Subgroups — C₆₇×#22751 #30702002

Order 17152.

976 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1923	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1917	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1870	C₂	1	2	yes	maximal
7	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1753	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#432×C₆₇	—	2	4	no	—
19	#423×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	2	yes	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×#97×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₆₇	C₄	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	C₄³×C₆₇	C₂²	1	1	yes	characteristic
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₆₇	—	2	4	no	—
53	C₂×M₁₆×C₆₇	—	2	4	no	—
54	C₂×M₁₆×C₆₇	—	4	2	no	—
55	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
59	C₄×D₄×C₆₇	—	2	2	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	2	no	—
74	C₂×C₄²×C₆₇	—	2	2	no	—
75	C₂×C₄²×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	C₄⋊C₈×C₆₇	—	2	4	no	—
90	C₄⋊C₈×C₆₇	—	2	2	no	—
91	C₄⋊C₈×C₆₇	—	2	2	no	—
92	C₄⋊C₈×C₆₇	—	2	2	no	—
93	C₄⋊C₈×C₆₇	D₄	1	4	yes	complemented
94	C₂²⋊C₈×C₆₇	—	2	4	no	—
95	C₂²⋊C₈×C₆₇	—	2	4	no	—
96	C₂²⋊C₈×C₆₇	—	4	2	no	—
97	#97×C₆₇	—	2	4	no	—
98	C₄×C₈×C₆₇	—	4	2	no	—
99	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
100	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
101	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
102	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
103	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
104	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×D₄×C₆₇	—	2	2	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	2	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	—	2	1	no	—
128	C₂²×C₄×C₆₇	—	2	1	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	4	1	no	—
131	C₂²×C₄×C₆₇	—	4	1	no	—
132	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	#41	1	2	yes	—
145	C₂²×C₄×C₆₇	#41	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
147	M₁₆×C₆₇	—	4	8	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	—	2	2	no	—
151	C₂×C₈×C₆₇	—	2	2	no	—
152	C₂×C₈×C₆₇	—	2	2	no	—
153	C₂×C₈×C₆₇	—	2	2	no	—
154	C₂×C₈×C₆₇	—	2	2	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
160	C₄⋊C₄×C₆₇	—	2	2	no	—
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₂²⋊C₄×C₆₇	—	2	2	no	—
164	C₂²⋊C₄×C₆₇	—	2	2	no	—
165	C₂²⋊C₄×C₆₇	—	2	2	no	—
166	C₂²⋊C₄×C₆₇	—	2	2	no	—
167	C₂²⋊C₄×C₆₇	—	2	2	no	—
168	C₂²⋊C₄×C₆₇	M₁₆	1	4	yes	complemented
169	C₄²×C₆₇	—	2	2	no	—
170	C₄²×C₆₇	—	2	2	no	—
171	C₄²×C₆₇	—	2	2	no	—
172	C₄²×C₆₇	—	2	2	no	—
173	C₄²×C₆₇	—	2	2	no	—
174	C₄²×C₆₇	—	2	2	no	—
175	C₄²×C₆₇	—	2	1	no	—
176	C₄²×C₆₇	—	2	1	no	—
177	C₄²×C₆₇	—	2	1	no	—
178	C₄²×C₆₇	—	2	1	no	—
179	C₄²×C₆₇	—	2	1	no	—
180	C₄²×C₆₇	—	2	1	no	—
181	C₄²×C₆₇	—	2	1	no	—
182	C₄²×C₆₇	—	2	1	no	—
183	C₄²×C₆₇	—	4	1	no	—
184	C₄²×C₆₇	M₁₆	1	4	yes	complemented
185	C₄²×C₆₇	C₂×D₄	1	2	yes	—
186	C₄²×C₆₇	#41	1	2	yes	—
187	C₂³×C₆₇	—	2	1	no	—
188	C₂³×C₆₇	—	2	1	no	—
189	C₂³×C₆₇	—	2	1	no	—
190	C₂³×C₆₇	—	2	1	no	—
191	C₂³×C₆₇	—	2	1	no	—
192	C₂³×C₆₇	—	2	1	no	—
193	C₂³×C₆₇	—	2	1	no	—
194	C₂³×C₆₇	—	2	1	no	—
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	2	1	no	—
197	C₂³×C₆₇	—	2	1	no	—
198	C₂³×C₆₇	—	2	1	no	—
199	C₂³×C₆₇	#117	1	2	yes	—
200	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
201	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
202	D₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	1	no	—
204	C₂×C₄×C₆₇	—	2	1	no	—
205	C₂×C₄×C₆₇	—	2	1	no	—
206	C₂×C₄×C₆₇	—	2	1	no	—
207	C₂×C₄×C₆₇	—	2	1	no	—
208	C₂×C₄×C₆₇	—	2	1	no	—
209	C₂×C₄×C₆₇	—	2	1	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	2	1	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	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	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	—	4	4	no	—
235	C₂×C₄×C₆₇	—	4	4	no	—
236	C₂×C₄×C₆₇	—	4	1	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	1	no	—
239	C₂×C₄×C₆₇	—	4	1	no	—
240	C₂×C₄×C₆₇	—	4	4	no	—
241	C₂×C₄×C₆₇	—	4	4	no	—
242	C₂×C₄×C₆₇	#117	1	2	yes	—
243	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
244	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
245	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
246	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
247	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
248	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
249	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
250	C₂×C₄×C₆₇	#131	1	2	yes	—
251	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
255	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
257	C₈×C₆₇	—	2	4	no	—
258	C₈×C₆₇	—	4	2	no	—
259	C₈×C₆₇	—	4	2	no	—
260	C₈×C₆₇	—	4	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	4	1	no	—
268	C₂²×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	C₂×#117	1	1	yes	characteristic
