆₇	—	2	2	no	—
108	C₄⋊C₈×C₆₇	—	2	2	no	—
109	C₄⋊C₈×C₆₇	—	2	2	no	—
110	C₄⋊C₈×C₆₇	—	2	2	no	—
111	C₄⋊C₈×C₆₇	D₄	1	4	yes	complemented
112	C₂²⋊C₈×C₆₇	—	2	2	no	—
113	C₂²⋊C₈×C₆₇	—	2	2	no	—
114	C₂²⋊C₈×C₆₇	—	2	2	no	—
115	C₂²⋊C₈×C₆₇	—	2	2	no	—
116	C₂²⋊C₈×C₆₇	—	4	1	no	—
117	C₂²⋊C₈×C₆₇	—	4	1	no	—
118	#97×C₆₇	—	4	1	no	—
119	C₄×C₈×C₆₇	—	2	2	no	—
120	C₄×C₈×C₆₇	—	2	2	no	—
121	C₄×C₈×C₆₇	—	4	1	no	—
122	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
123	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
124	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
125	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
126	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
127	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
128	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
129	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
130	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
131	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
132	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×D₄×C₆₇	—	2	2	no	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	4	1	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	1	no	—
147	C₂²×C₄×C₆₇	—	2	1	no	—
148	C₂²×C₄×C₆₇	—	2	1	no	—
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	4	1	no	—
153	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
159	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	#41	1	2	yes	—
166	C₂²×C₄×C₆₇	#41	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	M₁₆×C₆₇	—	4	4	no	—
169	C₂×C₈×C₆₇	—	2	1	no	—
170	C₂×C₈×C₆₇	—	2	1	no	—
171	C₂×C₈×C₆₇	—	2	1	no	—
172	C₂×C₈×C₆₇	—	2	1	no	—
173	C₂×C₈×C₆₇	—	2	1	no	—
174	C₂×C₈×C₆₇	—	2	1	no	—
175	C₂×C₈×C₆₇	—	2	1	no	—
176	C₂×C₈×C₆₇	—	2	1	no	—
177	C₂×C₈×C₆₇	—	2	1	no	—
178	C₂×C₈×C₆₇	—	2	1	no	—
179	C₂×C₈×C₆₇	—	2	1	no	—
180	C₂×C₈×C₆₇	—	2	1	no	—
181	C₂×C₈×C₆₇	—	2	1	no	—
182	C₂×C₈×C₆₇	—	2	1	no	—
183	C₂×C₈×C₆₇	—	2	1	no	—
184	C₂×C₈×C₆₇	—	2	1	no	—
185	C₂×C₈×C₆₇	—	2	1	no	—
186	C₂×C₈×C₆₇	—	2	1	no	—
187	C₂×C₈×C₆₇	—	2	1	no	—
188	C₂×C₈×C₆₇	—	2	1	no	—
189	C₂×C₈×C₆₇	—	2	1	no	—
190	C₂×C₈×C₆₇	—	2	1	no	—
191	C₂×C₈×C₆₇	—	4	4	no	—
192	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
193	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	2	2	no	—
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	2	2	no	—
199	C₂²⋊C₄×C₆₇	—	2	2	no	—
200	C₂²⋊C₄×C₆₇	—	2	2	no	—
201	C₂²⋊C₄×C₆₇	—	2	2	no	—
202	C₂²⋊C₄×C₆₇	—	2	2	no	—
203	C₂²⋊C₄×C₆₇	—	2	2	no	—
204	C₂²⋊C₄×C₆₇	—	2	2	no	—
205	C₂²⋊C₄×C₆₇	—	2	2	no	—
206	C₂²⋊C₄×C₆₇	—	4	1	no	—
207	C₄²×C₆₇	—	2	2	no	—
208	C₄²×C₆₇	—	2	2	no	—
209	C₄²×C₆₇	—	2	1	no	—
210	C₄²×C₆₇	—	2	1	no	—
211	C₄²×C₆₇	—	2	1	no	—
212	C₄²×C₆₇	—	2	1	no	—
213	C₄²×C₆₇	—	2	1	no	—
214	C₄²×C₆₇	—	2	1	no	—
215	C₄²×C₆₇	—	2	1	no	—
216	C₄²×C₆₇	—	2	1	no	—
217	C₄²×C₆₇	C₂×D₄	1	2	yes	—
218	C₄²×C₆₇	#41	1	2	yes	—
219	C₂³×C₆₇	—	2	1	no	—
220	C₂³×C₆₇	—	2	1	no	—
221	C₂³×C₆₇	—	2	1	no	—
222	C₂³×C₆₇	—	2	1	no	—
223	C₂³×C₆₇	—	2	1	no	—
224	C₂³×C₆₇	—	2	1	no	—
225	C₂³×C₆₇	—	2	1	no	—
226	C₂³×C₆₇	—	2	1	no	—
227	C₂³×C₆₇	—	2	1	no	—
228	C₂³×C₆₇	—	2	1	no	—
229	C₂³×C₆₇	—	2	1	no	—
230	C₂³×C₆₇	—	2	1	no	—
231	C₂³×C₆₇	#117	1	2	yes	—
232	C₂³×C₆₇	#131	1	2	yes	—
233	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
234	D₄×C₆₇	—	4	4	no	—
235	C₂×C₄×C₆₇	—	2	8	no	—
236	C₂×C₄×C₆₇	—	2	1	no	—
237	C₂×C₄×C₆₇	—	2	1	no	—
238	C₂×C₄×C₆₇	—	2	1	no	—
239	C₂×C₄×C₆₇	—	2	1	no	—
