Subgroups — C₆₇×#22537 #30701788

Order 17152.

1002 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#556×C₆₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1967	C₂	1	4	yes	maximal
4	C₆₇×#1966	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1801	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1800	C₂	1	1	yes	maximal, characteristic
7	C₄×#101×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₄×#100×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#556×C₆₇	—	2	2	no	—
10	#556×C₆₇	—	2	2	no	—
11	#556×C₆₇	—	2	4	no	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#126×C₆₇	C₄	1	4	yes	—
14	C₂×#124×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₄	1	2	yes	—
20	#446×C₆₇	—	2	2	no	—
21	#446×C₆₇	—	2	4	no	—
22	C₈×Q₈×C₆₇	—	2	2	no	—
23	#423×C₆₇	C₂²	1	2	yes	—
24	#423×C₆₇	C₂²	1	4	yes	—
25	C₂×#101×C₆₇	C₂²	1	4	yes	—
26	C₂×#101×C₆₇	C₂²	1	4	yes	—
27	C₂×#100×C₆₇	C₂²	1	2	yes	—
28	C₂×#100×C₆₇	C₂²	1	2	yes	—
29	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
30	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
31	#343×C₆₇	C₂²	1	1	yes	characteristic
32	#343×C₆₇	C₂²	1	1	yes	characteristic
33	#343×C₆₇	C₂²	1	2	yes	—
34	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
36	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
39	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
40	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₆₇	—	2	2	no	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	#126×C₆₇	—	2	8	no	—
44	#126×C₆₇	—	2	8	no	—
45	#124×C₆₇	—	2	4	no	—
46	#124×C₆₇	—	2	4	no	—
47	C₂²⋊Q₈×C₆₇	—	2	4	no	—
48	C₂²⋊Q₈×C₆₇	—	2	4	no	—
49	C₄×Q₈×C₆₇	—	2	4	no	—
50	C₄×Q₈×C₆₇	—	2	4	no	—
51	C₄×Q₈×C₆₇	D₄	1	2	yes	—
52	C₄×Q₈×C₆₇	D₄	1	2	yes	—
53	C₄×Q₈×C₆₇	D₄	1	4	yes	—
54	C₄×D₄×C₆₇	—	2	2	no	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	#117×C₆₇	—	2	4	no	—
57	#117×C₆₇	—	2	2	no	—
58	#117×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
66	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
68	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
71	C₄⋊C₈×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₈×C₆₇	—	2	2	no	—
74	C₄⋊C₈×C₆₇	—	2	4	no	—
75	#101×C₆₇	—	2	8	no	—
76	#101×C₆₇	—	4	4	no	—
77	#100×C₆₇	—	2	4	no	—
78	#100×C₆₇	—	4	2	no	—
79	#97×C₆₇	—	2	2	no	—
80	#97×C₆₇	—	2	4	no	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	C₄×C₈×C₆₇	—	2	4	no	—
83	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×Q₈×C₆₇	—	2	4	no	—
85	C₂×Q₈×C₆₇	—	2	4	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	—	4	2	no	—
89	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
90	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
91	C₂×D₄×C₆₇	—	2	2	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	4	4	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₆₇	—	2	2	no	—
106	M₁₆×C₆₇	—	2	2	no	—
107	M₁₆×C₆₇	—	2	4	no	—
108	M₁₆×C₆₇	—	2	4	no	—
109	M₁₆×C₆₇	—	2	2	no	—
110	M₁₆×C₆₇	#41	1	4	yes	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
120	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
121	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
122	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
123	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
124	C₂²⋊C₄×C₆₇	—	2	4	no	—
125	C₂²⋊C₄×C₆₇	—	2	4	no	—
126	C₂²⋊C₄×C₆₇	—	2	4	no	—
127	C₂²⋊C₄×C₆₇	—	2	8	no	—
128	C₄²×C₆₇	—	2	4	no	—
129	C₄²×C₆₇	—	2	4	no	—
130	C₄²×C₆₇	C₂×D₄	1	2	yes	—
131	C₄²×C₆₇	C₂×D₄	1	2	yes	—
132	C₄²×C₆₇	C₂×D₄	1	4	yes	—
133	C₄²×C₆₇	C₂×D₄	1	4	yes	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	4	2	no	—
138	C₂³×C₆₇	—	4	2	no	—
139	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
140	Q₈×C₆₇	—	2	4	no	—
141	Q₈×C₆₇	—	4	2	no	—
142	Q₈×C₆₇	—	4	4	no	—
143	Q₈×C₆₇	—	4	4	no	—
144	D₄×C₆₇	—	4	2	no	—
145	D₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	4	no	—
155	C₂×C₄×C₆₇	—	4	4	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