276	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
277	C₂²×C₆₇	#520	1	1	yes	characteristic
278	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
279	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
281	C₂²×C₆₇	#568	1	1	yes	characteristic
282	C₄×C₆₇	—	2	2	no	—
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	—	2	2	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	#436	1	4	yes	—
294	C₄×C₆₇	#546	1	1	yes	characteristic
295	C₄×C₆₇	#548	1	1	yes	characteristic
296	C₄×C₆₇	#552	1	1	yes	characteristic
297	C₄×C₆₇	#553	1	1	yes	characteristic
298	#22751	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
299	C₂×C₆₇	—	4	2	no	—
300	C₂×C₆₇	—	4	2	no	—
301	C₂×C₆₇	#2352	1	1	yes	characteristic
302	C₂×C₆₇	#2920	1	1	yes	characteristic
303	C₂×C₆₇	#2925	1	1	yes	characteristic
304	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
305	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
306	C₂×C₆₇	#2989	1	1	yes	characteristic
307	C₂×C₆₇	#2990	1	1	yes	characteristic
308	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
310	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
311	#1923	C₂×C₆₇	1	2	yes	complemented
312	#1917	C₂×C₆₇	1	2	yes	complemented
313	#1870	C₂×C₆₇	1	2	yes	—
314	#1831	C₂×C₆₇	1	1	yes	characteristic, complemented
315	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
316	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
317	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
318	#1753	C₂×C₆₇	1	2	yes	complemented
319	C₆₇	#22751	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
321	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
322	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
323	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
324	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
325	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
326	#432	—	2	4	no	—
327	#423	—	2	4	no	—
328	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
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	2	yes	—
332	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
334	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
335	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
336	C₂×#97	C₂²×C₆₇	1	2	yes	—
337	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
338	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
339	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
340	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
341	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
342	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
343	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
344	C₄×C₂²⋊C₄	C₄×C₆₇	1	1	yes	characteristic
345	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
346	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
347	C₄³	C₂²×C₆₇	1	1	yes	characteristic
348	#336	C₂²×C₆₇	1	2	yes	—
349	#336	C₂²×C₆₇	1	2	yes	—
350	#336	C₂²×C₆₇	1	2	yes	—
351	#336	C₂²×C₆₇	1	2	yes	—
352	#336	C₂²×C₆₇	1	2	yes	—
353	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
354	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
355	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
356	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
357	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
359	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
360	C₂×M₁₆	—	2	4	no	—
361	C₂×M₁₆	—	2	4	no	—
362	C₂×M₁₆	—	4	2	no	—
363	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
364	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
365	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
366	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	4	no	—
371	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
376	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
380	C₂×C₄²	—	2	2	no	—
381	C₂×C₄²	—	2	2	no	—
382	C₂×C₄²	—	2	2	no	—
383	C₂×C₄²	—	2	2	no	—
384	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
385	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
386	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
387	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
388	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
389	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
394	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
397	C₄⋊C₈	—	2	4	no	—
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	—	2	2	no	—
400	C₄⋊C₈	—	2	2	no	—
401	C₄⋊C₈	D₄×C₆₇	1	4	yes	complemented
402	C₂²⋊C₈	—	2	4	no	—
403	C₂²⋊C₈	—	2	4	no	—
404	C₂²⋊C₈	—	4	2	no	—
405	#97	—	2	4	no	—
406	C₄×C₈	—	4	2	no	—
407	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
408	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
409	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
410	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
