Subgroups — C₆₇×#34635 #30713886

Order 17152.

2514 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2625	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2624	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2623	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2622	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2395	C₂	1	8	yes	maximal, complemented
8	C₆₇×#2297	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2296	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2280	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	#525×C₆₇	—	2	8	no	—
14	C₂×#123×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#121×C₆₇	C₂²	1	2	yes	—
17	C₂×#120×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
19	C₂×#117×C₆₇	—	2	8	no	—
20	C₂×#117×C₆₇	C₂²	1	2	yes	—
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	4	yes	—
23	C₂×#117×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
27	#400×C₆₇	C₂²	1	8	yes	complemented
28	#399×C₆₇	C₂²	1	8	yes	complemented
29	#398×C₆₇	C₂²	1	8	yes	complemented
30	#397×C₆₇	C₂²	1	16	yes	—
31	#397×C₆₇	C₂²	1	8	yes	complemented
32	#394×C₆₇	C₂²	1	16	yes	—
33	#391×C₆₇	C₂²	1	4	yes	complemented
34	#390×C₆₇	C₂²	1	4	yes	complemented
35	#389×C₆₇	C₂²	1	8	yes	complemented
36	#386×C₆₇	C₂²	1	8	yes	—
37	#385×C₆₇	C₂²	1	8	yes	—
38	#381×C₆₇	C₂²	1	2	yes	—
39	#380×C₆₇	C₂²	1	4	yes	—
40	#379×C₆₇	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
42	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
43	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
44	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×#41×C₆₇	—	2	4	no	—
46	C₂×#41×C₆₇	—	2	8	no	—
47	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
49	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₇	—	2	8	no	—
52	C₂³×C₄×C₆₇	D₄	1	8	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	#123×C₆₇	—	2	8	no	—
60	C₂²⋊Q₈×C₆₇	—	2	8	no	—
61	#121×C₆₇	—	2	8	no	—
62	#120×C₆₇	—	2	8	no	—
63	C₄×D₄×C₆₇	—	2	16	no	—
64	#117×C₆₇	—	2	16	no	—
65	#117×C₆₇	—	2	16	no	—
66	#117×C₆₇	—	2	4	no	—
67	#117×C₆₇	—	2	8	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
83	C₂×C₄²×C₆₇	—	2	8	no	—
84	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
85	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
86	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
88	#95×C₆₇	C₂³	1	8	yes	—
89	#95×C₆₇	C₂³	1	16	yes	—
90	#95×C₆₇	C₂³	1	16	yes	—
91	C₂⁴×C₆₇	—	2	4	no	—
92	C₂⁴×C₆₇	—	4	4	no	—
93	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
94	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
95	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
96	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
97	#41×C₆₇	—	4	16	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	2	8	no	—
101	C₂×D₄×C₆₇	—	2	4	no	—
102	C₂×D₄×C₆₇	—	2	8	no	—
103	C₂×D₄×C₆₇	—	2	2	no	—
104	C₂×D₄×C₆₇	—	2	4	no	—
105	C₂×D₄×C₆₇	—	4	8	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	32	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	8	no	—
110	C₂²×C₄×C₆₇	—	2	8	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	8	no	—
113	C₂²×C₄×C₆₇	—	4	8	no	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₆₇	#41	1	8	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
126	C₄⋊C₄×C₆₇	—	2	8	no	—
127	C₄⋊C₄×C₆₇	—	2	8	no	—
128	C₄⋊C₄×C₆₇	—	2	4	no	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	4	16	no	—
132	C₂²⋊C₄×C₆₇	—	2	16	no	—
133	C₂²⋊C₄×C₆₇	—	2	4	no	—
134	C₂²⋊C₄×C₆₇	—	2	8	no	—
135	C₂²⋊C₄×C₆₇	—	2	8	no	—
136	C₂²⋊C₄×C₆₇	—	2	8	no	—
137	C₂²⋊C₄×C₆₇	—	4	8	no	—
138	C₂²⋊C₄×C₆₇	—	4	16	no	—
139	C₄²×C₆₇	—	2	8	no	—
140	C₄²×C₆₇	—	2	8	no	—
141	C₄²×C₆₇	—	2	4	no	—
142	C₄²×C₆₇	—	2	8	no	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	2	8	no	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	4	no	—
150	C₂³×C₆₇	—	2	8	no	—
151	C₂³×C₆₇	—	4	8	no	—
152	C₂³×C₆₇	—	4	16	no	—
153	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
154	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
155	C₂³×C₆₇	C₂×#41	1	4	yes	—
156	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
157	Q₈×C₆₇	—	4	4	no	—
158	D₄×C₆₇	—	4	16	no	—
159	D₄×C₆₇	—	4	8	no	—
160	D₄×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	16	no	—
164	C₂×C₄×C₆₇	—	2	16	no	—
165	C₂×C₄×C₆₇	—	2	8	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	8	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	16	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	8	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	4	16	no	—
175	C₂×C₄×C₆₇	—	4	8	no	—
176	C₂×C₄×C₆₇	#123	1	16	yes	—
177	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
