Subgroups — C₆₇×#35086 #30714337

Order 17152.

1974 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2857	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2851	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2848	C₂	1	2	yes	maximal
4	C₆₇×#2652	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2505	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2496	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2489	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2480	C₂	1	4	yes	maximal
9	C₆₇×#2429	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2421	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2334	C₂	1	4	yes	maximal, complemented
12	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
16	C₂×#126×C₆₇	C₂²	1	4	yes	—
17	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#400×C₆₇	C₂²	1	4	yes	complemented
26	#400×C₆₇	C₂²	1	4	yes	complemented
27	#400×C₆₇	C₂²	1	4	yes	complemented
28	#398×C₆₇	C₂²	1	4	yes	complemented
29	#397×C₆₇	C₂²	1	4	yes	—
30	#397×C₆₇	C₂²	1	4	yes	—
31	#397×C₆₇	C₂²	1	4	yes	—
32	#396×C₆₇	C₂²	1	4	yes	—
33	#391×C₆₇	C₂²	1	2	yes	complemented
34	#391×C₆₇	C₂²	1	2	yes	complemented
35	#389×C₆₇	C₂²	1	4	yes	complemented
36	#389×C₆₇	C₂²	1	2	yes	complemented
37	#389×C₆₇	C₂²	1	4	yes	complemented
38	#389×C₆₇	C₂²	1	2	yes	complemented
39	#388×C₆₇	C₂²	1	4	yes	—
40	#388×C₆₇	C₂²	1	4	yes	—
41	#388×C₆₇	C₂²	1	4	yes	—
42	#387×C₆₇	C₂²	1	4	yes	complemented
43	#387×C₆₇	C₂²	1	4	yes	complemented
44	#387×C₆₇	C₂²	1	4	yes	complemented
45	#387×C₆₇	C₂²	1	4	yes	complemented
46	#385×C₆₇	C₂²	1	4	yes	—
47	#385×C₆₇	C₂²	1	4	yes	—
48	#385×C₆₇	C₂²	1	4	yes	—
49	#385×C₆₇	C₂²	1	4	yes	—
50	#385×C₆₇	C₂²	1	4	yes	—
51	#384×C₆₇	C₂²	1	1	yes	characteristic, complemented
52	#382×C₆₇	C₂²	1	4	yes	complemented
53	#382×C₆₇	C₂²	1	1	yes	characteristic, complemented
54	#380×C₆₇	C₂²	1	4	yes	—
55	#380×C₆₇	C₂²	1	2	yes	—
56	#380×C₆₇	C₂²	1	2	yes	—
57	#380×C₆₇	C₂²	1	2	yes	—
58	#379×C₆₇	C₂²	1	4	yes	—
59	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
60	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
61	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
66	C₂×#95×C₆₇	C₂²	1	2	yes	—
67	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
69	C₂³×C₄×C₆₇	—	2	4	no	—
70	C₂³×C₄×C₆₇	—	4	2	no	—
71	C₂³×C₄×C₆₇	—	4	2	no	—
72	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
74	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₆₇	—	4	2	no	—
79	#126×C₆₇	—	2	16	no	—
80	#125×C₆₇	—	4	2	no	—
81	C₂²⋊Q₈×C₆₇	—	2	8	no	—
82	C₂²⋊Q₈×C₆₇	—	4	4	no	—
83	C₂²⋊Q₈×C₆₇	—	4	4	no	—
84	C₄×Q₈×C₆₇	—	4	4	no	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄²×C₆₇	—	4	2	no	—
109	C₂×C₄²×C₆₇	—	4	2	no	—
110	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
118	#95×C₆₇	C₂³	1	2	yes	—
119	#95×C₆₇	C₂³	1	2	yes	—
120	#95×C₆₇	C₂³	1	2	yes	—
121	#95×C₆₇	C₂³	1	2	yes	—
122	#95×C₆₇	C₂³	1	4	yes	—
123	#95×C₆₇	C₂³	1	4	yes	—
124	#95×C₆₇	C₂³	1	4	yes	—
125	#95×C₆₇	C₂³	1	4	yes	—
126	#95×C₆₇	C₂³	1	4	yes	—
127	#95×C₆₇	C₂³	1	4	yes	—
128	#95×C₆₇	C₂³	1	4	yes	—
129	#95×C₆₇	C₂³	1	4	yes	—
130	#95×C₆₇	C₂³	1	4	yes	—
131	#95×C₆₇	C₂³	1	4	yes	—
132	#95×C₆₇	C₂³	1	4	yes	—
133	C₂⁴×C₆₇	—	2	4	no	—
134	C₂⁴×C₆₇	—	2	2	no	—
135	C₂⁴×C₆₇	—	2	4	no	—
136	C₂⁴×C₆₇	—	4	1	no	—
137	C₂⁴×C₆₇	—	4	1	no	—
138	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
139	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂×Q₈×C₆₇	—	2	4	no	—
