Subgroups — C₂×C₈×D₃₇₆ #26185636

Order 12032.

910 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₁₈₈	C₂	1	2	yes	maximal, complemented
3	C₈×D₃₇₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15688876	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15688852	C₂	1	2	yes	maximal, complemented
6	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₃₇₆	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₄₇	—	2	2	no	—
9	C₂×C₄×D₁₈₈	C₂²	1	2	yes	—
10	C₄×D₃₇₆	C₄	1	4	yes	—
11	C₄×D₃₇₆	C₂²	1	4	yes	—
12	C₈×D₁₈₈	—	2	4	no	—
13	C₈×D₁₈₈	C₂²	1	8	yes	complemented
14	C₂×#13093958	C₄	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×#13093946	C₄	1	2	yes	—
17	C₂×(C₂×D₄₇)⋊C₈	—	2	2	no	—
18	#15688876	C₂²	1	4	yes	complemented
19	#15688852	C₂²	1	8	yes	complemented
20	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₈²×C₄₇	C₂²	1	4	yes	complemented
24	C₂²×D₁₈₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₄₇	—	2	2	no	—
26	C₂×D₃₇₆	C₈	1	8	yes	direct product factor
27	C₂×D₃₇₆	C₂×C₄	1	4	yes	—
28	C₂²×C₄₇⋊C₈	—	2	2	no	—
29	C₂×C₈×D₄₇	—	2	8	no	—
30	C₂×C₈×D₄₇	—	4	8	no	—
31	C₄×D₁₈₈	—	2	2	no	—
32	C₄×D₁₈₈	—	2	2	no	—
33	C₄×D₁₈₈	C₂×C₄	1	4	yes	—
34	C₄×D₁₈₈	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
36	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
37	#13093958	C₂×C₄	1	4	yes	—
38	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
39	C₄⋊(C₄₇⋊C₈)	C₂³	1	4	yes	—
40	#13093946	C₂×C₄	1	8	yes	—
41	(C₂×D₄₇)⋊C₈	—	2	8	no	—
42	C₂²×C₈×C₄₇	—	2	1	no	—
43	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₄₇	—	2	1	no	—
49	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₄₇	D₄	1	2	yes	—
51	C₄×C₈×C₄₇	C₂³	1	2	yes	—
52	C₄×C₈×C₄₇	C₂³	1	2	yes	—
53	C₄×C₈×C₄₇	C₂³	1	2	yes	—
54	C₂³×D₄₇	—	2	2	no	—
55	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
58	C₂²×Dic₄₇	—	2	2	no	—
59	C₂×D₁₈₈	—	2	2	no	—
60	C₂×D₁₈₈	—	2	4	no	—
61	C₂×D₁₈₈	C₂×C₈	1	8	yes	complemented
62	C₂×D₁₈₈	C₂²×C₄	1	4	yes	—
63	C₂×C₄×D₄₇	—	2	4	no	—
64	C₂×C₄×D₄₇	—	2	4	no	—
65	C₂×C₄×D₄₇	—	4	4	no	—
66	C₂×C₄×D₄₇	—	4	4	no	—
67	C₂×C₈×C₄₇	—	2	4	no	—
68	C₂×C₈×C₄₇	—	2	2	no	—
69	C₂×C₈×C₄₇	—	2	2	no	—
70	C₂×C₈×C₄₇	—	2	2	no	—
71	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	complemented
72	C₂×C₈×C₄₇	D₈	1	4	yes	complemented
73	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
77	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₄₇	#41	1	1	yes	characteristic
80	C₄²×C₄₇	C₂⁴	1	1	yes	characteristic
81	(C₂×D₄₇)⋊C₄	—	2	4	no	—
82	(C₂×D₄₇)⋊C₄	—	2	4	no	—
83	C₄⋊Dic₄₇	—	2	2	no	—
84	C₄⋊Dic₄₇	C₂²×C₄	1	4	yes	—
85	C₂×C₄₇⋊C₈	—	2	4	no	—
86	C₂×C₄₇⋊C₈	—	4	4	no	—
87	D₃₇₆	C₂×C₈	1	16	yes	direct product factor
88	C₈×D₄₇	—	4	16	no	—
89	C₂³×C₄₇	C₄×D₄	1	1	yes	characteristic
90	C₂²×D₄₇	—	2	4	no	—
91	C₂²×D₄₇	—	2	8	no	—
92	C₂²×D₄₇	—	4	4	no	—
93	C₂²×D₄₇	—	4	4	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
99	C₂×C₄×C₄₇	#131	1	2	yes	—
100	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
101	C₂×C₄×C₄₇	#135	1	2	yes	—
102	C₂×C₄×C₄₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₄₇	C₂×#41	1	1	yes	characteristic
105	C₂×Dic₄₇	—	2	4	no	—
106	C₂×Dic₄₇	—	4	4	no	—
107	D₁₈₈	—	2	4	no	—
108	D₁₈₈	—	2	8	no	—
109	D₁₈₈	C₂²×C₈	1	8	yes	complemented
110	C₄×D₄₇	—	4	4	no	—
111	C₄×D₄₇	—	4	4	no	—