411	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
412	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
413	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
414	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	4	1	no	—
439	C₂²×C₄	—	4	1	no	—
440	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
441	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
442	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
443	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
444	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
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	#41×C₆₇	1	2	yes	—
453	C₂²×C₄	#41×C₆₇	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
455	M₁₆	—	4	8	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	2	2	no	—
465	C₂×C₈	—	2	2	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	2	2	no	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	M₁₆×C₆₇	1	4	yes	complemented
477	C₄²	—	2	2	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	2	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	2	2	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	4	1	no	—
492	C₄²	M₁₆×C₆₇	1	4	yes	complemented
493	C₄²	C₂×D₄×C₆₇	1	2	yes	—
494	C₄²	#41×C₆₇	1	2	yes	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	1	no	—
501	C₂³	—	2	1	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	—	2	1	no	—
506	C₂³	—	2	1	no	—
507	C₂³	#117×C₆₇	1	2	yes	—
508	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
509	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
510	D₄	—	2	8	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	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	—	4	4	no	—
543	C₂×C₄	—	4	4	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	4	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	#117×C₆₇	1	2	yes	—
551	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
552	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
553	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
554	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
555	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
556	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
557	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
558	C₂×C₄	#131×C₆₇	1	2	yes	—
559	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
560	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
561	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
562	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
563	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
564	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
565	C₈	—	2	4	no	—
566	C₈	—	4	2	no	—
567	C₈	—	4	2	no	—
568	C₈	—	4	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	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	C₂×#117×C₆₇	1	1	yes	characteristic
584	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
585	C₂²	#520×C₆₇	1	1	yes	characteristic
586	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
587	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
588	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
589	C₂²	#568×C₆₇	1	1	yes	characteristic
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
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	#436×C₆₇	1	4	yes	—
602	C₄	#546×C₆₇	1	1	yes	characteristic
603	C₄	#548×C₆₇	1	1	yes	characteristic
604	C₄	#552×C₆₇	1	1	yes	characteristic
605	C₄	#553×C₆₇	1	1	yes	characteristic
606	C₂	—	4	2	no	—
607	C₂	—	4	2	no	—
608	C₂	C₆₇×#2352	1	1	yes	characteristic
609	C₂	C₆₇×#2920	1	1	yes	characteristic
610	C₂	C₆₇×#2925	1	1	yes	characteristic
611	C₂	C₂×#436×C₆₇	1	1	yes	characteristic
612	C₂	D₄×M₁₆×C₆₇	1	1	yes	characteristic
613	C₂	C₆₇×#2989	1	1	yes	characteristic
614	C₂	C₆₇×#2990	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×#432×C₆₇	C₂	semidirect
2. C₂×#423×C₆₇	C₂	semidirect
4. C₆₇×#1923	C₂	semidirect
5. C₆₇×#1917	C₂	semidirect
7. C₆₇×#1831	C₂	semidirect
8. C₆₇×#1797	C₂	semidirect
9. C₆₇×#1797	C₂	semidirect
10. C₄×C₄⋊C₈×C₆₇	C₂	semidirect
11. C₆₇×#1753	C₂	semidirect
93. C₄⋊C₈×C₆₇	D₄	semidirect
168. C₂²⋊C₄×C₆₇	M₁₆	semidirect
184. C₄²×C₆₇	M₁₆	semidirect
298. #22751	C₆₇	direct (N × H)
308. C₂×#432	C₂×C₆₇	semidirect
309. C₂×#423	C₂×C₆₇	semidirect
311. #1923	C₂×C₆₇	semidirect
312. #1917	C₂×C₆₇	semidirect
314. #1831	C₂×C₆₇	semidirect
315. #1797	C₂×C₆₇	semidirect
316. #1797	C₂×C₆₇	semidirect
317. C₄×C₄⋊C₈	C₂×C₆₇	semidirect
318. #1753	C₂×C₆₇	semidirect
319. C₆₇	#22751	direct (N × H)
401. C₄⋊C₈	D₄×C₆₇	semidirect
476. C₂²⋊C₄	M₁₆×C₆₇	semidirect
492. C₄²	M₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₆₇	2
C₂×#423×C₆₇	2
C₄²×D₄×C₆₇	2
C₆₇×#1923	2
C₆₇×#1917	2
C₆₇×#1870	2
C₆₇×#1831	2
C₆₇×#1797	2
C₄×C₄⋊C₈×C₆₇	2
C₆₇×#1753	2
#22751	67

Subgroups — C67×#22751 #30702002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22751 #30702002