141	C₂×Q₈×C₆₇	—	2	4	no	—
142	C₂×Q₈×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	4	no	—
151	C₂²×C₄×C₆₇	—	2	4	no	—
152	C₂²×C₄×C₆₇	—	2	4	no	—
153	C₂²×C₄×C₆₇	—	2	4	no	—
154	C₂²×C₄×C₆₇	—	4	2	no	—
155	C₂²×C₄×C₆₇	—	4	2	no	—
156	C₂²×C₄×C₆₇	—	4	2	no	—
157	C₂²×C₄×C₆₇	—	4	2	no	—
158	C₂²×C₄×C₆₇	—	4	1	no	—
159	C₂²×C₄×C₆₇	—	4	1	no	—
160	C₂²×C₄×C₆₇	—	4	16	no	—
161	C₂²×C₄×C₆₇	#41	1	4	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
173	C₄⋊C₄×C₆₇	—	2	4	no	—
174	C₄⋊C₄×C₆₇	—	2	8	no	—
175	C₄⋊C₄×C₆₇	—	2	8	no	—
176	C₄⋊C₄×C₆₇	—	2	8	no	—
177	C₄⋊C₄×C₆₇	—	2	8	no	—
178	C₄⋊C₄×C₆₇	—	2	4	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	4	4	no	—
181	C₄⋊C₄×C₆₇	—	4	4	no	—
182	C₄⋊C₄×C₆₇	—	4	4	no	—
183	C₄⋊C₄×C₆₇	—	4	4	no	—
184	C₄⋊C₄×C₆₇	—	4	4	no	—
185	C₂²⋊C₄×C₆₇	—	2	4	no	—
186	C₂²⋊C₄×C₆₇	—	2	8	no	—
187	C₂²⋊C₄×C₆₇	—	2	8	no	—
188	C₂²⋊C₄×C₆₇	—	2	8	no	—
189	C₂²⋊C₄×C₆₇	—	2	8	no	—
190	C₂²⋊C₄×C₆₇	—	2	4	no	—
191	C₂²⋊C₄×C₆₇	—	2	4	no	—
192	C₂²⋊C₄×C₆₇	—	4	4	no	—
193	C₂²⋊C₄×C₆₇	—	4	4	no	—
194	C₂²⋊C₄×C₆₇	#41	1	8	yes	—
195	C₄²×C₆₇	—	2	8	no	—
196	C₄²×C₆₇	—	2	4	no	—
197	C₄²×C₆₇	—	2	2	no	—
198	C₄²×C₆₇	—	2	2	no	—
199	C₄²×C₆₇	—	4	1	no	—
200	C₄²×C₆₇	—	4	2	no	—
201	C₄²×C₆₇	—	4	1	no	—
202	C₄²×C₆₇	—	4	4	no	—
203	C₂³×C₆₇	—	2	2	no	—
204	C₂³×C₆₇	—	2	1	no	—
205	C₂³×C₆₇	—	2	1	no	—
206	C₂³×C₆₇	—	2	8	no	—
207	C₂³×C₆₇	—	2	2	no	—
208	C₂³×C₆₇	—	2	2	no	—
209	C₂³×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	2	no	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	4	8	no	—
214	C₂³×C₆₇	—	4	8	no	—
215	C₂³×C₆₇	—	4	4	no	—
216	C₂³×C₆₇	—	4	4	no	—
217	C₂³×C₆₇	C₂×#41	1	4	yes	—
218	C₂³×C₆₇	C₂×#41	1	2	yes	—
219	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
220	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
221	Q₈×C₆₇	—	4	8	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	1	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	8	no	—
229	C₂×C₄×C₆₇	—	2	4	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	2	4	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	2	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	4	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	4	8	no	—
251	C₂×C₄×C₆₇	—	4	8	no	—
252	C₂×C₄×C₆₇	—	4	8	no	—
253	C₂×C₄×C₆₇	—	4	4	no	—
254	C₂×C₄×C₆₇	—	4	8	no	—
255	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
256	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
257	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
258	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
259	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
260	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	4	no	—
265	C₂²×C₆₇	—	2	4	no	—
266	C₂²×C₆₇	—	2	4	no	—
267	C₂²×C₆₇	—	2	4	no	—
268	C₂²×C₆₇	—	4	16	no	—
269	C₂²×C₆₇	—	4	2	no	—
270	C₂²×C₆₇	—	4	2	no	—
271	C₂²×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	4	2	no	—
273	C₂²×C₆₇	#541	1	4	yes	—
274	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
275	C₂²×C₆₇	C₂²×#41	1	2	yes	—
276	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
277	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	2	4	no	—
281	C₄×C₆₇	—	2	4	no	—
282	C₄×C₆₇	—	4	4	no	—
283	C₄×C₆₇	—	4	2	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	—	4	4	no	—
286	C₄×C₆₇	—	4	4	no	—
287	C₄×C₆₇	—	4	2	no	—
288	C₄×C₆₇	—	4	2	no	—
289	C₄×C₆₇	#538	1	4	yes	—
290	#35086	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂×C₆₇	—	2	4	no	—
292	C₂×C₆₇	—	4	4	no	—
