Subgroups — C₆₇×#35157 #30714408

Order 17152.

2514 conjugacy classes of subgroups.

The subgroup lattice has 414 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×#2853	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2622	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2500	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2499	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2485	C₂	1	8	yes	maximal, complemented
8	C₆₇×#2317	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2311	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	4	yes	—
21	C₂×#117×C₆₇	C₂²	1	2	yes	—
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
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	#397×C₆₇	C₂²	1	16	yes	—
28	#394×C₆₇	C₂²	1	16	yes	—
29	#391×C₆₇	C₂²	1	4	yes	complemented
30	#390×C₆₇	C₂²	1	4	yes	complemented
31	#389×C₆₇	C₂²	1	8	yes	complemented
32	#388×C₆₇	C₂²	1	16	yes	complemented
33	#387×C₆₇	C₂²	1	16	yes	complemented
34	#386×C₆₇	C₂²	1	8	yes	—
35	#385×C₆₇	C₂²	1	8	yes	—
36	#384×C₆₇	C₂²	1	4	yes	complemented
37	#382×C₆₇	C₂²	1	4	yes	complemented
38	#381×C₆₇	C₂²	1	2	yes	—
39	#380×C₆₇	C₂²	1	4	yes	—
40	#380×C₆₇	C₂²	1	8	yes	complemented
41	#379×C₆₇	C₂²	1	2	yes	—
42	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
43	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×#41×C₆₇	—	2	4	no	—
45	C₂×#41×C₆₇	—	2	8	no	—
46	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
48	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
50	C₂³×C₄×C₆₇	—	2	8	no	—
51	C₂³×C₄×C₆₇	D₄	1	8	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	#123×C₆₇	—	2	8	no	—
59	C₂²⋊Q₈×C₆₇	—	2	8	no	—
60	#121×C₆₇	—	2	8	no	—
61	#120×C₆₇	—	2	8	no	—
62	C₄×D₄×C₆₇	—	2	16	no	—
63	#117×C₆₇	—	2	16	no	—
64	#117×C₆₇	—	2	16	no	—
65	#117×C₆₇	—	2	4	no	—
66	#117×C₆₇	—	2	8	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	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	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
82	C₂×C₄²×C₆₇	—	2	8	no	—
83	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
84	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
85	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
86	#95×C₆₇	C₂³	1	8	yes	—
87	#95×C₆₇	C₂³	1	16	yes	—
88	#95×C₆₇	C₂³	1	16	yes	—
89	C₂⁴×C₆₇	—	2	4	no	—
90	C₂⁴×C₆₇	—	4	4	no	—
91	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
92	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
93	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
94	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
95	#41×C₆₇	—	4	16	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	8	no	—
99	C₂×D₄×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	2	8	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	4	8	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	32	no	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	2	8	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	8	no	—
111	C₂²×C₄×C₆₇	—	4	8	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
114	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
115	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
116	C₂²×C₄×C₆₇	#41	1	8	yes	—
117	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
118	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
124	C₄⋊C₄×C₆₇	—	2	8	no	—
125	C₄⋊C₄×C₆₇	—	2	8	no	—
126	C₄⋊C₄×C₆₇	—	2	4	no	—
127	C₄⋊C₄×C₆₇	—	2	8	no	—
128	C₄⋊C₄×C₆₇	—	2	8	no	—
129	C₄⋊C₄×C₆₇	—	4	16	no	—
130	C₂²⋊C₄×C₆₇	—	2	16	no	—
131	C₂²⋊C₄×C₆₇	—	2	4	no	—
