Subgroups — #30745540

Order 17152.

660 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
3	#22558060	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₄	1	2	yes	—
6	#16863453	—	2	1	no	—
7	#16863453	C₄	1	2	yes	—
8	#16863449	—	2	1	no	—
9	#16863449	C₂²	1	2	yes	—
10	#16863420	—	2	1	no	—
11	#16863420	—	2	1	no	—
12	#16863420	—	2	1	no	—
13	#16863420	—	2	1	no	—
14	#16863420	—	2	1	no	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	#16863360	C₄	1	2	yes	—
18	C₄×#11761612	—	2	1	no	—
19	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₆₇	—	2	1	no	—
22	C₄×M₁₆×C₆₇	—	2	1	no	—
23	C₄×M₁₆×C₆₇	C₄	1	2	yes	—
24	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
25	C₂×#97×C₆₇	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	—	2	1	no	—
27	C₂×#11761612	—	2	2	no	—
28	C₂×#11761612	—	2	2	no	—
29	C₂×#11761612	—	2	2	no	—
30	C₂×#11761612	—	4	4	no	—
31	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
39	C₂²⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
40	#14523954	—	2	2	no	—
41	C₄×C₆₇⋊C₈	—	2	2	no	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	—	2	2	no	—
46	C₂×M₁₆×C₆₇	—	2	8	no	—
47	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#97×C₆₇	—	2	2	no	—
62	#97×C₆₇	—	2	2	no	—
63	C₄×C₈×C₆₇	—	2	2	no	—
64	C₄×C₈×C₆₇	—	2	2	no	—
65	#11761612	—	4	4	no	—
66	#11761612	—	4	4	no	—
67	#11761612	—	4	4	no	—
68	C₂×C₆₇⋊C₈	—	2	4	no	—
69	C₂×C₆₇⋊C₈	—	2	4	no	—
70	C₂×C₆₇⋊C₈	—	4	4	no	—
71	C₂⁴×C₆₇	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	M₁₆×C₆₇	—	2	8	no	—
95	M₁₆×C₆₇	—	2	4	no	—
96	M₁₆×C₆₇	M₁₆	1	8	yes	complemented
97	C₂×C₈×C₆₇	—	2	8	no	—
98	C₂×C₈×C₆₇	—	2	4	no	—
99	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
102	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
105	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
115	C₂³×C₆₇	—	2	2	no	—
116	C₂³×C₆₇	—	2	2	no	—
117	C₂³×C₆₇	—	4	1	no	—
118	C₂³×C₆₇	#95	1	1	yes	characteristic
119	C₂³×C₆₇	#95	1	1	yes	characteristic
120	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
142	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
143	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
144	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
145	C₂×C₄×C₆₇	#100	1	2	yes	—
146	C₂×C₄×C₆₇	#101	1	2	yes	—
147	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
154	C₈×C₆₇	—	2	8	no	—
155	C₆₇⋊C₈	—	4	4	no	—
156	C₆₇⋊C₈	—	4	4	no	—
157	C₂²×C₆₇	—	2	1	no	—
158	C₂²×C₆₇	—	2	1	no	—
159	C₂²×C₆₇	—	2	1	no	—
160	C₂²×C₆₇	—	2	1	no	—
161	C₂²×C₆₇	—	2	1	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	2	1	no	—
164	C₂²×C₆₇	—	4	4	no	—
165	C₂²×C₆₇	#343	1	2	yes	—
166	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
167	C₂²×C₆₇	C₄×M₁₆	1	2	yes	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	2	1	no	—
170	C₄×C₆₇	—	2	1	no	—
171	C₄×C₆₇	—	2	1	no	—
172	C₄×C₆₇	—	2	1	no	—
173	C₄×C₆₇	—	2	1	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	#343	1	2	yes	—
176	C₄×C₆₇	#344	1	2	yes	—
177	C₄×C₆₇	#407	1	2	yes	—
178	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
179	C₄×C₆₇	C₂×#101	1	1	yes	characteristic
180	C₄×C₆₇	#423	1	2	yes	—
181	#7938	—	67	1	no	maximal, Sylow
182	C₂×C₆₇	—	2	2	no	—
183	C₂×C₆₇	—	2	2	no	—
184	C₂×C₆₇	—	4	1	no	—
185	C₂×C₆₇	#1728	1	1	yes	characteristic
186	C₂×C₆₇	C₂×#343	1	1	yes	characteristic
187	C₂×C₆₇	C₂×#344	1	1	yes	characteristic
188	#2920	—	67	1	no	—
189	C₂×#423	—	67	1	no	—
190	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
191	#1283	—	67	4	no	—
192	C₆₇	#7938	1	1	yes	characteristic, complemented, Sylow
193	C₂²×M₁₆	—	67	2	no	—
194	C₂²×M₁₆	Dic₆₇	1	2	yes	—
195	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
196	#432	—	67	2	no	—
197	#432	—	134	1	no	—
198	#431	—	67	2	no	—
199	#431	—	134	1	no	—
200	#423	—	134	1	no	—
201	#423	—	134	1	no	—
202	#423	—	134	1	no	—
203	#423	—	134	1	no	—
204	#423	—	134	1	no	—
205	C₂×C₄⋊C₈	—	67	2	no	—
206	C₂×C₄⋊C₈	—	67	2	no	—
207	#407	—	67	2	no	—
208	C₄×M₁₆	—	2	1	no	—
209	C₄×M₁₆	—	2	1	no	—
210	C₄×M₁₆	—	134	1	no	—
211	C₄×M₁₆	Dic₆₇	1	2	yes	—
212	C₄×M₁₆	C₂×D₆₇	1	2	yes	—
213	C₂×#97	—	2	1	no	—
214	C₂×C₄×C₈	—	2	1	no	—
215	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
216	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
217	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
218	C₂×M₁₆	—	2	2	no	—
219	C₂×M₁₆	—	2	8	no	—
220	C₂×M₁₆	—	268	4	no	—
221	C₂×M₁₆	—	134	2	no	—
222	C₂×M₁₆	—	134	2	no	—
223	C₂×M₁₆	—	134	2	no	—
224	C₂×M₁₆	C₂×Dic₆₇	1	4	yes	—
225	C₂²×C₈	—	2	2	no	—
226	C₂²×C₈	—	67	4	no	—
227	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
