Subgroups — #28383306

Order 14272.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₂₃⋊C₈	—	2	2	no	—
4	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₂₂₃	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×C₂₂₃	—	2	2	no	—
8	C₂₂₃⋊C₈	—	4	2	no	—
9	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₂₂₃	—	2	1	no	—
11	C₂×C₄×C₂₂₃	—	2	1	no	—
12	C₂×C₄×C₂₂₃	—	2	1	no	—
13	C₂×C₄×C₂₂₃	—	4	1	no	—
14	C₂×C₄×C₂₂₃	—	4	1	no	—
15	C₂×C₄×C₂₂₃	C₈	1	2	yes	complemented
16	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
17	C₂²×C₂₂₃	—	2	1	no	—
18	C₂²×C₂₂₃	—	2	1	no	—
19	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₂₂₃	C₂×C₈	1	1	yes	characteristic
21	C₂²×C₂₂₃	M₁₆	1	1	yes	characteristic
22	C₄×C₂₂₃	—	2	2	no	—
23	C₄×C₂₂₃	—	2	2	no	—
24	C₄×C₂₂₃	—	4	1	no	—
25	C₂×C₂₂₃	—	2	1	no	—
26	C₂×C₂₂₃	—	2	1	no	—
27	C₂×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
28	C₂×C₂₂₃	#99	1	1	yes	characteristic
29	C₂×C₂₂₃	#101	1	1	yes	characteristic
30	C₂₂₃	#324	1	1	yes	characteristic, complemented, Sylow
31	#324	—	223	1	no	maximal, Sylow
32	C₂×C₄⋊C₄	D₂₂₃	1	1	yes	characteristic
33	C₂²⋊C₈	—	223	2	no	—
34	C₂²×C₄	Dic₂₂₃	1	1	yes	characteristic
35	C₂²×C₄	Dic₂₂₃	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
37	C₂×C₈	—	446	2	no	—
38	C₄⋊C₄	—	2	2	no	—
39	C₂³	C₂×Dic₂₂₃	1	1	yes	characteristic
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	—	4	1	no	—
44	C₂×C₄	—	4	1	no	—
45	C₂×C₄	C₂₂₃⋊C₈	1	2	yes	complemented
46	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
47	C₈	—	892	2	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
51	C₂²	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
52	C₂²	#16813154	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	2	2	no	—
55	C₄	—	4	1	no	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂²⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
59	C₂	#20297219	1	1	yes	characteristic
60	C₂	#20297223	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₂₂₃
commutator	C₂²×C₂₂₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂×C₄⋊C₄×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂×C₄⋊C₄	D₂₂₃
hypocenter	C₂₂₃	#324
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
15. C₂×C₄×C₂₂₃	C₈	semidirect
30. C₂₂₃	#324	semidirect
45. 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
C₂²⋊(C₂₂₃⋊C₈)	2
C₂×C₄⋊C₄×C₂₂₃	2
#324	223