240	C₂×C₄×C₆₇	—	2	1	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	1	no	—
243	C₂×C₄×C₆₇	—	2	1	no	—
244	C₂×C₄×C₆₇	—	2	1	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	1	no	—
251	C₂×C₄×C₆₇	—	2	1	no	—
252	C₂×C₄×C₆₇	—	2	1	no	—
253	C₂×C₄×C₆₇	—	2	1	no	—
254	C₂×C₄×C₆₇	—	2	1	no	—
255	C₂×C₄×C₆₇	—	2	1	no	—
256	C₂×C₄×C₆₇	—	2	1	no	—
257	C₂×C₄×C₆₇	—	2	1	no	—
258	C₂×C₄×C₆₇	—	2	1	no	—
259	C₂×C₄×C₆₇	—	2	1	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	2	1	no	—
263	C₂×C₄×C₆₇	—	2	1	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	2	1	no	—
266	C₂×C₄×C₆₇	—	2	1	no	—
267	C₂×C₄×C₆₇	—	2	1	no	—
268	C₂×C₄×C₆₇	—	2	1	no	—
269	C₂×C₄×C₆₇	—	2	1	no	—
270	C₂×C₄×C₆₇	—	2	1	no	—
271	C₂×C₄×C₆₇	—	4	4	no	—
272	C₂×C₄×C₆₇	—	4	1	no	—
273	C₂×C₄×C₆₇	—	4	1	no	—
274	C₂×C₄×C₆₇	—	4	1	no	—
275	C₂×C₄×C₆₇	—	4	1	no	—
276	C₂×C₄×C₆₇	#117	1	2	yes	—
277	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
278	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
279	C₂×C₄×C₆₇	#131	1	2	yes	—
280	C₂×C₄×C₆₇	#131	1	2	yes	—
281	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
287	C₈×C₆₇	—	2	2	no	—
288	C₈×C₆₇	—	2	2	no	—
289	C₈×C₆₇	—	4	1	no	—
290	C₈×C₆₇	—	4	1	no	—
291	C₈×C₆₇	—	4	1	no	—
292	C₈×C₆₇	—	4	1	no	—
293	C₈×C₆₇	—	4	1	no	—
294	C₈×C₆₇	—	4	1	no	—
295	C₂²×C₆₇	—	2	2	no	—
296	C₂²×C₆₇	—	2	2	no	—
297	C₂²×C₆₇	—	2	2	no	—
298	C₂²×C₆₇	—	2	2	no	—
299	C₂²×C₆₇	—	2	2	no	—
300	C₂²×C₆₇	—	2	2	no	—
301	C₂²×C₆₇	—	4	1	no	—
302	C₂²×C₆₇	—	4	1	no	—
303	C₂²×C₆₇	—	4	1	no	—
304	C₂²×C₆₇	—	4	1	no	—
305	C₂²×C₆₇	—	4	1	no	—
306	C₂²×C₆₇	—	4	1	no	—
307	C₂²×C₆₇	—	4	1	no	—
308	C₂²×C₆₇	—	4	1	no	—
309	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
310	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
311	C₂²×C₆₇	#520	1	1	yes	characteristic
312	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
313	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
314	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
315	C₂²×C₆₇	#568	1	1	yes	characteristic
316	C₄×C₆₇	—	2	2	no	—
317	C₄×C₆₇	—	2	2	no	—
318	C₄×C₆₇	—	2	2	no	—
319	C₄×C₆₇	—	2	2	no	—
320	C₄×C₆₇	—	2	2	no	—
321	C₄×C₆₇	—	4	1	no	—
322	C₄×C₆₇	—	4	1	no	—
323	C₄×C₆₇	—	4	1	no	—
324	C₄×C₆₇	—	4	1	no	—
325	C₄×C₆₇	—	4	1	no	—
326	C₄×C₆₇	—	4	1	no	—
327	C₄×C₆₇	—	4	1	no	—
328	C₄×C₆₇	#546	1	1	yes	characteristic
329	C₄×C₆₇	#548	1	1	yes	characteristic
330	C₄×C₆₇	#552	1	1	yes	characteristic
331	C₄×C₆₇	#553	1	1	yes	characteristic
332	#22752	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
333	C₂×C₆₇	—	4	2	no	—
334	C₂×C₆₇	—	4	2	no	—
335	C₂×C₆₇	#2352	1	1	yes	characteristic
336	C₂×C₆₇	#2922	1	1	yes	characteristic
337	C₂×C₆₇	#2924	1	1	yes	characteristic
338	C₂×C₆₇	#2929	1	1	yes	characteristic
339	C₂×C₆₇	#2933	1	1	yes	characteristic
340	C₂×C₆₇	#2993	1	1	yes	characteristic
341	C₂×C₆₇	#2994	1	1	yes	characteristic
342	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
343	C₂×#424	C₂×C₆₇	1	1	yes	characteristic, complemented
344	#2290	C₂×C₆₇	1	1	yes	characteristic
345	#1924	C₂×C₆₇	1	1	yes	characteristic, complemented
346	#1923	C₂×C₆₇	1	1	yes	characteristic, complemented
347	#1916	C₂×C₆₇	1	1	yes	characteristic, complemented
348	#1916	C₂×C₆₇	1	1	yes	characteristic, complemented
349	#1870	C₂×C₆₇	1	1	yes	characteristic
350	#1869	C₂×C₆₇	1	1	yes	characteristic
351	#1832	C₂×C₆₇	1	1	yes	characteristic, complemented
352	#1798	C₂×C₆₇	1	1	yes	characteristic, complemented
353	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