228	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
229	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
230	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
231	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
232	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
235	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
237	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
239	C₄⋊C₈	—	134	2	no	—
240	C₄⋊C₈	—	134	2	no	—
241	C₄⋊C₈	—	134	2	no	—
242	C₄⋊C₈	—	134	2	no	—
243	C₄⋊C₈	—	134	2	no	—
244	C₄⋊C₈	—	134	2	no	—
245	C₂²⋊C₈	—	67	4	no	—
246	C₂²⋊C₈	—	134	2	no	—
247	#97	—	2	2	no	—
248	#97	—	2	2	no	—
249	#97	—	134	2	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	C₄×C₈	—	134	2	no	—
253	C₂⁴	C₄×Dic₆₇	1	1	yes	characteristic
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
259	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
260	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
261	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
262	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
263	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
264	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
267	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
268	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
269	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
270	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
271	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
272	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
276	M₁₆	—	2	8	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	268	4	no	—
279	M₁₆	—	268	4	no	—
280	M₁₆	—	268	4	no	—
281	M₁₆	#11761612	1	8	yes	complemented
282	C₂×C₈	—	2	8	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	268	4	no	—
285	C₂×C₈	—	134	4	no	—
286	C₂×C₈	—	134	4	no	—
287	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
288	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
289	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
290	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
291	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
292	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
293	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
294	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
295	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
296	C₄²	Dic₆₇⋊C₄	1	1	yes	characteristic
297	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
298	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
301	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
302	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	C₆₇⋊#95	1	1	yes	characteristic
307	C₂³	C₆₇⋊#95	1	1	yes	characteristic
308	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
328	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
329	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
330	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
331	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
332	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
333	C₂×C₄	#14523959	1	2	yes	—
334	C₂×C₄	#14523961	1	2	yes	—
335	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
336	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
337	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
338	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
339	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂×#11761612	1	4	yes	—
342	C₈	—	2	8	no	—
343	C₈	—	268	4	no	—
344	C₈	—	268	4	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	4	4	no	—
353	C₂²	#16863164	1	2	yes	—
354	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
355	C₂²	C₄×#11761612	1	2	yes	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	#16863164	1	2	yes	—
364	C₄	#16863168	1	2	yes	—
365	C₄	#16863360	1	2	yes	—
366	C₄	C₂×#14523959	1	1	yes	characteristic
367	C₄	C₂×#14523961	1	1	yes	characteristic
368	C₄	#16863420	1	2	yes	—
369	C₂	—	2	2	no	—
370	C₂	—	2	2	no	—
371	C₂	—	4	1	no	—
372	C₂	#22559903	1	1	yes	characteristic
373	C₂	C₂×#16863164	1	1	yes	characteristic
374	C₂	C₂×#16863168	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄
commutator	C₂³×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	#7938
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22558060	C₂	semidirect
96. M₁₆×C₆₇	M₁₆	semidirect
192. C₆₇	#7938	semidirect
281. M₁₆	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
C₂×#16863420	2
#22558060	2
C₂×C₄×M₁₆×C₆₇	2
#7938	67