Subgroups — C₆₇×#22515 #30701766

Order 17152.

1014 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1998	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1997	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1800	C₂	1	2	yes	maximal
6	C₆₇×#1800	C₂	1	1	yes	maximal, characteristic
7	C₄×#100×C₆₇	C₂	1	2	yes	maximal
8	C₄×#100×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#554×C₆₇	—	2	2	no	—
10	#554×C₆₇	—	2	4	no	—
11	#554×C₆₇	—	2	2	no	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₆₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊Q₈×C₆₇	C₄	1	4	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
20	#446×C₆₇	—	2	4	no	—
21	#446×C₆₇	—	2	2	no	—
22	C₈×Q₈×C₆₇	—	2	2	no	—
23	#423×C₆₇	C₂²	1	2	yes	—
24	#423×C₆₇	C₂²	1	4	yes	—
25	C₂×#100×C₆₇	C₂²	1	2	yes	—
26	C₂×#100×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#100×C₆₇	C₂²	1	1	yes	characteristic
28	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
29	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
30	#343×C₆₇	C₂²	1	4	yes	—
31	#343×C₆₇	C₂²	1	4	yes	—
32	#343×C₆₇	C₂²	1	2	yes	—
33	#343×C₆₇	C₂²	1	2	yes	—
34	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
40	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
41	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	C₄⋊Q₈×C₆₇	—	2	2	no	—
44	#125×C₆₇	—	2	4	no	—
45	#125×C₆₇	—	2	4	no	—
46	C₂²⋊Q₈×C₆₇	—	2	4	no	—
47	C₂²⋊Q₈×C₆₇	—	2	4	no	—
48	C₂²⋊Q₈×C₆₇	—	2	8	no	—
49	C₂²⋊Q₈×C₆₇	—	2	8	no	—
50	C₄×Q₈×C₆₇	D₄	1	2	yes	—
51	C₄×Q₈×C₆₇	D₄	1	4	yes	—
52	C₄×Q₈×C₆₇	D₄	1	2	yes	—
53	C₄×D₄×C₆₇	—	2	4	no	—
54	C₄×D₄×C₆₇	—	2	4	no	—
55	C₄×D₄×C₆₇	—	2	2	no	—
56	C₄×D₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
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	4	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₆₇	—	2	4	no	—
72	C₄⋊C₈×C₆₇	—	2	2	no	—
73	C₄⋊C₈×C₆₇	—	2	4	no	—
74	C₄⋊C₈×C₆₇	—	2	2	no	—
75	#100×C₆₇	—	2	4	no	—
76	#100×C₆₇	—	2	2	no	—
77	#100×C₆₇	Q₈	1	4	yes	complemented
78	#97×C₆₇	—	2	4	no	—
79	#97×C₆₇	—	2	2	no	—
80	C₄×C₈×C₆₇	—	2	4	no	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×Q₈×C₆₇	—	4	2	no	—
85	C₂×Q₈×C₆₇	—	8	2	no	—
86	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
87	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₂×D₄×C₆₇	—	2	2	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂×D₄×C₆₇	#41	1	2	yes	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	4	4	no	—
97	C₂²×C₄×C₆₇	—	8	4	no	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	4	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	4	no	—
107	M₁₆×C₆₇	—	2	2	no	—
108	M₁₆×C₆₇	—	2	4	no	—
109	M₁₆×C₆₇	—	2	2	no	—
110	M₁₆×C₆₇	C₂×Q₈	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	4	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	8	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
122	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
123	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
124	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
125	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
126	C₂²⋊C₄×C₆₇	—	2	4	no	—
127	C₂²⋊C₄×C₆₇	—	2	8	no	—
128	C₄²×C₆₇	C₂×D₄	1	4	yes	—
129	C₄²×C₆₇	C₂×D₄	1	4	yes	—
130	C₄²×C₆₇	C₂×D₄	1	2	yes	—
131	C₄²×C₆₇	C₂×D₄	1	2	yes	—
132	C₂³×C₆₇	—	2	2	no	—
133	C₂³×C₆₇	—	2	2	no	—
134	C₂³×C₆₇	—	4	2	no	—
135	C₂³×C₆₇	—	4	2	no	—
136	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
137	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
138	Q₈×C₆₇	—	2	4	no	—
139	Q₈×C₆₇	—	4	2	no	—
140	Q₈×C₆₇	—	8	4	no	—
141	D₄×C₆₇	—	2	4	no	—
142	D₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	4	4	no	—
150	C₂×C₄×C₆₇	—	4	4	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	4	no	—
153	C₂×C₄×C₆₇	—	8	4	no	—
154	C₂×C₄×C₆₇	—	8	4	no	—