354	#1768	C₂×C₆₇	1	1	yes	characteristic, complemented
355	#1753	C₂×C₆₇	1	1	yes	characteristic, complemented
356	C₈×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
357	C₆₇	#22752	1	1	yes	characteristic, direct product factor, Sylow
358	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
359	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
361	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
362	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
363	#431	—	2	4	no	—
364	#424	—	2	4	no	—
365	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
366	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
367	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
368	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
369	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
370	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
371	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
372	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
373	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
374	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
375	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
376	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
377	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
378	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
379	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
380	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
381	#388	C₄×C₆₇	1	1	yes	characteristic
382	#388	C₄×C₆₇	1	1	yes	characteristic
383	#387	C₄×C₆₇	1	1	yes	characteristic
384	#387	C₄×C₆₇	1	1	yes	characteristic
385	#386	C₄×C₆₇	1	1	yes	characteristic
386	#386	C₄×C₆₇	1	1	yes	characteristic
387	#385	C₄×C₆₇	1	1	yes	characteristic
388	#385	C₄×C₆₇	1	1	yes	characteristic
389	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
390	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
391	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
392	#376	C₂²×C₆₇	1	1	yes	characteristic
393	#336	C₂²×C₆₇	1	1	yes	characteristic
394	#336	C₂²×C₆₇	1	1	yes	characteristic
395	#336	C₂²×C₆₇	1	1	yes	characteristic
396	#336	C₂²×C₆₇	1	1	yes	characteristic
397	#336	C₂²×C₆₇	1	1	yes	characteristic
398	#336	C₂²×C₆₇	1	1	yes	characteristic
399	#336	C₂²×C₆₇	1	1	yes	characteristic
400	#336	C₂²×C₆₇	1	1	yes	characteristic
401	#336	C₂²×C₆₇	1	1	yes	characteristic
402	#336	C₂²×C₆₇	1	1	yes	characteristic
403	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
404	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
405	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
406	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
407	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂×M₁₆	—	2	2	no	—
409	C₂×M₁₆	—	2	2	no	—
410	C₂×M₁₆	—	4	1	no	—
411	C₂²×C₈	—	2	2	no	—
412	C₂²×C₈	—	2	2	no	—
413	C₂²×C₈	—	4	1	no	—
414	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
415	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
416	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
417	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
418	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
419	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
420	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
422	C₄×D₄	—	2	2	no	—
423	C₄×D₄	—	2	2	no	—
424	#117	—	2	2	no	—
425	#117	—	2	2	no	—
