Subgroups — #28383645

Order 14272.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₂₃	C₂	1	2	yes	maximal, complemented
2	C₄×D₈₉₂	C₂	1	1	yes	maximal, characteristic
3	(C₂×D₂₂₃)⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₄×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
8	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₈×D₂₂₃	—	2	4	no	—
11	C₄⋊Dic₂₂₃	C₄	1	1	yes	characteristic
12	(C₂×D₂₂₃)⋊C₄	C₄	1	2	yes	—
13	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
14	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×D₂₂₃	C₂×C₄	1	2	yes	—
16	D₈₉₂	C₈	1	4	yes	complemented
17	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
18	C₄×D₂₂₃	—	2	2	no	—
19	C₄×D₂₂₃	—	2	2	no	—
20	C₂₂₃⋊C₈	—	2	1	no	—
21	C₂₂₃⋊C₈	D₄	1	2	yes	complemented
22	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₈×C₂₂₃	—	2	2	no	—
26	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₂₃	—	2	2	no	—
28	C₂×D₂₂₃	—	2	2	no	—
29	C₂×D₂₂₃	C₂×C₈	1	4	yes	complemented
30	C₄×C₂₂₃	—	2	1	no	—
31	C₄×C₂₂₃	C₂×C₈	1	2	yes	complemented
32	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₂₃	#41	1	1	yes	characteristic
34	Dic₂₂₃	—	2	2	no	—
35	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
36	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₂₂₃	#131	1	1	yes	characteristic
38	D₂₂₃	—	2	4	no	—
39	C₂₂₃	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₈×D₄	—	223	1	no	maximal, Sylow
41	C₂²×C₈	—	223	2	no	—
42	C₄×D₄	—	223	1	no	—
43	C₄⋊C₈	D₂₂₃	1	1	yes	characteristic, complemented
44	C₂²⋊C₈	—	223	2	no	—
45	C₄×C₈	—	223	1	no	—
46	C₂×D₄	—	223	1	no	—
47	C₂²×C₄	—	223	2	no	—
48	C₂×C₈	—	446	4	no	—
49	C₂×C₈	—	223	1	no	—
50	C₂×C₈	—	223	1	no	—
51	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
52	C₄⋊C₄	—	223	1	no	—
53	C₂²⋊C₄	—	223	2	no	—
54	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
55	C₂³	—	223	2	no	—
56	D₄	—	223	4	no	—
57	C₂×C₄	—	446	2	no	—
58	C₂×C₄	—	446	2	no	—
59	C₂×C₄	—	223	2	no	—
60	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
61	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	446	1	no	—
65	C₈	—	223	2	no	—
66	C₂²	—	446	2	no	—
67	C₂²	—	446	2	no	—
68	C₂²	—	223	4	no	—
69	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	446	2	no	—
72	C₄	C₈×D₂₂₃	1	2	yes	complemented
73	C₄	D₄×D₂₂₃	1	1	yes	characteristic
74	C₄	#16813171	1	1	yes	characteristic
75	C₂	—	446	4	no	—
76	C₂	#20297266	1	1	yes	characteristic
77	C₂	C₂×C₈×D₂₂₃	1	1	yes	characteristic
78	C₂	#20297312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator	C₂×C₂₂₃	C₂²×C₈
Fitting	C₄⋊C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²×C₄
prime radical	C₂×D₈₉₂	C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	C₈×D₄
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₂₂₃	C₂	semidirect
3. (C₂×D₂₂₃)⋊C₈	C₂	semidirect
4. C₄×C₂₂₃⋊C₈	C₂	semidirect
5. C₄⋊C₈×C₂₂₃	C₂	semidirect
16. D₈₉₂	C₈	semidirect
21. C₂₂₃⋊C₈	D₄	semidirect
29. C₂×D₂₂₃	C₂×C₈	semidirect
31. C₄×C₂₂₃	C₂×C₈	semidirect
39. C₂₂₃	C₈×D₄	semidirect
43. C₄⋊C₈	D₂₂₃	semidirect
72. C₄	C₈×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₂₃	2
C₄×D₈₉₂	2
(C₂×D₂₂₃)⋊C₈	2
C₄×C₂₂₃⋊C₈	2
C₄⋊C₈×C₂₂₃	2
C₈×D₄	223