132	C₂²⋊C₄×C₆₇	—	2	8	no	—
133	C₂²⋊C₄×C₆₇	—	2	8	no	—
134	C₂²⋊C₄×C₆₇	—	2	8	no	—
135	C₂²⋊C₄×C₆₇	—	4	8	no	—
136	C₂²⋊C₄×C₆₇	—	4	16	no	—
137	C₄²×C₆₇	—	2	16	no	—
138	C₄²×C₆₇	—	2	4	no	—
139	C₄²×C₆₇	—	2	8	no	—
140	C₂³×C₆₇	—	2	4	no	—
141	C₂³×C₆₇	—	2	8	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	2	4	no	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	4	no	—
147	C₂³×C₆₇	—	2	8	no	—
148	C₂³×C₆₇	—	4	8	no	—
149	C₂³×C₆₇	—	4	16	no	—
150	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
151	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
152	C₂³×C₆₇	C₂×#41	1	4	yes	—
153	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
154	Q₈×C₆₇	—	4	4	no	—
155	D₄×C₆₇	—	4	16	no	—
156	D₄×C₆₇	—	4	8	no	—
157	D₄×C₆₇	—	4	4	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	16	no	—
161	C₂×C₄×C₆₇	—	2	16	no	—
162	C₂×C₄×C₆₇	—	2	8	no	—
163	C₂×C₄×C₆₇	—	2	4	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	8	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	8	no	—
169	C₂×C₄×C₆₇	—	2	16	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	4	16	no	—
172	C₂×C₄×C₆₇	—	4	8	no	—
173	C₂×C₄×C₆₇	#123	1	16	yes	—
174	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
175	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
176	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
178	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
179	C₂²×C₆₇	—	2	4	no	—
180	C₂²×C₆₇	—	2	8	no	—
181	C₂²×C₆₇	—	2	4	no	—
182	C₂²×C₆₇	—	2	4	no	—
183	C₂²×C₆₇	—	2	4	no	—
184	C₂²×C₆₇	—	4	16	no	—
185	C₂²×C₆₇	—	4	8	no	—
186	C₂²×C₆₇	—	4	2	no	—
187	C₂²×C₆₇	—	4	4	no	—
188	C₂²×C₆₇	—	4	8	no	—
189	C₂²×C₆₇	C₂×#123	1	4	yes	—
190	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
191	C₂²×C₆₇	C₂²×#41	1	2	yes	—
192	C₂²×C₆₇	#585	1	4	yes	—
193	C₄×C₆₇	—	2	4	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	—	4	8	no	—
196	C₄×C₆₇	—	4	4	no	—
197	C₄×C₆₇	—	4	8	no	—
198	C₄×C₆₇	C₂×#123	1	4	yes	—
199	C₄×C₆₇	C₂×#123	1	4	yes	—
200	#35157	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₆₇	—	2	4	no	—
202	C₂×C₆₇	—	4	4	no	—
203	C₂×C₆₇	—	8	2	no	—
204	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
205	C₂×C₆₇	#3449	1	2	yes	—
206	C₂×C₆₇	#3493	1	4	yes	—
207	C₂×#525	C₂×C₆₇	1	1	yes	characteristic
208	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#2853	C₂×C₆₇	1	4	yes	complemented
210	#2622	C₂×C₆₇	1	4	yes	complemented
211	#2500	C₂×C₆₇	1	4	yes	complemented
212	#2499	C₂×C₆₇	1	4	yes	complemented
213	#2485	C₂×C₆₇	1	8	yes	complemented
214	#2317	C₂×C₆₇	1	2	yes	complemented
215	#2311	C₂×C₆₇	1	2	yes	complemented
216	#2280	C₂×C₆₇	1	1	yes	characteristic, complemented
217	C₆₇	#35157	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
219	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
220	#525	—	2	8	no	—
221	C₂×#123	C₂²×C₆₇	1	2	yes	—
222	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
223	C₂×#121	C₂²×C₆₇	1	2	yes	—
224	C₂×#120	C₂²×C₆₇	1	2	yes	—
225	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
226	C₂×#117	—	2	8	no	—
227	C₂×#117	C₂²×C₆₇	1	2	yes	—
228	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×#117	C₂²×C₆₇	1	4	yes	—
230	C₂×#117	C₂²×C₆₇	1	4	yes	—
231	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
232	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
233	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
234	#397	C₂²×C₆₇	1	16	yes	—
235	#394	C₂²×C₆₇	1	16	yes	—
236	#391	C₂²×C₆₇	1	4	yes	complemented
237	#390	C₂²×C₆₇	1	4	yes	complemented