155	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
156	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
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	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
164	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
165	C₈×C₆₇	—	2	4	no	—
166	C₈×C₆₇	—	4	2	no	—
167	C₈×C₆₇	—	4	4	no	—
168	C₂²×C₆₇	—	2	1	no	—
169	C₂²×C₆₇	—	2	1	no	—
170	C₂²×C₆₇	—	2	4	no	—
171	C₂²×C₆₇	—	4	4	no	—
172	C₂²×C₆₇	—	4	4	no	—
173	C₂²×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	—	4	2	no	—
175	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
176	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
177	C₂²×C₆₇	#519	1	1	yes	characteristic
178	C₄×C₆₇	—	2	1	no	—
179	C₄×C₆₇	—	2	4	no	—
180	C₄×C₆₇	—	2	2	no	—
181	C₄×C₆₇	—	2	2	no	—
182	C₄×C₆₇	—	2	2	no	—
183	C₄×C₆₇	—	4	2	no	—
184	C₄×C₆₇	—	4	4	no	—
185	C₄×C₆₇	#413	1	2	yes	—
186	C₄×C₆₇	#548	1	2	yes	—
187	C₄×C₆₇	D₄×Q₈	1	2	yes	—
188	#22515	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂×C₆₇	—	2	1	no	—
190	C₂×C₆₇	—	2	1	no	—
191	C₂×C₆₇	—	4	4	no	—
192	C₂×C₆₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₆₇	C₂×#413	1	1	yes	characteristic
194	C₂×C₆₇	#2888	1	1	yes	characteristic
195	C₂×#554	C₂×C₆₇	1	1	yes	characteristic
196	Q₈×M₁₆	C₂×C₆₇	1	2	yes	complemented
197	#1998	C₂×C₆₇	1	4	yes	complemented
198	#1997	C₂×C₆₇	1	2	yes	complemented
199	#1800	C₂×C₆₇	1	2	yes	—
200	#1800	C₂×C₆₇	1	1	yes	characteristic
201	C₄×#100	C₂×C₆₇	1	2	yes	—
202	C₄×#100	C₂×C₆₇	1	1	yes	characteristic
203	C₆₇	#22515	1	1	yes	characteristic, direct product factor, Sylow
204	#554	—	2	2	no	—
205	#554	—	2	4	no	—
206	#554	—	2	2	no	—
207	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×#125	C₂²×C₆₇	1	2	yes	—
209	C₂×C₂²⋊Q₈	C₄×C₆₇	1	4	yes	—
210	C₂×C₂²⋊Q₈	C₄×C₆₇	1	2	yes	—
211	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
212	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
213	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
214	C₂²×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
215	#446	—	2	4	no	—
216	#446	—	2	2	no	—
217	C₈×Q₈	—	2	2	no	—
218	#423	C₂²×C₆₇	1	2	yes	—
219	#423	C₂²×C₆₇	1	4	yes	—
220	C₂×#100	C₂²×C₆₇	1	1	yes	characteristic
221	C₂×#100	C₂²×C₆₇	1	2	yes	—
222	C₂×#100	C₂²×C₆₇	1	1	yes	characteristic
223	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
224	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
225	#343	C₂²×C₆₇	1	4	yes	—
226	#343	C₂²×C₆₇	1	2	yes	—
227	#343	C₂²×C₆₇	1	4	yes	—
228	#343	C₂²×C₆₇	1	2	yes	—
229	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
230	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
231	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
232	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
233	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
234	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
235	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
236	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
237	C₄⋊Q₈	—	2	2	no	—
238	C₄⋊Q₈	—	2	2	no	—
239	#125	—	2	4	no	—
240	#125	—	2	4	no	—
241	C₂²⋊Q₈	—	2	4	no	—
242	C₂²⋊Q₈	—	2	4	no	—
243	C₂²⋊Q₈	—	2	8	no	—
244	C₂²⋊Q₈	—	2	8	no	—
245	C₄×Q₈	D₄×C₆₇	1	2	yes	—
246	C₄×Q₈	D₄×C₆₇	1	4	yes	—
247	C₄×Q₈	D₄×C₆₇	1	2	yes	—
248	C₄×D₄	—	2	4	no	—
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	—	2	4	no	—
251	C₄×D₄	—	2	2	no	—
252	C₂×C₄⋊C₄	—	2	4	no	—
253	C₂×C₄⋊C₄	—	2	2	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
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	1	yes	characteristic
259	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
260	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
262	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
