Subgroups — C₆₇×#22107 #30701358

Order 17152.

728 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#433×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2396	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1924	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1924	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1923	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1923	C₂	1	1	yes	maximal, characteristic
12	C₆₇×#1917	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#1916	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#1798	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#1771	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
18	#446×C₆₇	—	2	4	no	—
19	#433×C₆₇	—	2	4	no	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
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×C₂²⋊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×#97×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×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	#400×C₆₇	C₄	1	1	yes	characteristic
39	#399×C₆₇	C₄	1	1	yes	characteristic
40	#398×C₆₇	C₄	1	1	yes	characteristic
41	#398×C₆₇	C₄	1	1	yes	characteristic
42	#396×C₆₇	C₄	1	1	yes	characteristic
43	#396×C₆₇	C₄	1	1	yes	characteristic
44	#395×C₆₇	C₄	1	1	yes	characteristic
45	#393×C₆₇	C₄	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
49	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×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×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄×Q₈×C₆₇	—	2	2	no	—
72	C₄×Q₈×C₆₇	—	2	2	no	—
73	#117×C₆₇	—	2	2	no	—
74	#117×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊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×C₄²×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₆₇	—	2	2	no	—
99	C₄⋊C₈×C₆₇	—	2	2	no	—
100	C₄⋊C₈×C₆₇	—	2	2	no	—
101	C₄⋊C₈×C₆₇	—	2	2	no	—
102	C₄⋊C₈×C₆₇	—	2	2	no	—
103	C₄⋊C₈×C₆₇	—	2	2	no	—
104	C₄⋊C₈×C₆₇	—	4	1	no	—
105	C₄⋊C₈×C₆₇	—	4	1	no	—
106	C₂²⋊C₈×C₆₇	—	2	2	no	—
107	C₂²⋊C₈×C₆₇	—	2	2	no	—
108	C₂²⋊C₈×C₆₇	—	4	1	no	—
109	C₂²⋊C₈×C₆₇	—	4	1	no	—
110	#97×C₆₇	—	2	2	no	—
111	#97×C₆₇	—	4	1	no	—
112	#97×C₆₇	D₄	1	4	yes	—
113	C₄×C₈×C₆₇	—	2	2	no	—
114	C₄×C₈×C₆₇	—	2	2	no	—
115	C₄×C₈×C₆₇	—	4	1	no	—
116	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
117	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
118	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
119	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
120	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
121	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
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	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×Q₈×C₆₇	—	2	2	no	—
128	C₂×Q₈×C₆₇	—	2	2	no	—
129	C₂×Q₈×C₆₇	—	4	1	no	—
130	C₂²×C₄×C₆₇	—	2	1	no	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	1	no	—
134	C₂²×C₄×C₆₇	—	2	1	no	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
150	C₂×C₈×C₆₇	—	2	1	no	—
151	C₂×C₈×C₆₇	—	2	1	no	—
152	C₂×C₈×C₆₇	—	2	1	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	1	no	—
156	C₂×C₈×C₆₇	—	2	1	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	1	no	—
160	C₂×C₈×C₆₇	—	2	1	no	—
161	C₂×C₈×C₆₇	—	2	1	no	—
162	C₂×C₈×C₆₇	—	2	1	no	—
163	C₂×C₈×C₆₇	—	2	1	no	—
164	C₂×C₈×C₆₇	—	2	1	no	—
165	C₂×C₈×C₆₇	—	2	1	no	—
166	C₂×C₈×C₆₇	—	2	1	no	—
167	C₂×C₈×C₆₇	—	2	1	no	—
168	C₂×C₈×C₆₇	—	2	1	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	C₂×D₄	1	2	yes	—
173	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	—	2	2	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	2	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	4	1	no	—
183	C₄⋊C₄×C₆₇	—	4	1	no	—
184	C₄⋊C₄×C₆₇	—	4	1	no	—
185	C₂²⋊C₄×C₆₇	—	2	2	no	—