238	#389	C₂²×C₆₇	1	8	yes	complemented
239	#388	C₂²×C₆₇	1	16	yes	complemented
240	#387	C₂²×C₆₇	1	16	yes	complemented
241	#386	C₂²×C₆₇	1	8	yes	—
242	#385	C₂²×C₆₇	1	8	yes	—
243	#384	C₂²×C₆₇	1	4	yes	complemented
244	#382	C₂²×C₆₇	1	4	yes	complemented
245	#381	C₂²×C₆₇	1	2	yes	—
246	#380	C₂²×C₆₇	1	4	yes	—
247	#380	C₂²×C₆₇	1	8	yes	complemented
248	#379	C₂²×C₆₇	1	2	yes	—
249	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
250	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
251	C₂×#41	—	2	4	no	—
252	C₂×#41	—	2	8	no	—
253	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
254	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
255	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
256	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
257	C₂³×C₄	—	2	8	no	—
258	C₂³×C₄	D₄×C₆₇	1	8	yes	—
259	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
260	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
261	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
262	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
263	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
264	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
265	#123	—	2	8	no	—
266	C₂²⋊Q₈	—	2	8	no	—
267	#121	—	2	8	no	—
268	#120	—	2	8	no	—
269	C₄×D₄	—	2	16	no	—
270	#117	—	2	16	no	—
271	#117	—	2	16	no	—
272	#117	—	2	4	no	—
273	#117	—	2	8	no	—
274	C₂×C₄⋊C₄	—	2	8	no	—
275	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
276	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
277	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
281	C₂×C₂²⋊C₄	—	2	8	no	—
282	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
283	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
285	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
286	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
289	C₂×C₄²	—	2	8	no	—
290	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
291	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
292	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
293	#95	C₂³×C₆₇	1	8	yes	—
294	#95	C₂³×C₆₇	1	16	yes	—
295	#95	C₂³×C₆₇	1	16	yes	—
296	C₂⁴	—	2	4	no	—
297	C₂⁴	—	4	4	no	—
298	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
299	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
300	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
301	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
302	#41	—	4	16	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	4	no	—
305	C₂×D₄	—	2	8	no	—
306	C₂×D₄	—	2	4	no	—
307	C₂×D₄	—	2	8	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	4	no	—
310	C₂×D₄	—	4	8	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	32	no	—
313	C₂²×C₄	—	2	8	no	—
314	C₂²×C₄	—	2	8	no	—
315	C₂²×C₄	—	2	8	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	8	no	—
318	C₂²×C₄	—	4	8	no	—
319	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
320	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
321	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
322	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
323	C₂²×C₄	#41×C₆₇	1	8	yes	—
324	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
326	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
327	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
328	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
329	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
330	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	2	8	no	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	8	no	—