426	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
431	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
433	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
435	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
437	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
438	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
439	C₂×C₄²	—	2	2	no	—
440	C₂×C₄²	—	2	2	no	—
441	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
442	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
448	C₄⋊C₈	—	2	2	no	—
449	C₄⋊C₈	—	2	2	no	—
450	C₄⋊C₈	—	2	2	no	—
451	C₄⋊C₈	—	2	2	no	—
452	C₄⋊C₈	—	2	2	no	—
453	C₄⋊C₈	D₄×C₆₇	1	4	yes	complemented
454	C₂²⋊C₈	—	2	2	no	—
455	C₂²⋊C₈	—	2	2	no	—
456	C₂²⋊C₈	—	2	2	no	—
457	C₂²⋊C₈	—	2	2	no	—
458	C₂²⋊C₈	—	4	1	no	—
459	C₂²⋊C₈	—	4	1	no	—
460	#97	—	4	1	no	—
461	C₄×C₈	—	2	2	no	—
462	C₄×C₈	—	2	2	no	—
463	C₄×C₈	—	4	1	no	—
464	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
465	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
466	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
467	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
468	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
469	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
470	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
471	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
472	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
473	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
474	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
475	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
476	C₂×D₄	—	2	2	no	—
477	C₂×D₄	—	2	2	no	—
478	C₂×D₄	—	4	1	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	4	1	no	—
495	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
498	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
499	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
500	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
501	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
502	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
503	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
504	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
505	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
506	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
507	C₂²×C₄	#41×C₆₇	1	2	yes	—
508	C₂²×C₄	#41×C₆₇	1	2	yes	—
509	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
510	M₁₆	—	4	4	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	1	no	—
533	C₂×C₈	—	4	4	no	—
534	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
535	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₂²⋊C₄	—	2	2	no	—
542	C₂²⋊C₄	—	2	2	no	—
543	C₂²⋊C₄	—	2	2	no	—
544	C₂²⋊C₄	—	2	2	no	—
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	2	no	—
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	4	1	no	—
549	C₄²	—	2	2	no	—
550	C₄²	—	2	2	no	—
551	C₄²	—	2	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	2	1	no	—
554	C₄²	—	2	1	no	—
555	C₄²	—	2	1	no	—
556	C₄²	—	2	1	no	—
557	C₄²	—	2	1	no	—
558	C₄²	—	2	1	no	—
559	C₄²	C₂×D₄×C₆₇	1	2	yes	—
560	C₄²	#41×C₆₇	1	2	yes	—
561	C₂³	—	2	1	no	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	1	no	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	1	no	—
568	C₂³	—	2	1	no	—
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	#117×C₆₇	1	2	yes	—
574	C₂³	#131×C₆₇	1	2	yes	—
