Subgroups — #30745934

Order 17152.

790 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863449	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
3	#22558092	C₂	1	4	yes	maximal, complemented
4	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	1	no	—
6	#16863449	—	2	2	no	—
7	#16863449	C₄	1	4	yes	complemented
8	#16863420	—	2	2	no	—
9	#16863420	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
14	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₂²×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	#384×C₆₇	—	2	1	no	—
20	#384×C₆₇	—	2	1	no	—
21	#384×C₆₇	C₄	1	2	yes	—
22	#384×C₆₇	C₂²	1	2	yes	—
23	C₂×#11761612	—	2	4	no	—
24	C₂×#11761612	—	4	2	no	—
25	C₂×#11761612	—	4	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	complemented
32	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	complemented
33	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
34	C₄×C₆₇⋊C₈	—	2	4	no	—
35	C₂³×C₄×C₆₇	—	2	2	no	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	D₄	1	2	yes	—
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
57	#11761612	—	4	8	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	2	4	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	—	4	2	no	—
62	C₂×C₆₇⋊C₈	—	4	2	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	—	4	4	no	—
71	C₂²×C₄×C₆₇	—	4	4	no	—
72	C₂²×C₄×C₆₇	—	4	4	no	—
73	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
82	C₂²×C₄×C₆₇	D₈	1	2	yes	—
83	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
84	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
85	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄⋊C₄×C₆₇	—	4	4	no	—
90	C₄⋊C₄×C₆₇	—	4	4	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₄⋊C₄×C₆₇	—	4	4	no	—
93	C₄⋊C₄×C₆₇	—	4	2	no	—
94	C₄⋊C₄×C₆₇	—	4	2	no	—
95	C₄⋊C₄×C₆₇	—	4	4	no	—
96	C₄²×C₆₇	C₄²	1	2	yes	—
97	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
100	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
101	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
102	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
106	C₂³×C₆₇	—	2	1	no	—
107	C₂³×C₆₇	—	2	1	no	—
108	C₂³×C₆₇	—	2	1	no	—
109	C₂³×C₆₇	—	2	1	no	—
110	C₂³×C₆₇	#95	1	2	yes	—
111	C₂³×C₆₇	#102	1	2	yes	—
112	C₂³×C₆₇	#103	1	2	yes	—
113	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	4	no	—
121	C₂×C₄×C₆₇	—	2	4	no	—
122	C₂×C₄×C₆₇	—	4	4	no	—
123	C₂×C₄×C₆₇	—	4	4	no	—
124	C₂×C₄×C₆₇	—	4	4	no	—
125	C₂×C₄×C₆₇	—	8	8	no	—
126	C₂×C₄×C₆₇	#95	1	2	yes	—
127	C₂×C₄×C₆₇	#95	1	2	yes	—
128	C₂×C₄×C₆₇	#95	1	2	yes	—
129	C₂×C₄×C₆₇	#100	1	2	yes	—
130	C₂×C₄×C₆₇	#101	1	2	yes	—
131	C₂×C₄×C₆₇	#102	1	4	yes	—
132	C₂×C₄×C₆₇	#102	1	2	yes	—
133	C₂×C₄×C₆₇	#103	1	4	yes	—
134	C₂×C₄×C₆₇	#103	1	2	yes	—
135	C₂×C₄×C₆₇	#106	1	4	yes	—
136	C₂×C₄×C₆₇	#107	1	4	yes	—
137	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
147	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₆₇	#137	1	2	yes	—
149	C₆₇⋊C₈	—	4	4	no	—
150	C₆₇⋊C₈	—	4	4	no	—
151	C₂²×C₆₇	—	2	2	no	—
152	C₂²×C₆₇	—	2	2	no	—
153	C₂²×C₆₇	—	2	2	no	—
154	C₂²×C₆₇	—	2	2	no	—
155	C₂²×C₆₇	—	2	2	no	—
156	C₂²×C₆₇	—	2	2	no	—
157	C₂²×C₆₇	#340	1	4	yes	—
158	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
159	C₂²×C₆₇	C₂×#100	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