112	C₄×D₄₇	—	4	8	no	—
113	C₈×C₄₇	—	2	4	no	—
114	C₈×C₄₇	—	2	2	no	—
115	C₈×C₄₇	—	2	2	no	—
116	C₈×C₄₇	C₂²×C₈	1	4	yes	complemented
117	C₈×C₄₇	C₂×D₈	1	4	yes	complemented
118	C₄₇⋊C₈	—	4	4	no	—
119	C₂×C₈×D₈	—	47	1	no	maximal, Sylow
120	C₂²×C₄₇	C₈×D₄	1	2	yes	—
121	C₂²×C₄₇	C₄×D₈	1	2	yes	—
122	C₂²×C₄₇	#443	1	2	yes	—
123	C₂²×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂×D₄₇	—	2	8	no	—
125	C₂×D₄₇	—	4	16	no	—
126	C₂×D₄₇	—	4	4	no	—
127	C₂×D₄₇	—	4	4	no	—
128	C₄×C₄₇	—	2	1	no	—
129	C₄×C₄₇	—	2	1	no	—
130	C₄×C₄₇	C₈×D₄	1	2	yes	—
131	C₄×C₄₇	C₄×D₈	1	2	yes	—
132	C₄×C₄₇	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₄₇	C₂×#131	1	1	yes	characteristic
134	C₄×C₄₇	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₄₇	C₂×#135	1	1	yes	characteristic
136	Dic₄₇	—	4	4	no	—
137	C₂×C₄×D₈	—	47	1	no	—
138	C₂×C₈×D₄	—	47	2	no	—
139	C₈×D₈	—	47	8	no	—
140	C₂×#335	—	47	1	no	—
141	C₂×D₄⋊C₈	—	47	2	no	—
142	C₂×C₈²	D₄₇	1	1	yes	characteristic, complemented
143	C₂×C₄₇	C₈×D₈	1	4	yes	complemented
144	C₂×C₄₇	C₂×C₈×D₄	1	1	yes	characteristic
145	C₂×C₄₇	C₂×C₄×D₈	1	1	yes	characteristic
146	C₂×C₄₇	C₂×#443	1	1	yes	characteristic
147	D₄₇	—	4	8	no	—
148	C₂²×D₈	—	47	1	no	—
149	C₂³×C₈	—	94	2	no	—
150	C₂×C₄×D₄	—	47	2	no	—
151	C₄×D₈	—	47	4	no	—
152	C₄×D₈	—	47	4	no	—
153	C₈×D₄	—	47	8	no	—
154	C₈×D₄	—	94	4	no	—
155	C₂×#107	—	47	1	no	—
156	C₂×C₄⋊C₈	—	47	2	no	—
157	C₂×#102	—	47	2	no	—
158	C₂×C₂²⋊C₈	—	94	2	no	—
159	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
162	#335	—	47	4	no	—
163	D₄⋊C₈	—	47	8	no	—
164	C₈²	C₂×D₄₇	1	4	yes	complemented
165	C₄₇	C₂×C₈×D₈	1	1	yes	characteristic, complemented, Sylow
166	C₂²×D₄	—	47	2	no	—
167	C₂³×C₄	—	94	2	no	—
168	C₂×D₈	—	47	8	no	—
169	C₂×D₈	—	47	4	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	94	2	no	—
172	C₂²×C₈	—	94	8	no	—
173	C₂²×C₈	—	188	8	no	—
174	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
175	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
176	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
177	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
178	C₄×D₄	—	47	4	no	—
179	C₄×D₄	—	47	4	no	—
180	C₄×D₄	—	94	2	no	—
181	C₄×D₄	—	94	2	no	—
182	C₂×C₄⋊C₄	—	47	2	no	—
183	C₂×C₂²⋊C₄	—	94	2	no	—
184	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
185	#107	—	47	4	no	—
186	C₄⋊C₈	—	47	4	no	—
187	C₄⋊C₈	—	94	2	no	—
188	#102	—	47	8	no	—
189	C₂²⋊C₈	—	94	8	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	C₄×D₄₇	1	2	yes	—
192	C₄×C₈	D₁₈₈	1	2	yes	—
193	C₄×C₈	C₂²×D₄₇	1	2	yes	—
194	C₄×C₈	C₂²×D₄₇	1	2	yes	—
195	C₄×C₈	C₂²×D₄₇	1	2	yes	—
196	C₂⁴	—	94	2	no	—
197	C₂×D₄	—	47	8	no	—
198	C₂×D₄	—	47	4	no	—
199	C₂×D₄	—	94	2	no	—
200	C₂×D₄	—	94	4	no	—
201	C₂²×C₄	—	94	2	no	—
202	C₂²×C₄	—	94	4	no	—
203	C₂²×C₄	—	94	4	no	—
204	C₂²×C₄	—	188	4	no	—
205	C₂²×C₄	—	188	4	no	—
206	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
208	C₂²×C₄	#10625350	1	1	yes	characteristic
209	D₈	—	47	16	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	94	4	no	—
215	C₂×C₈	—	188	4	no	—
216	C₂×C₈	—	188	16	no	—
217	C₂×C₈	C₈×D₄₇	1	4	yes	complemented
218	C₂×C₈	D₃₇₆	1	4	yes	direct product factor
219	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
221	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
222	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