186	C₂²⋊C₄×C₆₇	—	2	2	no	—
187	C₂²⋊C₄×C₆₇	—	2	2	no	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₂²⋊C₄×C₆₇	—	4	1	no	—
190	C₂²⋊C₄×C₆₇	—	4	1	no	—
191	C₄²×C₆₇	—	2	1	no	—
192	C₄²×C₆₇	—	2	1	no	—
193	C₄²×C₆₇	—	2	1	no	—
194	C₄²×C₆₇	—	2	1	no	—
195	C₄²×C₆₇	—	2	1	no	—
196	C₄²×C₆₇	—	2	1	no	—
197	C₄²×C₆₇	—	2	1	no	—
198	C₄²×C₆₇	—	2	1	no	—
199	C₄²×C₆₇	—	2	1	no	—
200	C₄²×C₆₇	—	2	1	no	—
201	C₄²×C₆₇	C₂×D₄	1	2	yes	—
202	C₄²×C₆₇	#41	1	2	yes	—
203	C₂³×C₆₇	—	2	1	no	—
204	C₂³×C₆₇	—	2	1	no	—
205	C₂³×C₆₇	—	2	1	no	—
206	C₂³×C₆₇	—	2	1	no	—
207	C₂³×C₆₇	—	2	1	no	—
208	C₂³×C₆₇	—	2	1	no	—
209	C₂³×C₆₇	—	2	1	no	—
210	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
211	Q₈×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	1	no	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	1	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	1	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	1	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	1	no	—
235	C₂×C₄×C₆₇	—	2	1	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	1	no	—
247	C₂×C₄×C₆₇	—	4	1	no	—
248	C₂×C₄×C₆₇	—	4	1	no	—
249	C₂×C₄×C₆₇	—	4	1	no	—
250	C₂×C₄×C₆₇	—	4	1	no	—
251	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
252	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
253	C₂×C₄×C₆₇	#131	1	2	yes	—
254	C₂×C₄×C₆₇	#131	1	2	yes	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
257	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
258	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
259	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
260	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
261	C₈×C₆₇	—	2	2	no	—
262	C₈×C₆₇	—	2	2	no	—
263	C₈×C₆₇	—	4	1	no	—
264	C₈×C₆₇	—	4	1	no	—
265	C₈×C₆₇	—	4	1	no	—
266	C₈×C₆₇	—	4	1	no	—
267	C₈×C₆₇	—	4	1	no	—
268	C₈×C₆₇	—	4	1	no	—
269	C₂²×C₆₇	—	4	1	no	—
270	C₂²×C₆₇	—	4	1	no	—
271	C₂²×C₆₇	—	4	1	no	—
272	C₂²×C₆₇	—	4	1	no	—
273	C₂²×C₆₇	—	4	1	no	—
274	C₂²×C₆₇	—	4	1	no	—
275	C₂²×C₆₇	—	4	1	no	—
276	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
277	C₂²×C₆₇	#519	1	1	yes	characteristic
278	C₂²×C₆₇	#520	1	1	yes	characteristic
279	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
281	C₂²×C₆₇	#568	1	1	yes	characteristic
282	C₂²×C₆₇	#568	1	1	yes	characteristic
283	C₄×C₆₇	—	2	2	no	—
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	2	2	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	4	1	no	—
288	C₄×C₆₇	—	4	1	no	—
289	C₄×C₆₇	—	4	1	no	—
290	C₄×C₆₇	—	4	1	no	—
291	C₄×C₆₇	—	4	1	no	—
292	C₄×C₆₇	—	4	1	no	—
293	C₄×C₆₇	—	4	1	no	—
294	C₄×C₆₇	—	4	1	no	—
295	C₄×C₆₇	—	4	1	no	—
296	C₄×C₆₇	#536	1	1	yes	characteristic
297	C₄×C₆₇	#537	1	1	yes	characteristic
298	C₄×C₆₇	#541	1	1	yes	characteristic
299	C₄×C₆₇	#542	1	1	yes	characteristic
300	#22107	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
301	C₂×C₆₇	—	8	1	no	—
302	C₂×C₆₇	#2323	1	1	yes	characteristic
303	C₂×C₆₇	#2933	1	1	yes	characteristic
304	C₂×C₆₇	#2935	1	1	yes	characteristic
305	C₂×C₆₇	#2985	1	1	yes	characteristic
306	C₂×C₆₇	#2985	1	1	yes	characteristic
307	C₂×C₆₇	#2994	1	1	yes	characteristic
308	C₂×C₆₇	#2995	1	1	yes	characteristic
309	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
310	C₂×#433	C₂×C₆₇	1	1	yes	characteristic
311	#2396	C₂×C₆₇	1	1	yes	characteristic
312	#1984	C₂×C₆₇	1	1	yes	characteristic, complemented
313	#1940	C₂×C₆₇	1	1	yes	characteristic, complemented
314	#1939	C₂×C₆₇	1	1	yes	characteristic, complemented
315	#1939	C₂×C₆₇	1	1	yes	characteristic, complemented
316	#1924	C₂×C₆₇	1	1	yes	characteristic
317	#1924	C₂×C₆₇	1	1	yes	characteristic
318	#1923	C₂×C₆₇	1	1	yes	characteristic
319	#1923	C₂×C₆₇	1	1	yes	characteristic
320	#1917	C₂×C₆₇	1	1	yes	characteristic, complemented
