Subgroups — C₂×#22559053 #30825336

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16863474	C₂	1	1	yes	maximal, characteristic
3	#22559053	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863147	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₂×#16863118	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11761612	—	2	1	no	—
10	Q₈×C₆₇⋊C₈	—	2	2	no	—
11	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	complemented
12	#16863474	—	2	2	no	—
13	#16863474	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
18	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
19	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	#16863147	C₂²	1	4	yes	complemented
22	Q₈⋊(C₆₇⋊C₈)	C₂²	1	4	yes	complemented
23	D₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
24	#16863118	C₂²	1	4	yes	complemented
25	C₂²×SD₁₆×C₆₇	C₄	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₄×SD₁₆×C₆₇	C₄	1	4	yes	—
29	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	—
30	C₂×#106×C₆₇	C₄	1	1	yes	characteristic
31	C₂×#103×C₆₇	C₄	1	1	yes	characteristic
32	C₂×#102×C₆₇	C₄	1	1	yes	characteristic
33	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#11761612	—	2	4	no	—
35	C₂×#11761612	—	4	4	no	—
36	C₂²×C₆₇⋊C₈	—	2	1	no	—
37	C₂²×C₆₇⋊C₈	—	2	1	no	—
38	C₂²×C₆₇⋊C₈	—	2	1	no	—
39	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
41	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
44	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
45	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
46	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
47	C₄×C₆₇⋊C₈	—	2	4	no	—
48	C₄×C₆₇⋊C₈	—	2	1	no	—
49	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
50	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
51	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
52	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	—	2	1	no	—
55	C₂×SD₁₆×C₆₇	—	2	4	no	—
56	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
57	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₄×Q₈×C₆₇	—	2	1	no	—
60	C₄×Q₈×C₆₇	—	2	1	no	—
61	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
62	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₄×D₄×C₆₇	—	2	1	no	—
64	C₄×D₄×C₆₇	—	2	1	no	—
65	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₄×D₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₄²×C₆₇	—	2	1	no	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	#106×C₆₇	C₂×C₄	1	4	yes	—
74	#103×C₆₇	C₂×C₄	1	4	yes	—
75	#102×C₆₇	C₂×C₄	1	4	yes	—
76	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₆₇	C₂³	1	2	yes	—
78	#11761612	—	4	8	no	—
79	C₂×C₆₇⋊C₈	—	2	2	no	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₂×C₆₇⋊C₈	—	2	2	no	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	4	2	no	—
84	C₂×C₆₇⋊C₈	—	4	2	no	—
85	C₂×C₆₇⋊C₈	—	4	2	no	—
86	C₂×C₆₇⋊C₈	SD₁₆	1	4	yes	complemented
87	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
88	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
89	C₂⁴×C₆₇	—	2	1	no	—
90	C₂×Q₈×C₆₇	—	2	1	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	M₁₆	1	4	yes	complemented
93	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	—	2	2	no	—
96	C₂×D₄×C₆₇	—	2	2	no	—
97	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	4	2	no	—
104	C₂²×C₄×C₆₇	—	4	2	no	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
108	SD₁₆×C₆₇	—	2	8	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₆₇	—	2	2	no	—
114	C₄⋊C₄×C₆₇	—	2	1	no	—
115	C₄⋊C₄×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂²⋊C₄×C₆₇	—	2	2	no	—
120	C₂²⋊C₄×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	2	no	—
123	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₆₇	#41	1	1	yes	characteristic
