Subgroups — #28383641

Order 14272.

96 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	#20297215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#117×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₂₃	C₄	1	2	yes	—
10	C₂²⋊C₄×C₂₂₃	C₄	1	2	yes	—
11	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂₂₃⋊C₈	—	2	2	no	—
14	C₂₂₃⋊C₈	—	2	2	no	—
15	C₂³×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₂₃	—	2	2	no	—
17	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₂²×C₂₂₃	—	2	1	no	—
22	C₂²×C₂₂₃	—	2	2	no	—
23	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₂₃	—	2	1	no	—
25	C₄×C₂₂₃	—	2	2	no	—
26	C₄×C₂₂₃	—	2	2	no	—
27	C₄×C₂₂₃	#41	1	2	yes	—
28	C₂×C₂₂₃	—	4	1	no	—
29	C₂×C₂₂₃	#117	1	1	yes	characteristic
30	C₂×C₂₂₃	#131	1	2	yes	—
31	C₂₂₃	#433	1	1	yes	characteristic, complemented, Sylow
32	#433	—	223	1	no	maximal, Sylow
33	#117	D₂₂₃	1	1	yes	characteristic
34	C₄⋊C₈	—	223	2	no	—
35	C₂²⋊C₈	—	223	2	no	—
36	#97	—	223	1	no	—
37	C₄×C₈	—	223	1	no	—
38	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
39	C₂×C₈	—	223	2	no	—
40	C₂×C₈	—	223	2	no	—
41	C₄⋊C₄	Dic₂₂₃	1	2	yes	—
42	C₂²⋊C₄	Dic₂₂₃	1	2	yes	—
43	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
44	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂³	C₂×Dic₂₂₃	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
48	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
49	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
51	C₈	—	446	2	no	—
52	C₈	—	446	2	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	2	no	—
55	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	2	2	no	—
58	C₄	—	2	2	no	—
59	C₄	#16813170	1	2	yes	—
60	C₂	—	4	1	no	—
61	C₂	#20297263	1	1	yes	characteristic
62	C₂	#20297311	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂²×C₄
Fitting	#117×C₂₂₃	C₂
prime radical	C₂³×C₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#117	D₂₂₃
hypocenter	C₂₂₃	#433
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₂₂₃⋊C₈)	C₂	semidirect
3. #20297215	C₂	semidirect
4. C₄×C₂₂₃⋊C₈	C₂	semidirect
31. C₂₂₃	#433	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
#20297215	2
C₄×C₂₂₃⋊C₈	2
#117×C₂₂₃	2
#433	223