161	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
162	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
163	C₂²×C₆₇	#418	1	1	yes	characteristic
164	C₂²×C₆₇	#419	1	1	yes	characteristic
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	1	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	8	4	no	—
169	C₄×C₆₇	#340	1	4	yes	—
170	C₄×C₆₇	#343	1	4	yes	—
171	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
172	C₄×C₆₇	C₂×#103	1	1	yes	characteristic
173	C₄×C₆₇	#417	1	2	yes	—
174	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
175	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
176	C₄×C₆₇	#428	1	2	yes	—
177	#8043	—	67	1	no	maximal, Sylow
178	C₂×C₆₇	—	2	4	no	—
179	C₂×C₆₇	#1545	1	1	yes	characteristic
180	C₂×C₆₇	#1546	1	1	yes	characteristic
181	C₂×C₆₇	#1548	1	1	yes	characteristic
182	C₂×C₆₇	#1549	1	1	yes	characteristic
183	C₂×C₆₇	C₂×#340	1	1	yes	characteristic
184	C₂×C₆₇	#1736	1	1	yes	characteristic
185	C₂×C₆₇	C₂×#343	1	1	yes	characteristic
186	C₂×#431	—	67	1	no	—
187	C₂×#423	—	67	1	no	—
188	C₂×#384	D₆₇	1	1	yes	characteristic
189	#1296	—	67	4	no	—
190	C₆₇	#8043	1	1	yes	characteristic, complemented, Sylow
191	C₂²×M₁₆	—	134	1	no	—
192	C₂²×C₄⋊C₄	—	2	1	no	—
193	C₂²×C₄⋊C₄	—	2	1	no	—
194	C₂²×C₄⋊C₄	Dic₆₇	1	2	yes	—
195	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
196	#431	—	67	4	no	—
197	#431	—	134	2	no	—
198	#423	—	67	4	no	—
199	#423	—	134	2	no	—
200	C₂×C₄⋊C₈	—	67	2	no	—
201	C₂×C₄⋊C₈	—	67	2	no	—
202	C₂×C₄⋊C₈	—	134	1	no	—
203	C₂×C₂²⋊C₈	—	134	1	no	—
204	C₂×C₄×C₈	—	134	1	no	—
205	#384	—	2	1	no	—
206	#384	—	2	1	no	—
207	#384	Dic₆₇	1	2	yes	—
208	#384	C₂×D₆₇	1	2	yes	—
209	C₂³×C₄	—	2	2	no	—
210	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
211	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
212	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂×M₁₆	—	268	2	no	—
214	C₂×M₁₆	—	268	2	no	—
215	C₂×M₁₆	—	134	4	no	—
216	C₂²×C₈	—	134	2	no	—
217	C₂²×C₈	—	134	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	4	4	no	—
224	C₂×C₄⋊C₄	—	4	2	no	—
225	C₂×C₄⋊C₄	—	4	2	no	—
226	C₂×C₄⋊C₄	—	4	4	no	—
227	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
228	C₂×C₄²	Dic₁₃₄	1	2	yes	—
229	C₂×C₄²	C₄×D₆₇	1	2	yes	—
230	C₂×C₄²	C₄×D₆₇	1	2	yes	—
231	C₂×C₄²	D₂₆₈	1	2	yes	—
232	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
233	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
236	C₄⋊C₈	—	67	4	no	—
237	C₄⋊C₈	—	67	4	no	—
238	C₄⋊C₈	—	134	4	no	—
239	C₄⋊C₈	—	134	2	no	—
240	C₄⋊C₈	—	134	2	no	—
241	C₂²⋊C₈	—	134	4	no	—
242	C₄×C₈	—	134	4	no	—
243	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	4	4	no	—
252	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
253	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
254	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
255	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
258	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
259	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
260	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
261	C₂²×C₄	#11761613	1	2	yes	—
262	C₂²×C₄	#11761615	1	2	yes	—
263	C₂²×C₄	#11761616	1	2	yes	—
264	C₂²×C₄	#11761618	1	2	yes	—
265	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
268	M₁₆	—	268	8	no	—
269	C₂×C₈	—	268	2	no	—
270	C₂×C₈	—	268	2	no	—
271	C₂×C₈	—	268	2	no	—
