Subgroups — #28383367

Order 14272.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297221	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297219	C₂	1	1	yes	maximal, characteristic, complemented
3	#120×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#16813154	—	2	1	no	—
5	C₂²⋊Dic₂₂₃	—	2	1	no	—
6	C₂⁴×C₂₂₃	C₄	1	1	yes	characteristic, complemented
7	C₂×D₄×C₂₂₃	—	2	1	no	—
8	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
10	C₂²⋊C₄×C₂₂₃	C₄	1	1	yes	characteristic, complemented
11	C₂×Dic₂₂₃	—	4	1	no	—
12	C₂₂₃⋊C₈	—	4	1	no	—
13	C₂³×C₂₂₃	—	2	1	no	—
14	C₂³×C₂₂₃	—	4	2	no	—
15	C₂³×C₂₂₃	—	4	1	no	—
16	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
18	D₄×C₂₂₃	—	4	1	no	—
19	D₄×C₂₂₃	—	4	2	no	—
20	C₂×C₄×C₂₂₃	—	2	1	no	—
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₂₂₃	—	2	2	no	—
23	C₂²×C₂₂₃	—	2	1	no	—
24	C₂²×C₂₂₃	—	2	1	no	—
25	C₂²×C₂₂₃	—	4	1	no	—
26	C₂²×C₂₂₃	—	4	1	no	—
27	C₂²×C₂₂₃	—	4	1	no	—
28	C₂²×C₂₂₃	—	4	2	no	—
29	C₂²×C₂₂₃	—	4	2	no	—
30	C₂²×C₂₂₃	—	4	1	no	—
31	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₂₂₃	—	2	1	no	—
33	C₄×C₂₂₃	—	4	1	no	—
34	Dic₂₂₃	—	8	1	no	—
35	C₂×C₂₂₃	—	2	1	no	—
36	C₂×C₂₂₃	—	4	1	no	—
37	C₂×C₂₂₃	—	4	2	no	—
38	C₂×C₂₂₃	—	4	1	no	—
39	C₂×C₂₂₃	#99	1	1	yes	characteristic
40	C₂₂₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	223	1	no	maximal, Sylow
42	#120	D₂₂₃	1	1	yes	characteristic
43	#100	—	223	1	no	—
44	#99	—	223	1	no	—
45	C₂⁴	Dic₂₂₃	1	1	yes	characteristic, complemented
46	C₂×D₄	—	2	1	no	—
47	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
48	M₁₆	—	446	1	no	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	446	1	no	—
51	C₂²⋊C₄	Dic₂₂₃	1	1	yes	characteristic, complemented
52	C₂³	—	2	1	no	—
53	C₂³	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	C₂³	C₂×Dic₂₂₃	1	1	yes	characteristic
56	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
57	D₄	—	4	1	no	—
58	D₄	—	4	2	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	892	1	no	—
61	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
62	C₈	—	892	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	1784	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	2	no	—
79	C₂	—	4	1	no	—
80	C₂	#20297219	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20297219
commutator	C₂³×C₂₂₃	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₂₃
Fitting, prime radical	#120×C₂₂₃	C₂
socle	C₂×C₂₂₃	#99
cosocle	C₂×D₄×C₂₂₃	C₂²
hypercenter	#120	D₂₂₃
hypocenter	C₂₂₃	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #120

Complements

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

N	H ≅ G/N	type
1. #20297221	C₂	semidirect
2. #20297219	C₂	semidirect
6. C₂⁴×C₂₂₃	C₄	semidirect
10. C₂²⋊C₄×C₂₂₃	C₄	semidirect
40. C₂₂₃	AΣL(1,𝔽₁₆)	semidirect
45. C₂⁴	Dic₂₂₃	semidirect
51. C₂²⋊C₄	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297221	2
#20297219	2
#120×C₂₂₃	2
AΣL(1,𝔽₁₆)	223