Subgroups — #28383633

Order 14272.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₂₃⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	2	yes	maximal
3	C₄×C₂₂₃⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₂₃⋊C₈	C₂²	1	4	yes	—
6	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₄²×C₂₂₃	C₄	1	1	yes	characteristic
10	C₄²×C₂₂₃	C₄	1	1	yes	characteristic
11	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂₂₃⋊C₈	—	2	4	no	—
14	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₂₃	—	2	2	no	—
16	C₂×C₄×C₂₂₃	—	2	2	no	—
17	C₂×C₄×C₂₂₃	C₈	1	4	yes	complemented
18	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₂²×C₂₂₃	—	2	1	no	—
25	C₂²×C₂₂₃	—	2	1	no	—
26	C₂²×C₂₂₃	C₂×C₈	1	2	yes	—
27	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₂₂₃	—	2	1	no	—
29	C₄×C₂₂₃	—	2	1	no	—
30	C₄×C₂₂₃	C₂×C₈	1	2	yes	—
31	C₄×C₂₂₃	C₂×C₈	1	2	yes	—
32	C₄×C₂₂₃	M₁₆	1	2	yes	—
33	C₄×C₂₂₃	#41	1	2	yes	—
34	C₂×C₂₂₃	—	2	2	no	—
35	C₂×C₂₂₃	#117	1	1	yes	characteristic
36	C₂×C₂₂₃	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₂₂₃	C₂×M₁₆	1	1	yes	characteristic
38	C₂₂₃	#431	1	1	yes	characteristic, complemented, Sylow
39	#431	—	223	1	no	maximal, Sylow
40	C₂×C₄²	D₂₂₃	1	1	yes	characteristic
41	C₄⋊C₈	—	223	2	no	—
42	C₂²⋊C₈	—	223	2	no	—
43	C₄×C₈	—	223	2	no	—
44	C₂²×C₄	Dic₂₂₃	1	1	yes	characteristic
45	C₂²×C₄	Dic₂₂₃	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
47	C₂×C₈	—	223	4	no	—
48	C₄²	Dic₂₂₃	1	1	yes	characteristic
49	C₄²	Dic₂₂₃	1	1	yes	characteristic
50	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
51	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
52	C₂³	C₂×Dic₂₂₃	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂₂₃⋊C₈	1	4	yes	complemented
56	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
57	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
58	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
59	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
62	C₈	—	446	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	C₂×C₂₂₃⋊C₈	1	2	yes	—
66	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	C₂×C₂₂₃⋊C₈	1	2	yes	—
70	C₄	C₂×C₂₂₃⋊C₈	1	2	yes	—
71	C₄	#16813154	1	2	yes	—
72	C₄	#16813170	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#20297263	1	1	yes	characteristic
75	C₂	C₂²×C₂₂₃⋊C₈	1	1	yes	characteristic
76	C₂	C₂×#16813154	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂×C₄²×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²×C₄
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂×C₄²	D₂₂₃
hypocenter	C₂₂₃	#431
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₂₂₃⋊C₈)	C₂	semidirect
3. C₄×C₂₂₃⋊C₈	C₂	semidirect
17. C₂×C₄×C₂₂₃	C₈	semidirect
38. C₂₂₃	#431	semidirect
55. C₂×C₄	C₂₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₂₃⋊C₈)	2
C₂²⋊(C₂₂₃⋊C₈)	2
C₄×C₂₂₃⋊C₈	2
C₂×C₄²×C₂₂₃	2
#431	223