272	C₂×C₈	—	268	2	no	—
273	C₂×C₈	—	134	4	no	—
274	C₂×C₈	—	134	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	4	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄²	C₄×Dic₆₇	1	2	yes	—
283	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
284	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
285	C₄²	Dic₆₇⋊C₄	1	2	yes	—
286	C₄²	Dic₆₇⋊C₄	1	2	yes	—
287	C₄²	C₄⋊Dic₆₇	1	2	yes	—
288	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
289	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
290	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
291	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	C₆₇⋊#95	1	2	yes	—
297	C₂³	#14523964	1	2	yes	—
298	C₂³	#14523967	1	2	yes	—
299	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	8	8	no	—
312	C₂×C₄	C₆₇⋊#95	1	2	yes	—
313	C₂×C₄	C₆₇⋊#95	1	2	yes	—
314	C₂×C₄	C₆₇⋊#95	1	2	yes	—
315	C₂×C₄	#14523960	1	2	yes	—
316	C₂×C₄	#14523962	1	2	yes	—
317	C₂×C₄	#14523963	1	4	yes	—
318	C₂×C₄	#14523964	1	2	yes	—
319	C₂×C₄	#14523966	1	4	yes	—
320	C₂×C₄	#14523967	1	2	yes	—
321	C₂×C₄	#14523974	1	4	yes	—
322	C₂×C₄	#14523976	1	4	yes	—
323	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
324	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
325	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
327	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
328	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
329	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
330	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
331	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
332	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
333	C₂×C₄	#14524069	1	2	yes	—
334	C₂×C₄	#14524074	1	2	yes	—
335	C₈	—	268	4	no	—
336	C₈	—	268	4	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	#16863158	1	4	yes	—
344	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
345	C₂²	C₂×#14523960	1	1	yes	characteristic
346	C₂²	C₂×#14523962	1	1	yes	characteristic
347	C₂²	C₂×#14523964	1	1	yes	characteristic
348	C₂²	C₂×#14523967	1	1	yes	characteristic
349	C₂²	#16863401	1	1	yes	characteristic
350	C₂²	#16863405	1	1	yes	characteristic
351	C₄	—	2	2	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	8	4	no	—
355	C₄	#16863158	1	4	yes	—
356	C₄	#16863165	1	4	yes	—
357	C₄	C₂×#14523963	1	1	yes	characteristic
358	C₄	C₂×#14523966	1	1	yes	characteristic
359	C₄	#16863395	1	2	yes	—
360	C₄	C₂×#14523974	1	1	yes	characteristic
361	C₄	C₂×#14523976	1	1	yes	characteristic
362	C₄	#16863436	1	2	yes	—
363	C₂	—	2	4	no	—
364	C₂	#22558877	1	1	yes	characteristic
365	C₂	#22558881	1	1	yes	characteristic
366	C₂	#22558890	1	1	yes	characteristic
367	C₂	#22558894	1	1	yes	characteristic
368	C₂	C₂×#16863158	1	1	yes	characteristic
369	C₂	#22559928	1	1	yes	characteristic
370	C₂	C₂×#16863165	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#384×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#384	D₆₇
hypocenter	C₆₇	#8043
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
3. #22558092	C₂	semidirect
7. #16863449	C₄	semidirect
9. #16863420	C₄	semidirect
31. C₄⋊(C₆₇⋊C₈)	C₂×C₄	semidirect
32. C₄⋊(C₆₇⋊C₈)	C₂×C₄	semidirect
190. C₆₇	#8043	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863449	2
C₂×#16863420	2
#22558092	2
C₂×#384×C₆₇	2
#8043	67