321	#1916	C₂×C₆₇	1	1	yes	characteristic, complemented
322	#1798	C₂×C₆₇	1	1	yes	characteristic
323	#1771	C₂×C₆₇	1	1	yes	characteristic, complemented
324	C₆₇	#22107	1	1	yes	characteristic, direct product factor, Sylow
325	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
326	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
327	#446	—	2	4	no	—
328	#433	—	2	4	no	—
329	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
330	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
334	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
340	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
341	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
342	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
344	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
345	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
346	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
347	#400	C₄×C₆₇	1	1	yes	characteristic
348	#399	C₄×C₆₇	1	1	yes	characteristic
349	#398	C₄×C₆₇	1	1	yes	characteristic
350	#398	C₄×C₆₇	1	1	yes	characteristic
351	#396	C₄×C₆₇	1	1	yes	characteristic
352	#396	C₄×C₆₇	1	1	yes	characteristic
353	#395	C₄×C₆₇	1	1	yes	characteristic
354	#393	C₄×C₆₇	1	1	yes	characteristic
355	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
356	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
357	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
358	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
359	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
360	#336	C₂²×C₆₇	1	1	yes	characteristic
361	#336	C₂²×C₆₇	1	1	yes	characteristic
362	#336	C₂²×C₆₇	1	1	yes	characteristic
363	#336	C₂²×C₆₇	1	1	yes	characteristic
364	#336	C₂²×C₆₇	1	1	yes	characteristic
365	#336	C₂²×C₆₇	1	1	yes	characteristic
366	#336	C₂²×C₆₇	1	1	yes	characteristic
367	#336	C₂²×C₆₇	1	1	yes	characteristic
368	#336	C₂²×C₆₇	1	1	yes	characteristic
369	#336	C₂²×C₆₇	1	1	yes	characteristic
370	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
371	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
372	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
373	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
374	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
377	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	—	2	2	no	—
382	#117	—	2	2	no	—
383	#117	—	2	2	no	—
384	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
394	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
395	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
396	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
397	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
398	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
399	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
400	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
403	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
406	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
407	C₄⋊C₈	—	2	2	no	—
408	C₄⋊C₈	—	2	2	no	—
409	C₄⋊C₈	—	2	2	no	—
410	C₄⋊C₈	—	2	2	no	—
411	C₄⋊C₈	—	2	2	no	—
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	—	4	1	no	—
414	C₄⋊C₈	—	4	1	no	—
415	C₂²⋊C₈	—	2	2	no	—
416	C₂²⋊C₈	—	2	2	no	—
417	C₂²⋊C₈	—	4	1	no	—
418	C₂²⋊C₈	—	4	1	no	—
419	#97	—	2	2	no	—
420	#97	—	4	1	no	—
421	#97	D₄×C₆₇	1	4	yes	—
422	C₄×C₈	—	2	2	no	—
423	C₄×C₈	—	2	2	no	—
424	C₄×C₈	—	4	1	no	—
425	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
426	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
427	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
428	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
429	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
430	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