293	C₂×C₆₇	#3444	1	1	yes	characteristic
294	C₂×C₆₇	C₂×#538	1	1	yes	characteristic
295	C₂×C₆₇	C₂×#541	1	1	yes	characteristic
296	C₂×C₆₇	#3520	1	2	yes	—
297	C₂×C₆₇	#3520	1	2	yes	—
298	#2857	C₂×C₆₇	1	1	yes	characteristic, complemented
299	#2851	C₂×C₆₇	1	1	yes	characteristic, complemented
300	#2848	C₂×C₆₇	1	2	yes	—
301	#2652	C₂×C₆₇	1	4	yes	complemented
302	#2505	C₂×C₆₇	1	4	yes	complemented
303	#2496	C₂×C₆₇	1	2	yes	complemented
304	#2489	C₂×C₆₇	1	2	yes	complemented
305	#2480	C₂×C₆₇	1	4	yes	—
306	#2429	C₂×C₆₇	1	2	yes	complemented
307	#2421	C₂×C₆₇	1	2	yes	complemented
308	#2334	C₂×C₆₇	1	4	yes	complemented
309	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	complemented
310	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
311	C₆₇	#35086	1	1	yes	characteristic, direct product factor, Sylow
312	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
314	C₂×#126	C₂²×C₆₇	1	4	yes	—
315	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
316	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
317	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
318	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
319	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
320	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
321	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
322	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
323	#400	C₂²×C₆₇	1	4	yes	complemented
324	#400	C₂²×C₆₇	1	4	yes	complemented
325	#400	C₂²×C₆₇	1	4	yes	complemented
326	#398	C₂²×C₆₇	1	4	yes	complemented
327	#397	C₂²×C₆₇	1	4	yes	—
328	#397	C₂²×C₆₇	1	4	yes	—
329	#397	C₂²×C₆₇	1	4	yes	—
330	#396	C₂²×C₆₇	1	4	yes	—
331	#391	C₂²×C₆₇	1	2	yes	complemented
332	#391	C₂²×C₆₇	1	2	yes	complemented
333	#389	C₂²×C₆₇	1	4	yes	complemented
334	#389	C₂²×C₆₇	1	4	yes	complemented
335	#389	C₂²×C₆₇	1	2	yes	complemented
336	#389	C₂²×C₆₇	1	2	yes	complemented
337	#388	C₂²×C₆₇	1	4	yes	—
338	#388	C₂²×C₆₇	1	4	yes	—
339	#388	C₂²×C₆₇	1	4	yes	—
340	#387	C₂²×C₆₇	1	4	yes	complemented
341	#387	C₂²×C₆₇	1	4	yes	complemented
342	#387	C₂²×C₆₇	1	4	yes	complemented
343	#387	C₂²×C₆₇	1	4	yes	complemented
344	#385	C₂²×C₆₇	1	4	yes	—
345	#385	C₂²×C₆₇	1	4	yes	—
346	#385	C₂²×C₆₇	1	4	yes	—
347	#385	C₂²×C₆₇	1	4	yes	—
348	#385	C₂²×C₆₇	1	4	yes	—
349	#384	C₂²×C₆₇	1	1	yes	characteristic, complemented
350	#382	C₂²×C₆₇	1	4	yes	complemented
351	#382	C₂²×C₆₇	1	1	yes	characteristic, complemented
352	#380	C₂²×C₆₇	1	2	yes	—
353	#380	C₂²×C₆₇	1	2	yes	—
354	#380	C₂²×C₆₇	1	2	yes	—
355	#380	C₂²×C₆₇	1	4	yes	—
356	#379	C₂²×C₆₇	1	4	yes	—
357	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
358	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
359	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
360	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
361	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
362	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
363	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
364	C₂×#95	C₂²×C₆₇	1	2	yes	—
365	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
366	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
367	C₂³×C₄	—	2	4	no	—
368	C₂³×C₄	—	4	2	no	—
369	C₂³×C₄	—	4	2	no	—
370	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
371	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
372	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
373	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