178	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
179	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
180	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
181	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	2	8	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	2	4	no	—
187	C₂²×C₆₇	—	4	16	no	—
188	C₂²×C₆₇	—	4	8	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	4	no	—
191	C₂²×C₆₇	—	4	8	no	—
192	C₂²×C₆₇	C₂×#123	1	4	yes	—
193	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
194	C₂²×C₆₇	C₂²×#41	1	2	yes	—
195	C₂²×C₆₇	#585	1	4	yes	—
196	C₄×C₆₇	—	2	4	no	—
197	C₄×C₆₇	—	4	2	no	—
198	C₄×C₆₇	—	4	8	no	—
199	C₄×C₆₇	—	4	4	no	—
200	C₄×C₆₇	—	4	8	no	—
201	C₄×C₆₇	C₂×#123	1	4	yes	—
202	C₄×C₆₇	C₂×#123	1	4	yes	—
203	#34635	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂×C₆₇	—	2	4	no	—
205	C₂×C₆₇	—	4	4	no	—
206	C₂×C₆₇	—	8	2	no	—
207	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
208	C₂×C₆₇	#3449	1	2	yes	—
209	C₂×C₆₇	#3455	1	4	yes	—
210	C₂×#525	C₂×C₆₇	1	1	yes	characteristic
211	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#2625	C₂×C₆₇	1	4	yes	complemented
213	#2624	C₂×C₆₇	1	4	yes	complemented
214	#2623	C₂×C₆₇	1	4	yes	complemented
215	#2622	C₂×C₆₇	1	4	yes	complemented
216	#2395	C₂×C₆₇	1	8	yes	complemented
217	#2297	C₂×C₆₇	1	2	yes	complemented
218	#2296	C₂×C₆₇	1	2	yes	complemented
219	#2280	C₂×C₆₇	1	1	yes	characteristic, complemented
220	C₆₇	#34635	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
222	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
223	#525	—	2	8	no	—
224	C₂×#123	C₂²×C₆₇	1	2	yes	—
225	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
226	C₂×#121	C₂²×C₆₇	1	2	yes	—
227	C₂×#120	C₂²×C₆₇	1	2	yes	—
228	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
229	C₂×#117	—	2	8	no	—
230	C₂×#117	C₂²×C₆₇	1	4	yes	—
231	C₂×#117	C₂²×C₆₇	1	4	yes	—
232	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
233	C₂×#117	C₂²×C₆₇	1	2	yes	—
234	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
235	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
236	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
237	#400	C₂²×C₆₇	1	8	yes	complemented
238	#399	C₂²×C₆₇	1	8	yes	complemented
239	#398	C₂²×C₆₇	1	8	yes	complemented
240	#397	C₂²×C₆₇	1	16	yes	—
241	#397	C₂²×C₆₇	1	8	yes	complemented
242	#394	C₂²×C₆₇	1	16	yes	—
243	#391	C₂²×C₆₇	1	4	yes	complemented
244	#390	C₂²×C₆₇	1	4	yes	complemented
245	#389	C₂²×C₆₇	1	8	yes	complemented
246	#386	C₂²×C₆₇	1	8	yes	—
247	#385	C₂²×C₆₇	1	8	yes	—
248	#381	C₂²×C₆₇	1	2	yes	—
249	#380	C₂²×C₆₇	1	4	yes	—
250	#379	C₂²×C₆₇	1	2	yes	—
251	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
252	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
253	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
254	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
255	C₂×#41	—	2	4	no	—
256	C₂×#41	—	2	8	no	—
257	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
258	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
259	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
260	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
261	C₂³×C₄	—	2	8	no	—
262	C₂³×C₄	D₄×C₆₇	1	8	yes	—
263	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
264	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
266	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
267	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
268	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
269	#123	—	2	8	no	—
270	C₂²⋊Q₈	—	2	8	no	—
271	#121	—	2	8	no	—
272	#120	—	2	8	no	—
273	C₄×D₄	—	2	16	no	—
274	#117	—	2	16	no	—
275	#117	—	2	16	no	—
276	#117	—	2	4	no	—
277	#117	—	2	8	no	—
278	C₂×C₄⋊C₄	—	2	8	no	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
285	C₂×C₂²⋊C₄	—	2	8	no	—
286	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
289	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
290	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
293	C₂×C₄²	—	2	8	no	—
294	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
295	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
296	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
297	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
298	#95	C₂³×C₆₇	1	8	yes	—
299	#95	C₂³×C₆₇	1	16	yes	—
300	#95	C₂³×C₆₇	1	16	yes	—
301	C₂⁴	—	2	4	no	—
302	C₂⁴	—	4	4	no	—
303	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
304	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
305	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
306	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
307	#41	—	4	16	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	4	no	—