335	C₄⋊C₄	—	2	8	no	—
336	C₄⋊C₄	—	4	16	no	—
337	C₂²⋊C₄	—	2	16	no	—
338	C₂²⋊C₄	—	2	4	no	—
339	C₂²⋊C₄	—	2	8	no	—
340	C₂²⋊C₄	—	2	8	no	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₂²⋊C₄	—	4	8	no	—
343	C₂²⋊C₄	—	4	16	no	—
344	C₄²	—	2	16	no	—
345	C₄²	—	2	4	no	—
346	C₄²	—	2	8	no	—
347	C₂³	—	2	4	no	—
348	C₂³	—	2	8	no	—
349	C₂³	—	2	4	no	—
350	C₂³	—	2	4	no	—
351	C₂³	—	2	4	no	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	4	no	—
354	C₂³	—	2	8	no	—
355	C₂³	—	4	8	no	—
356	C₂³	—	4	16	no	—
357	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
358	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
359	C₂³	C₂×#41×C₆₇	1	4	yes	—
360	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
361	Q₈	—	4	4	no	—
362	D₄	—	4	16	no	—
363	D₄	—	4	8	no	—
364	D₄	—	4	4	no	—
365	C₂×C₄	—	2	8	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	16	no	—
368	C₂×C₄	—	2	16	no	—
369	C₂×C₄	—	2	8	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	2	16	no	—
377	C₂×C₄	—	2	8	no	—
378	C₂×C₄	—	4	16	no	—
379	C₂×C₄	—	4	8	no	—
380	C₂×C₄	#123×C₆₇	1	16	yes	—
381	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
382	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
383	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
384	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
385	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
386	C₂²	—	2	4	no	—
387	C₂²	—	2	8	no	—
388	C₂²	—	2	4	no	—
389	C₂²	—	2	4	no	—
390	C₂²	—	2	4	no	—
391	C₂²	—	4	16	no	—
392	C₂²	—	4	8	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	8	no	—
396	C₂²	C₂×#123×C₆₇	1	4	yes	—
397	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
398	C₂²	C₂²×#41×C₆₇	1	2	yes	—
399	C₂²	#585×C₆₇	1	4	yes	—
400	C₄	—	2	4	no	—
401	C₄	—	4	2	no	—
402	C₄	—	4	8	no	—
403	C₄	—	4	4	no	—
404	C₄	—	4	8	no	—
405	C₄	C₂×#123×C₆₇	1	4	yes	—
406	C₄	C₂×#123×C₆₇	1	4	yes	—
407	C₂	—	2	4	no	—
408	C₂	—	4	4	no	—
409	C₂	—	8	2	no	—
410	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
411	C₂	C₆₇×#3449	1	2	yes	—
412	C₂	C₆₇×#3493	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂×#525×C₆₇	C₂
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
2. C₂²×#117×C₆₇	C₂	semidirect
3. C₆₇×#2853	C₂	semidirect
4. C₆₇×#2622	C₂	semidirect
5. C₆₇×#2500	C₂	semidirect
6. C₆₇×#2499	C₂	semidirect
7. C₆₇×#2485	C₂	semidirect
8. C₆₇×#2317	C₂	semidirect
9. C₆₇×#2311	C₂	semidirect
10. C₆₇×#2280	C₂	semidirect
29. #391×C₆₇	C₂²	semidirect
30. #390×C₆₇	C₂²	semidirect
31. #389×C₆₇	C₂²	semidirect
32. #388×C₆₇	C₂²	semidirect
33. #387×C₆₇	C₂²	semidirect
36. #384×C₆₇	C₂²	semidirect
37. #382×C₆₇	C₂²	semidirect
40. #380×C₆₇	C₂²	semidirect
200. #35157	C₆₇	direct (N × H)
208. C₂²×#117	C₂×C₆₇	semidirect
209. #2853	C₂×C₆₇	semidirect
210. #2622	C₂×C₆₇	semidirect
211. #2500	C₂×C₆₇	semidirect
212. #2499	C₂×C₆₇	semidirect
213. #2485	C₂×C₆₇	semidirect
214. #2317	C₂×C₆₇	semidirect
215. #2311	C₂×C₆₇	semidirect
216. #2280	C₂×C₆₇	semidirect
217. C₆₇	#35157	direct (N × H)
236. #391	C₂²×C₆₇	semidirect
237. #390	C₂²×C₆₇	semidirect
238. #389	C₂²×C₆₇	semidirect
239. #388	C₂²×C₆₇	semidirect
240. #387	C₂²×C₆₇	semidirect
243. #384	C₂²×C₆₇	semidirect
244. #382	C₂²×C₆₇	semidirect
247. #380	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×#2853	2
C₆₇×#2622	2
C₆₇×#2500	2
C₆₇×#2499	2
C₆₇×#2485	2
C₆₇×#2317	2
C₆₇×#2311	2
C₆₇×#2280	2
#35157	67

Subgroups — C67×#35157 #30714408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#35157 #30714408