374	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
375	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
376	C₄⋊Q₈	—	4	2	no	—
377	#126	—	2	16	no	—
378	#125	—	4	2	no	—
379	C₂²⋊Q₈	—	2	8	no	—
380	C₂²⋊Q₈	—	4	4	no	—
381	C₂²⋊Q₈	—	4	4	no	—
382	C₄×Q₈	—	4	4	no	—
383	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
387	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
388	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
395	C₂×C₂²⋊C₄	—	2	8	no	—
396	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
406	C₂×C₄²	—	4	2	no	—
407	C₂×C₄²	—	4	2	no	—
408	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
409	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
411	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
412	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
413	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
415	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
416	#95	C₂³×C₆₇	1	2	yes	—
417	#95	C₂³×C₆₇	1	2	yes	—
418	#95	C₂³×C₆₇	1	2	yes	—
419	#95	C₂³×C₆₇	1	2	yes	—
420	#95	C₂³×C₆₇	1	4	yes	—
421	#95	C₂³×C₆₇	1	4	yes	—
422	#95	C₂³×C₆₇	1	4	yes	—
423	#95	C₂³×C₆₇	1	4	yes	—
424	#95	C₂³×C₆₇	1	4	yes	—
425	#95	C₂³×C₆₇	1	4	yes	—
426	#95	C₂³×C₆₇	1	4	yes	—
427	#95	C₂³×C₆₇	1	4	yes	—
428	#95	C₂³×C₆₇	1	4	yes	—
429	#95	C₂³×C₆₇	1	4	yes	—
430	#95	C₂³×C₆₇	1	4	yes	—
431	C₂⁴	—	2	4	no	—
432	C₂⁴	—	2	2	no	—
433	C₂⁴	—	2	4	no	—
434	C₂⁴	—	4	1	no	—
435	C₂⁴	—	4	1	no	—
436	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
437	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	4	no	—
440	C₂×Q₈	—	2	4	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	2	no	—
454	C₂²×C₄	—	4	2	no	—
455	C₂²×C₄	—	4	2	no	—
456	C₂²×C₄	—	4	1	no	—
457	C₂²×C₄	—	4	1	no	—
458	C₂²×C₄	—	4	16	no	—
459	C₂²×C₄	#41×C₆₇	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
465	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
466	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
468	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
469	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	8	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	8	no	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	4	4	no	—
479	C₄⋊C₄	—	4	4	no	—
480	C₄⋊C₄	—	4	4	no	—
481	C₄⋊C₄	—	4	4	no	—
482	C₄⋊C₄	—	4	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	4	4	no	—
491	C₂²⋊C₄	—	4	4	no	—
492	C₂²⋊C₄	#41×C₆₇	1	8	yes	—
493	C₄²	—	2	8	no	—
494	C₄²	—	2	4	no	—
495	C₄²	—	2	2	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	4	1	no	—
498	C₄²	—	4	2	no	—
499	C₄²	—	4	1	no	—
500	C₄²	—	4	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	4	8	no	—
512	C₂³	—	4	8	no	—
513	C₂³	—	4	4	no	—
514	C₂³	—	4	4	no	—
515	C₂³	C₂×#41×C₆₇	1	4	yes	—
516	C₂³	C₂×#41×C₆₇	1	2	yes	—
517	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
518	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
519	Q₈	—	4	8	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	4	8	no	—
549	C₂×C₄	—	4	8	no	—
550	C₂×C₄	—	4	8	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	8	no	—
553	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
554	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
555	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
556	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
557	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