126	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	4	2	no	—
129	C₂³×C₆₇	—	4	2	no	—
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
132	Q₈×C₆₇	—	2	2	no	—
133	Q₈×C₆₇	—	2	4	no	—
134	Q₈×C₆₇	C₂×M₁₆	1	4	yes	complemented
135	D₄×C₆₇	—	2	2	no	—
136	D₄×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	4	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	4	no	—
148	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
152	C₂×C₄×C₆₇	#131	1	2	yes	—
153	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₆₇	#135	1	2	yes	—
155	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
158	C₈×C₆₇	—	2	2	no	—
159	C₈×C₆₇	—	2	2	no	—
160	C₆₇⋊C₈	—	2	2	no	—
161	C₆₇⋊C₈	—	4	2	no	—
162	C₆₇⋊C₈	—	4	2	no	—
163	C₆₇⋊C₈	—	4	2	no	—
164	C₆₇⋊C₈	C₂×SD₁₆	1	4	yes	complemented
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	—	8	4	no	—
169	C₂²×C₆₇	#435	1	2	yes	—
170	C₂²×C₆₇	C₄×SD₁₆	1	2	yes	—
171	C₂²×C₆₇	#444	1	2	yes	—
172	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₆₇	—	2	4	no	—
174	C₄×C₆₇	—	2	1	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	4	2	no	—
177	C₄×C₆₇	—	4	2	no	—
178	C₄×C₆₇	#435	1	2	yes	—
179	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
180	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
182	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
183	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
184	C₂×#1583	—	67	1	no	maximal, Sylow
185	C₂×C₆₇	—	8	2	no	—
186	C₂×C₆₇	#1583	1	4	yes	complemented
187	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
188	C₂×C₆₇	C₂×C₄×SD₁₆	1	1	yes	characteristic
189	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
190	C₂×C₈×Q₈	—	67	1	no	—
191	C₂×C₄×SD₁₆	D₆₇	1	1	yes	characteristic
192	C₂×#436	—	67	1	no	—
193	#1583	—	67	8	no	—
194	C₂×#334	—	67	1	no	—
195	C₂×Q₈⋊C₈	—	67	1	no	—
196	C₂×D₄⋊C₈	—	67	1	no	—
197	C₂×#322	—	67	1	no	—
198	C₆₇	C₂×#1583	1	1	yes	characteristic, complemented, Sylow
199	C₂²×SD₁₆	Dic₆₇	1	1	yes	characteristic
200	C₂²×M₁₆	—	134	1	no	—
201	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
202	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
203	C₈×Q₈	—	67	4	no	—
204	C₈×Q₈	—	134	2	no	—
205	C₄×SD₁₆	Dic₆₇	1	4	yes	—
206	C₄×SD₁₆	C₂×D₆₇	1	4	yes	—
207	#436	—	67	4	no	—
208	#436	—	134	2	no	—
209	C₂×#106	Dic₆₇	1	1	yes	characteristic
210	C₂×C₄⋊C₈	—	67	1	no	—
211	C₂×C₄⋊C₈	—	67	1	no	—
212	C₂×C₄⋊C₈	—	134	1	no	—
213	C₂×#103	Dic₆₇	1	1	yes	characteristic
214	C₂×#102	Dic₆₇	1	1	yes	characteristic
215	C₂×C₂²⋊C₈	—	134	1	no	—
216	C₂×C₄×C₈	—	67	1	no	—
217	C₂×C₄×C₈	—	67	1	no	—
218	C₂×C₄×C₈	—	134	1	no	—
219	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
220	#334	—	67	4	no	—
221	Q₈⋊C₈	—	67	4	no	—
222	D₄⋊C₈	—	67	4	no	—
223	#322	—	67	4	no	—
224	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
225	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
226	C₂³×C₄	—	2	1	no	—
227	C₂×SD₁₆	—	2	4	no	—
228	C₂×SD₁₆	C₂×Dic₆₇	1	4	yes	—
229	C₂×M₁₆	—	268	4	no	—
230	C₂×M₁₆	—	134	4	no	—
231	C₂²×C₈	—	67	1	no	—
232	C₂²×C₈	—	67	1	no	—
233	C₂²×C₈	—	134	1	no	—
234	C₂²×C₈	—	134	1	no	—
235	C₂²×C₈	—	134	1	no	—
236	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
237	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	C₂×Dic₆₇	1	2	yes	—
241	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
245	C₄×D₄	C₂²×D₆₇	1	2	yes	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
249	C₂×C₂²⋊C₄	—	2	1	no	—
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
252	#106	C₂×Dic₆₇	1	4	yes	—
253	C₄⋊C₈	—	67	2	no	—
254	C₄⋊C₈	—	67	2	no	—
255	C₄⋊C₈	—	134	4	no	—
256	C₄⋊C₈	—	134	1	no	—
257	C₄⋊C₈	—	134	1	no	—
258	#103	C₂×Dic₆₇	1	4	yes	—