223	C₄⋊C₄	—	47	4	no	—
224	C₄⋊C₄	—	94	2	no	—
225	C₂²⋊C₄	—	94	4	no	—
226	C₂²⋊C₄	—	94	4	no	—
227	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
228	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
229	C₄²	#10625350	1	1	yes	characteristic
230	C₄²	C₂³×D₄₇	1	1	yes	characteristic
231	C₂³	—	94	4	no	—
232	C₂³	—	94	8	no	—
233	C₂³	—	188	4	no	—
234	C₂³	—	188	4	no	—
235	C₂³	C₄×D₁₈₈	1	1	yes	characteristic
236	D₄	—	47	8	no	—
237	D₄	—	94	4	no	—
238	D₄	—	94	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	94	4	no	—
241	C₂×C₄	—	188	4	no	—
242	C₂×C₄	—	188	4	no	—
243	C₂×C₄	—	188	4	no	—
244	C₂×C₄	—	188	8	no	—
245	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
246	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
247	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
249	C₂×C₄	#13094030	1	2	yes	—
250	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
251	C₂×C₄	#13094046	1	2	yes	—
252	C₂×C₄	C₂²×C₄×D₄₇	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₁₈₈	1	1	yes	characteristic
254	C₂×C₄	C₂×#10625350	1	1	yes	characteristic
255	C₈	—	2	4	no	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	188	4	no	—
259	C₈	C₂×C₈×D₄₇	1	4	yes	complemented
260	C₈	C₂×D₃₇₆	1	4	yes	direct product factor
261	C₂²	—	94	8	no	—
262	C₂²	—	188	16	no	—
263	C₂²	—	188	4	no	—
264	C₂²	—	188	4	no	—
265	C₂²	C₈×D₁₈₈	1	2	yes	—
266	C₂²	C₄×D₃₇₆	1	2	yes	—
267	C₂²	#15689241	1	2	yes	—
268	C₂²	C₂×C₄×D₁₈₈	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	188	4	no	—
272	C₄	C₈×D₁₈₈	1	2	yes	—
273	C₄	C₄×D₃₇₆	1	2	yes	—
274	C₄	C₂²×C₈×D₄₇	1	1	yes	characteristic
275	C₄	C₂×#13094030	1	1	yes	characteristic
276	C₄	C₂²×D₃₇₆	1	1	yes	characteristic
277	C₄	C₂×#13094046	1	1	yes	characteristic
278	C₂	—	188	8	no	—
279	C₂	C₈×D₃₇₆	1	4	yes	direct product factor
280	C₂	C₂×C₈×D₁₈₈	1	1	yes	characteristic
281	C₂	C₂×C₄×D₃₇₆	1	1	yes	characteristic
282	C₂	C₂×#15689241	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₁₈₈
commutator	C₄×C₄₇	C₂³×C₈
Frattini	C₄²	C₂³×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle	C₂³×C₄₇	C₄×D₄
prime radical	C₂²×D₃₇₆	C₄
cosocle	C₄²×C₄₇	C₂⁴
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	C₂×C₈×D₈
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₁₈₈	C₂	semidirect
3. C₈×D₃₇₆	C₂	direct (N × H)
4. C₂×#15688876	C₂	semidirect
5. C₂×#15688852	C₂	semidirect
6. C₂×C₈²×C₄₇	C₂	semidirect
13. C₈×D₁₈₈	C₂²	semidirect
18. #15688876	C₂²	semidirect
19. #15688852	C₂²	semidirect
23. C₈²×C₄₇	C₂²	semidirect
26. C₂×D₃₇₆	C₈	direct (N × H)
61. C₂×D₁₈₈	C₂×C₈	semidirect
71. C₂×C₈×C₄₇	C₂×C₈	semidirect
72. C₂×C₈×C₄₇	D₈	semidirect
87. D₃₇₆	C₂×C₈	direct (N × H)
109. D₁₈₈	C₂²×C₈	semidirect
116. C₈×C₄₇	C₂²×C₈	semidirect
117. C₈×C₄₇	C₂×D₈	semidirect
142. C₂×C₈²	D₄₇	semidirect
143. C₂×C₄₇	C₈×D₈	semidirect
164. C₈²	C₂×D₄₇	semidirect
165. C₄₇	C₂×C₈×D₈	semidirect
217. C₂×C₈	C₈×D₄₇	semidirect
218. C₂×C₈	D₃₇₆	direct (N × H)
259. C₈	C₂×C₈×D₄₇	semidirect
260. C₈	C₂×D₃₇₆	direct (N × H)
279. C₂	C₈×D₃₇₆	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×C₄×D₃₇₆	2
C₂×C₈×D₁₈₈	2
C₈×D₃₇₆	2
C₂×#15688876	2
C₂×#15688852	2
C₂×C₈²×C₄₇	2
C₂×C₈×D₈	47

Subgroups — C2×C8×D376 #26185636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×D₃₇₆ #26185636