159	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
161	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
164	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
165	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
167	C₈×C₆₇	—	2	4	no	—
168	C₈×C₆₇	—	4	2	no	—
169	C₈×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	2	1	no	—
171	C₂²×C₆₇	—	2	1	no	—
172	C₂²×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	—	4	2	no	—
175	C₂²×C₆₇	—	8	2	no	—
176	C₂²×C₆₇	—	8	2	no	—
177	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
178	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
179	C₂²×C₆₇	#520	1	1	yes	characteristic
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	2	4	no	—
182	C₄×C₆₇	—	2	2	no	—
183	C₄×C₆₇	—	2	2	no	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	4	4	no	—
186	C₄×C₆₇	—	4	2	no	—
187	C₄×C₆₇	#413	1	2	yes	—
188	C₄×C₆₇	#548	1	2	yes	—
189	C₄×C₆₇	#550	1	2	yes	—
190	#22537	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
191	C₂×C₆₇	—	2	1	no	—
192	C₂×C₆₇	—	2	1	no	—
193	C₂×C₆₇	—	8	2	no	—
194	C₂×C₆₇	#2341	1	1	yes	characteristic
195	C₂×C₆₇	C₂×#413	1	1	yes	characteristic
196	C₂×C₆₇	#2889	1	1	yes	characteristic
197	C₂×#556	C₂×C₆₇	1	1	yes	characteristic
198	Q₈×M₁₆	C₂×C₆₇	1	2	yes	complemented
199	#1967	C₂×C₆₇	1	4	yes	—
200	#1966	C₂×C₆₇	1	2	yes	complemented
201	#1801	C₂×C₆₇	1	2	yes	complemented
202	#1800	C₂×C₆₇	1	1	yes	characteristic
203	C₄×#101	C₂×C₆₇	1	2	yes	complemented
204	C₄×#100	C₂×C₆₇	1	1	yes	characteristic
205	C₆₇	#22537	1	1	yes	characteristic, direct product factor, Sylow
206	#556	—	2	2	no	—
207	#556	—	2	2	no	—
208	#556	—	2	4	no	—
209	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×#126	C₄×C₆₇	1	4	yes	—
211	C₂×#124	C₂²×C₆₇	1	2	yes	—
212	C₂×C₂²⋊Q₈	C₄×C₆₇	1	2	yes	—
213	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
214	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
216	C₂×#117	C₄×C₆₇	1	2	yes	—
217	#446	—	2	2	no	—
218	#446	—	2	4	no	—
219	C₈×Q₈	—	2	2	no	—
220	#423	C₂²×C₆₇	1	2	yes	—
221	#423	C₂²×C₆₇	1	4	yes	—
222	C₂×#101	C₂²×C₆₇	1	4	yes	—
223	C₂×#101	C₂²×C₆₇	1	4	yes	—
224	C₂×#100	C₂²×C₆₇	1	2	yes	—
225	C₂×#100	C₂²×C₆₇	1	2	yes	—
226	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
227	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
228	#343	C₂²×C₆₇	1	2	yes	—
229	#343	C₂²×C₆₇	1	1	yes	characteristic
230	#343	C₂²×C₆₇	1	1	yes	characteristic
231	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
232	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
233	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
234	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
235	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
236	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
237	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
238	C₄⋊Q₈	—	2	2	no	—
239	C₄⋊Q₈	—	2	2	no	—
240	#126	—	2	8	no	—
241	#126	—	2	8	no	—
242	#124	—	2	4	no	—
243	#124	—	2	4	no	—
244	C₂²⋊Q₈	—	2	4	no	—
245	C₂²⋊Q₈	—	2	4	no	—
246	C₄×Q₈	—	2	4	no	—
247	C₄×Q₈	—	2	4	no	—
248	C₄×Q₈	D₄×C₆₇	1	2	yes	—
249	C₄×Q₈	D₄×C₆₇	1	2	yes	—
250	C₄×Q₈	D₄×C₆₇	1	4	yes	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	2	2	no	—
253	#117	—	2	4	no	—
254	#117	—	2	2	no	—
255	#117	—	2	2	no	—
256	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
257	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
258	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
259	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
260	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
261	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
262	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
263	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
264	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
265	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