431	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
432	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
433	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
434	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
435	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	4	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
447	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
448	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
449	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
450	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
456	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
457	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
458	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
482	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	4	1	no	—
492	C₄⋊C₄	—	4	1	no	—
493	C₄⋊C₄	—	4	1	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	4	1	no	—
499	C₂²⋊C₄	—	4	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	1	no	—
509	C₄²	—	2	1	no	—
510	C₄²	C₂×D₄×C₆₇	1	2	yes	—
511	C₄²	#41×C₆₇	1	2	yes	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
520	Q₈	—	4	4	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
561	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
562	C₂×C₄	#131×C₆₇	1	2	yes	—
563	C₂×C₄	#131×C₆₇	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
565	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
566	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
567	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
568	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
569	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
570	C₈	—	2	2	no	—
571	C₈	—	2	2	no	—
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	—	4	1	no	—
577	C₈	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
586	C₂²	#519×C₆₇	1	1	yes	characteristic
587	C₂²	#520×C₆₇	1	1	yes	characteristic
588	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
589	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
590	C₂²	#568×C₆₇	1	1	yes	characteristic
591	C₂²	#568×C₆₇	1	1	yes	characteristic
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	1	no	—
601	C₄	—	4	1	no	—
602	C₄	—	4	1	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	#536×C₆₇	1	1	yes	characteristic
606	C₄	#537×C₆₇	1	1	yes	characteristic
607	C₄	#541×C₆₇	1	1	yes	characteristic
608	C₄	#542×C₆₇	1	1	yes	characteristic
609	C₂	—	8	1	no	—
610	C₂	C₆₇×#2323	1	1	yes	characteristic
611	C₂	C₆₇×#2933	1	1	yes	characteristic
612	C₂	C₆₇×#2935	1	1	yes	characteristic
613	C₂	C₆₇×#2985	1	1	yes	characteristic
614	C₂	C₆₇×#2985	1	1	yes	characteristic
615	C₂	C₆₇×#2994	1	1	yes	characteristic
616	C₂	C₆₇×#2995	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	C₂⁴×C₆₇	C₂²×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×#446×C₆₇	C₂	semidirect
4. C₆₇×#1984	C₂	semidirect
5. C₆₇×#1940	C₂	semidirect
6. C₆₇×#1939	C₂	semidirect
7. C₆₇×#1939	C₂	semidirect
12. C₆₇×#1917	C₂	semidirect
13. C₆₇×#1916	C₂	semidirect
15. C₆₇×#1771	C₂	semidirect
300. #22107	C₆₇	direct (N × H)
309. C₂×#446	C₂×C₆₇	semidirect
312. #1984	C₂×C₆₇	semidirect
313. #1940	C₂×C₆₇	semidirect
314. #1939	C₂×C₆₇	semidirect
315. #1939	C₂×C₆₇	semidirect
320. #1917	C₂×C₆₇	semidirect
321. #1916	C₂×C₆₇	semidirect
323. #1771	C₂×C₆₇	semidirect
324. C₆₇	#22107	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#446×C₆₇	2
C₂×#433×C₆₇	2
C₆₇×#2396	2
C₆₇×#1984	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1924	2
C₆₇×#1923	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1798	2
C₆₇×#1771	2
#22107	67

Subgroups — C67×#22107 #30701358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22107 #30701358