558	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	2	4	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	4	16	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	2	no	—
570	C₂²	—	4	2	no	—
571	C₂²	#541×C₆₇	1	4	yes	—
572	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
573	C₂²	C₂²×#41×C₆₇	1	2	yes	—
574	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
575	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
576	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
577	C₄	—	2	2	no	—
578	C₄	—	2	4	no	—
579	C₄	—	2	4	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	4	no	—
586	C₄	—	4	4	no	—
587	C₄	#538×C₆₇	1	4	yes	—
588	C₂	—	2	4	no	—
589	C₂	—	4	4	no	—
590	C₂	C₆₇×#3444	1	1	yes	characteristic
591	C₂	C₂×#538×C₆₇	1	1	yes	characteristic
592	C₂	C₂×#541×C₆₇	1	1	yes	characteristic
593	C₂	C₆₇×#3520	1	2	yes	—
594	C₂	C₆₇×#3520	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2857	C₂	semidirect
2. C₆₇×#2851	C₂	semidirect
4. C₆₇×#2652	C₂	semidirect
5. C₆₇×#2505	C₂	semidirect
6. C₆₇×#2496	C₂	semidirect
7. C₆₇×#2489	C₂	semidirect
9. C₆₇×#2429	C₂	semidirect
10. C₆₇×#2421	C₂	semidirect
11. C₆₇×#2334	C₂	semidirect
12. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
15. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
17. C₂×#125×C₆₇	C₂²	semidirect
21. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
25. #400×C₆₇	C₂²	semidirect
26. #400×C₆₇	C₂²	semidirect
27. #400×C₆₇	C₂²	semidirect
28. #398×C₆₇	C₂²	semidirect
33. #391×C₆₇	C₂²	semidirect
34. #391×C₆₇	C₂²	semidirect
35. #389×C₆₇	C₂²	semidirect
36. #389×C₆₇	C₂²	semidirect
37. #389×C₆₇	C₂²	semidirect
38. #389×C₆₇	C₂²	semidirect
42. #387×C₆₇	C₂²	semidirect
43. #387×C₆₇	C₂²	semidirect
44. #387×C₆₇	C₂²	semidirect
45. #387×C₆₇	C₂²	semidirect
51. #384×C₆₇	C₂²	semidirect
52. #382×C₆₇	C₂²	semidirect
53. #382×C₆₇	C₂²	semidirect
59. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
60. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
290. #35086	C₆₇	direct (N × H)
298. #2857	C₂×C₆₇	semidirect
299. #2851	C₂×C₆₇	semidirect
301. #2652	C₂×C₆₇	semidirect
302. #2505	C₂×C₆₇	semidirect
303. #2496	C₂×C₆₇	semidirect
304. #2489	C₂×C₆₇	semidirect
306. #2429	C₂×C₆₇	semidirect
307. #2421	C₂×C₆₇	semidirect
308. #2334	C₂×C₆₇	semidirect
309. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
311. C₆₇	#35086	direct (N × H)
313. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
315. C₂×#125	C₂²×C₆₇	semidirect
319. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
323. #400	C₂²×C₆₇	semidirect
324. #400	C₂²×C₆₇	semidirect
325. #400	C₂²×C₆₇	semidirect
326. #398	C₂²×C₆₇	semidirect
331. #391	C₂²×C₆₇	semidirect
332. #391	C₂²×C₆₇	semidirect
333. #389	C₂²×C₆₇	semidirect
334. #389	C₂²×C₆₇	semidirect
335. #389	C₂²×C₆₇	semidirect
336. #389	C₂²×C₆₇	semidirect
340. #387	C₂²×C₆₇	semidirect
341. #387	C₂²×C₆₇	semidirect
342. #387	C₂²×C₆₇	semidirect
343. #387	C₂²×C₆₇	semidirect
349. #384	C₂²×C₆₇	semidirect
350. #382	C₂²×C₆₇	semidirect
351. #382	C₂²×C₆₇	semidirect
357. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
358. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2857	2
C₆₇×#2851	2
C₆₇×#2848	2
C₆₇×#2652	2
C₆₇×#2505	2
C₆₇×#2496	2
C₆₇×#2489	2
C₆₇×#2480	2
C₆₇×#2429	2
C₆₇×#2421	2
C₆₇×#2334	2
C₄×C₂²⋊Q₈×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
#35086	67

Subgroups — C67×#35086 #30714337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35086 #30714337