266	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	2	4	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	2	4	no	—
272	#101	—	2	8	no	—
273	#101	—	4	4	no	—
274	#100	—	2	4	no	—
275	#100	—	4	2	no	—
276	#97	—	2	2	no	—
277	#97	—	2	4	no	—
278	C₄×C₈	—	2	2	no	—
279	C₄×C₈	—	2	4	no	—
280	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
281	C₂×Q₈	—	2	4	no	—
282	C₂×Q₈	—	2	4	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	4	2	no	—
286	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
287	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	—	2	1	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	4	4	no	—
295	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
296	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
298	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
299	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
300	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
301	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
302	M₁₆	—	2	2	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	4	no	—
305	M₁₆	—	2	4	no	—
306	M₁₆	—	2	2	no	—
307	M₁₆	#41×C₆₇	1	4	yes	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	4	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	8	no	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
317	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
318	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
319	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
320	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
321	C₂²⋊C₄	—	2	4	no	—
322	C₂²⋊C₄	—	2	4	no	—
323	C₂²⋊C₄	—	2	4	no	—
324	C₂²⋊C₄	—	2	8	no	—
325	C₄²	—	2	4	no	—
326	C₄²	—	2	4	no	—
327	C₄²	C₂×D₄×C₆₇	1	2	yes	—
328	C₄²	C₂×D₄×C₆₇	1	4	yes	—
329	C₄²	C₂×D₄×C₆₇	1	2	yes	—
330	C₄²	C₂×D₄×C₆₇	1	4	yes	—
331	C₂³	—	2	2	no	—
332	C₂³	—	2	2	no	—
333	C₂³	—	2	2	no	—
334	C₂³	—	4	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
337	Q₈	—	2	4	no	—
338	Q₈	—	4	2	no	—
339	Q₈	—	4	4	no	—
340	Q₈	—	4	4	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	4	no	—
352	C₂×C₄	—	4	4	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
355	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
356	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
357	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
358	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
360	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
361	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
362	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
363	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
364	C₈	—	2	4	no	—
365	C₈	—	4	2	no	—
366	C₈	—	4	4	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	8	2	no	—
373	C₂²	—	8	2	no	—
374	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
375	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
376	C₂²	#520×C₆₇	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	4	no	—
379	C₄	—	2	2	no	—
380	C₄	—	2	2	no	—
381	C₄	—	2	2	no	—
382	C₄	—	4	4	no	—
383	C₄	—	4	2	no	—
384	C₄	#413×C₆₇	1	2	yes	—
385	C₄	#548×C₆₇	1	2	yes	—
386	C₄	#550×C₆₇	1	2	yes	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	8	2	no	—
390	C₂	C₆₇×#2341	1	1	yes	characteristic
391	C₂	C₂×#413×C₆₇	1	1	yes	characteristic
392	C₂	C₆₇×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1966	C₂	semidirect
5. C₆₇×#1801	C₂	semidirect
7. C₄×#101×C₆₇	C₂	semidirect
190. #22537	C₆₇	direct (N × H)
198. Q₈×M₁₆	C₂×C₆₇	semidirect
200. #1966	C₂×C₆₇	semidirect
201. #1801	C₂×C₆₇	semidirect
203. C₄×#101	C₂×C₆₇	semidirect
205. C₆₇	#22537	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×#556×C₆₇	2
Q₈×M₁₆×C₆₇	2
C₆₇×#1967	2
C₆₇×#1966	2
C₆₇×#1801	2
C₆₇×#1800	2
C₄×#101×C₆₇	2
C₄×#100×C₆₇	2
#22537	67

Subgroups — C67×#22537 #30701788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22537 #30701788