575	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
576	D₄	—	4	4	no	—
577	C₂×C₄	—	2	8	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	4	4	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	#117×C₆₇	1	2	yes	—
619	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
620	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
621	C₂×C₄	#131×C₆₇	1	2	yes	—
622	C₂×C₄	#131×C₆₇	1	2	yes	—
623	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
624	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
625	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
626	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
627	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
628	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
629	C₈	—	2	2	no	—
630	C₈	—	2	2	no	—
631	C₈	—	4	1	no	—
632	C₈	—	4	1	no	—
633	C₈	—	4	1	no	—
634	C₈	—	4	1	no	—
635	C₈	—	4	1	no	—
636	C₈	—	4	1	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	2	2	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	4	1	no	—
644	C₂²	—	4	1	no	—
645	C₂²	—	4	1	no	—
646	C₂²	—	4	1	no	—
647	C₂²	—	4	1	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
652	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
653	C₂²	#520×C₆₇	1	1	yes	characteristic
654	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
655	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
656	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
657	C₂²	#568×C₆₇	1	1	yes	characteristic
658	C₄	—	2	2	no	—
659	C₄	—	2	2	no	—
660	C₄	—	2	2	no	—
661	C₄	—	2	2	no	—
662	C₄	—	2	2	no	—
663	C₄	—	4	1	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	1	no	—
666	C₄	—	4	1	no	—
667	C₄	—	4	1	no	—
668	C₄	—	4	1	no	—
669	C₄	—	4	1	no	—
670	C₄	#546×C₆₇	1	1	yes	characteristic
671	C₄	#548×C₆₇	1	1	yes	characteristic
672	C₄	#552×C₆₇	1	1	yes	characteristic
673	C₄	#553×C₆₇	1	1	yes	characteristic
674	C₂	—	4	2	no	—
675	C₂	—	4	2	no	—
676	C₂	C₆₇×#2352	1	1	yes	characteristic
677	C₂	C₆₇×#2922	1	1	yes	characteristic
678	C₂	C₆₇×#2924	1	1	yes	characteristic
679	C₂	C₆₇×#2929	1	1	yes	characteristic
680	C₂	C₆₇×#2933	1	1	yes	characteristic
681	C₂	C₆₇×#2993	1	1	yes	characteristic
682	C₂	C₆₇×#2994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#431×C₆₇	C₂	semidirect
2. C₂×#424×C₆₇	C₂	semidirect
4. C₆₇×#1924	C₂	semidirect
5. C₆₇×#1923	C₂	semidirect
6. C₆₇×#1916	C₂	semidirect
7. C₆₇×#1916	C₂	semidirect
10. C₆₇×#1832	C₂	semidirect
11. C₆₇×#1798	C₂	semidirect
12. C₆₇×#1797	C₂	semidirect
13. C₆₇×#1768	C₂	semidirect
14. C₆₇×#1753	C₂	semidirect
15. C₈×C₂²⋊C₄×C₆₇	C₂	semidirect
111. C₄⋊C₈×C₆₇	D₄	semidirect
332. #22752	C₆₇	direct (N × H)
342. C₂×#431	C₂×C₆₇	semidirect
343. C₂×#424	C₂×C₆₇	semidirect
345. #1924	C₂×C₆₇	semidirect
346. #1923	C₂×C₆₇	semidirect
347. #1916	C₂×C₆₇	semidirect
348. #1916	C₂×C₆₇	semidirect
351. #1832	C₂×C₆₇	semidirect
352. #1798	C₂×C₆₇	semidirect
353. #1797	C₂×C₆₇	semidirect
354. #1768	C₂×C₆₇	semidirect
355. #1753	C₂×C₆₇	semidirect
356. C₈×C₂²⋊C₄	C₂×C₆₇	semidirect
357. C₆₇	#22752	direct (N × H)
453. C₄⋊C₈	D₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#431×C₆₇	2
C₂×#424×C₆₇	2
C₆₇×#2290	2
C₆₇×#1924	2
C₆₇×#1923	2
C₆₇×#1916	2
C₆₇×#1870	2
C₆₇×#1869	2
C₆₇×#1832	2
C₆₇×#1798	2
C₆₇×#1797	2
C₆₇×#1768	2
C₆₇×#1753	2
C₈×C₂²⋊C₄×C₆₇	2
#22752	67

Subgroups — C67×#22752 #30702003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22752 #30702003