263	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
264	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
265	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
266	C₄⋊C₈	—	2	4	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	—	2	4	no	—
269	C₄⋊C₈	—	2	2	no	—
270	#100	—	2	2	no	—
271	#100	—	2	4	no	—
272	#100	Q₈×C₆₇	1	4	yes	complemented
273	#97	—	2	4	no	—
274	#97	—	2	2	no	—
275	C₄×C₈	—	2	4	no	—
276	C₄×C₈	—	2	2	no	—
277	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
278	C₂×Q₈	—	2	4	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	8	2	no	—
281	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
282	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
286	C₂×D₄	#41×C₆₇	1	2	yes	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	4	no	—
290	C₂²×C₄	—	2	4	no	—
291	C₂²×C₄	—	4	4	no	—
292	C₂²×C₄	—	8	4	no	—
293	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
295	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
296	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
297	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
299	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	4	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	4	no	—
305	M₁₆	C₂×Q₈×C₆₇	1	4	yes	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₄⋊C₄	—	2	4	no	—
310	C₄⋊C₄	—	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	2	no	—
315	C₄⋊C₄	—	2	4	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	8	no	—
323	C₄²	C₂×D₄×C₆₇	1	4	yes	—
324	C₄²	C₂×D₄×C₆₇	1	2	yes	—
325	C₄²	C₂×D₄×C₆₇	1	4	yes	—
326	C₄²	C₂×D₄×C₆₇	1	2	yes	—
327	C₂³	—	2	2	no	—
328	C₂³	—	2	2	no	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	2	no	—
331	C₂³	C₄×Q₈×C₆₇	1	4	yes	—
332	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
333	Q₈	—	2	4	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	8	4	no	—
336	D₄	—	2	4	no	—
337	D₄	—	2	4	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	4	no	—
345	C₂×C₄	—	4	4	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	8	4	no	—
349	C₂×C₄	—	8	4	no	—
350	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
351	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
352	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
353	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
354	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
357	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
359	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
360	C₈	—	2	4	no	—
361	C₈	—	4	4	no	—
362	C₈	—	4	2	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	4	no	—
366	C₂²	—	4	4	no	—
367	C₂²	—	4	4	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
371	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
372	C₂²	#519×C₆₇	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	4	no	—
375	C₄	—	2	2	no	—
376	C₄	—	2	2	no	—
377	C₄	—	2	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	4	4	no	—
380	C₄	#413×C₆₇	1	2	yes	—
381	C₄	#548×C₆₇	1	2	yes	—
382	C₄	D₄×Q₈×C₆₇	1	2	yes	—
383	C₂	—	2	1	no	—
384	C₂	—	2	1	no	—
385	C₂	—	4	4	no	—
386	C₂	Q₈×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
387	C₂	C₂×#413×C₆₇	1	1	yes	characteristic
388	C₂	C₆₇×#2888	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
3. C₆₇×#1998	C₂	semidirect
4. C₆₇×#1997	C₂	semidirect
77. #100×C₆₇	Q₈	semidirect
188. #22515	C₆₇	direct (N × H)
196. Q₈×M₁₆	C₂×C₆₇	semidirect
197. #1998	C₂×C₆₇	semidirect
198. #1997	C₂×C₆₇	semidirect
203. C₆₇	#22515	direct (N × H)
272. #100	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#554×C₆₇	2
Q₈×M₁₆×C₆₇	2
C₆₇×#1998	2
C₆₇×#1997	2
C₆₇×#1800	2
C₄×#100×C₆₇	2
#22515	67

Subgroups — C67×#22515 #30701766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22515 #30701766