259	#102	C₂×Dic₆₇	1	4	yes	—
260	C₂²⋊C₈	—	134	4	no	—
261	C₄×C₈	—	67	2	no	—
262	C₄×C₈	—	67	2	no	—
263	C₄×C₈	—	67	2	no	—
264	C₄×C₈	—	134	4	no	—
265	C₄×C₈	—	134	1	no	—
266	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
267	C₄×C₈	C₂²×D₆₇	1	2	yes	—
268	C₂⁴	—	2	1	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	#11761612	1	4	yes	complemented
272	C₂×Q₈	C₂²×Dic₆₇	1	2	yes	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
285	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
286	C₂²×C₄	#11761627	1	1	yes	characteristic
287	SD₁₆	—	2	8	no	—
288	M₁₆	—	268	8	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	268	2	no	—
292	C₂×C₈	—	268	2	no	—
293	C₂×C₈	—	268	2	no	—
294	C₂×C₈	—	67	4	no	—
295	C₂×C₈	—	67	2	no	—
296	C₂×C₈	—	67	2	no	—
297	C₂×C₈	—	134	2	no	—
298	C₂×C₈	—	134	2	no	—
299	C₂×C₈	—	134	2	no	—
300	C₂×C₈	—	134	2	no	—
301	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
302	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
308	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
314	C₄²	D₄×D₆₇	1	1	yes	characteristic
315	C₄²	#11761627	1	1	yes	characteristic
316	C₄²	C₂³×D₆₇	1	1	yes	characteristic
317	C₂³	—	2	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	4	2	no	—
320	C₂³	—	4	2	no	—
321	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
322	Q₈	—	2	2	no	—
323	Q₈	—	2	4	no	—
324	Q₈	C₂×#11761612	1	4	yes	complemented
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
339	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
340	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂×#11761612	1	2	yes	—
342	C₂×C₄	#14524050	1	2	yes	—
343	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
344	C₂×C₄	#14524064	1	2	yes	—
345	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	268	2	no	—
351	C₈	—	268	2	no	—
352	C₈	—	268	2	no	—
353	C₈	—	67	4	no	—
354	C₈	—	134	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	8	4	no	—
359	C₂²	#16863468	1	2	yes	—
360	C₂²	SD₁₆×Dic₆₇	1	2	yes	—
361	C₂²	#16863518	1	2	yes	—
362	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	4	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	#16863468	1	2	yes	—
369	C₄	SD₁₆×Dic₆₇	1	2	yes	—
370	C₄	C₂²×#11761612	1	1	yes	characteristic
371	C₄	C₂×#14524050	1	1	yes	characteristic
372	C₄	C₂×SD₁₆×D₆₇	1	1	yes	characteristic
373	C₄	C₂×#14524064	1	1	yes	characteristic
374	C₂	—	8	2	no	—
375	C₂	#22559053	1	4	yes	direct product factor
376	C₂	C₂×#16863468	1	1	yes	characteristic
377	C₂	C₂×SD₁₆×Dic₆₇	1	1	yes	characteristic
378	C₂	C₂×#16863518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₂×C₄×SD₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂²×D₄×C₆₇	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₂×C₄×SD₁₆	D₆₇
hypocenter	C₆₇	C₂×#1583
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
3. #22559053	C₂	direct (N × H)
4. C₂×#16863147	C₂	semidirect
5. C₂×Q₈⋊(C₆₇⋊C₈)	C₂	semidirect
7. C₂×#16863118	C₂	semidirect
11. Q₈×C₆₇⋊C₈	C₂²	semidirect
21. #16863147	C₂²	semidirect
22. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
24. #16863118	C₂²	semidirect
86. C₂×C₆₇⋊C₈	SD₁₆	semidirect
92. C₂×Q₈×C₆₇	M₁₆	semidirect
134. Q₈×C₆₇	C₂×M₁₆	semidirect
164. C₆₇⋊C₈	C₂×SD₁₆	semidirect
186. C₂×C₆₇	#1583	semidirect
198. C₆₇	C₂×#1583	semidirect
271. C₂×Q₈	#11761612	semidirect
324. Q₈	C₂×#11761612	semidirect
375. C₂	#22559053	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×Q₈×C₆₇⋊C₈	2
C₂×#16863474	2
#22559053	2
C₂×#16863147	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
C₂×D₄⋊(C₆₇⋊C₈)	2
C₂×#16863118	2
C₂×C₄×SD₁₆×C₆₇	2
C₂×#1583	67

Subgroups — C2×#22559053 #30825336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22559053 #30825336