310	C₂×D₄	—	2	8	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	4	no	—
315	C₂×D₄	—	4	8	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	32	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	8	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	8	no	—
323	C₂²×C₄	—	4	8	no	—
324	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
325	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
326	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
327	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
328	C₂²×C₄	#41×C₆₇	1	8	yes	—
329	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
330	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
331	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
332	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
333	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	8	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	8	no	—
340	C₄⋊C₄	—	2	8	no	—
341	C₄⋊C₄	—	4	16	no	—
342	C₂²⋊C₄	—	2	16	no	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	2	8	no	—
345	C₂²⋊C₄	—	2	8	no	—
346	C₂²⋊C₄	—	2	8	no	—
347	C₂²⋊C₄	—	4	8	no	—
348	C₂²⋊C₄	—	4	16	no	—
349	C₄²	—	2	8	no	—
350	C₄²	—	2	8	no	—
351	C₄²	—	2	4	no	—
352	C₄²	—	2	8	no	—
353	C₂³	—	2	4	no	—
354	C₂³	—	2	8	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	2	4	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	2	8	no	—
361	C₂³	—	4	8	no	—
362	C₂³	—	4	16	no	—
363	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
364	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
365	C₂³	C₂×#41×C₆₇	1	4	yes	—
366	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
367	Q₈	—	4	4	no	—
368	D₄	—	4	16	no	—
369	D₄	—	4	8	no	—
370	D₄	—	4	4	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	16	no	—
374	C₂×C₄	—	2	16	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	8	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	16	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	4	16	no	—
385	C₂×C₄	—	4	8	no	—
386	C₂×C₄	#123×C₆₇	1	16	yes	—
387	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
388	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
389	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
390	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
391	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	8	no	—
394	C₂²	—	2	4	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	2	4	no	—
397	C₂²	—	4	16	no	—
398	C₂²	—	4	8	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	4	8	no	—
402	C₂²	C₂×#123×C₆₇	1	4	yes	—
403	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
404	C₂²	C₂²×#41×C₆₇	1	2	yes	—
405	C₂²	#585×C₆₇	1	4	yes	—
406	C₄	—	2	4	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	8	no	—
409	C₄	—	4	4	no	—
410	C₄	—	4	8	no	—
411	C₄	C₂×#123×C₆₇	1	4	yes	—
412	C₄	C₂×#123×C₆₇	1	4	yes	—
413	C₂	—	2	4	no	—
414	C₂	—	4	4	no	—
415	C₂	—	8	2	no	—
416	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
417	C₂	C₆₇×#3449	1	2	yes	—
418	C₂	C₆₇×#3455	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂×#525×C₆₇	C₂
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
2. C₂²×#117×C₆₇	C₂	semidirect
3. C₆₇×#2625	C₂	semidirect
4. C₆₇×#2624	C₂	semidirect
5. C₆₇×#2623	C₂	semidirect
6. C₆₇×#2622	C₂	semidirect
7. C₆₇×#2395	C₂	semidirect
8. C₆₇×#2297	C₂	semidirect
9. C₆₇×#2296	C₂	semidirect
10. C₆₇×#2280	C₂	semidirect
27. #400×C₆₇	C₂²	semidirect
28. #399×C₆₇	C₂²	semidirect
29. #398×C₆₇	C₂²	semidirect
31. #397×C₆₇	C₂²	semidirect
33. #391×C₆₇	C₂²	semidirect
34. #390×C₆₇	C₂²	semidirect
35. #389×C₆₇	C₂²	semidirect
41. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
43. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
203. #34635	C₆₇	direct (N × H)
211. C₂²×#117	C₂×C₆₇	semidirect
212. #2625	C₂×C₆₇	semidirect
213. #2624	C₂×C₆₇	semidirect
214. #2623	C₂×C₆₇	semidirect
215. #2622	C₂×C₆₇	semidirect
216. #2395	C₂×C₆₇	semidirect
217. #2297	C₂×C₆₇	semidirect
218. #2296	C₂×C₆₇	semidirect
219. #2280	C₂×C₆₇	semidirect
220. C₆₇	#34635	direct (N × H)
237. #400	C₂²×C₆₇	semidirect
238. #399	C₂²×C₆₇	semidirect
239. #398	C₂²×C₆₇	semidirect
241. #397	C₂²×C₆₇	semidirect
243. #391	C₂²×C₆₇	semidirect
244. #390	C₂²×C₆₇	semidirect
245. #389	C₂²×C₆₇	semidirect
251. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
253. C₄×C₂²⋊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
C₂×#525×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2625	2
C₆₇×#2624	2
C₆₇×#2623	2
C₆₇×#2622	2
C₆₇×#2395	2
C₆₇×#2297	2
C₆₇×#2296	2
C₆₇×#2280	2
#34635	67

Subgroups — C67×#34635 #30713886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34635 #30713886