Subgroups — #20297299

Order 7136.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	Dic₂₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₂₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	(C₂×D₂₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	(C₂×D₂₂₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₂₂₃	C₂³	1	1	yes	characteristic
16	C₂×D₂₂₃	—	2	1	no	—
17	C₂×D₂₂₃	—	2	1	no	—
18	C₂×D₂₂₃	—	2	1	no	—
19	C₄×C₂₂₃	—	2	1	no	—
20	C₄×C₂₂₃	—	2	1	no	—
21	C₄×C₂₂₃	—	2	1	no	—
22	Dic₂₂₃	—	2	1	no	—
23	Dic₂₂₃	—	2	1	no	—
24	Dic₂₂₃	—	2	1	no	—
25	C₂×C₂₂₃	#41	1	1	yes	characteristic
26	C₂×C₂₂₃	#41	1	1	yes	characteristic
27	C₂×C₂₂₃	#41	1	1	yes	characteristic
28	D₂₂₃	—	4	1	no	—
29	C₂₂₃	#126	1	1	yes	characteristic, complemented, Sylow
30	#126	—	223	1	no	maximal, Sylow
31	C₄⋊C₄	—	223	1	no	—
32	C₄⋊C₄	—	223	1	no	—
33	C₄⋊C₄	D₂₂₃	1	1	yes	characteristic, complemented
34	C₂²⋊C₄	—	223	1	no	—
35	C₂²⋊C₄	—	223	1	no	—
36	C₂²⋊C₄	—	223	1	no	—
37	C₄²	—	223	1	no	—
38	C₂³	—	223	1	no	—
39	C₂×C₄	—	223	1	no	—
40	C₂×C₄	—	223	1	no	—
41	C₂×C₄	—	223	1	no	—
42	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
43	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
44	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂²	—	446	1	no	—
46	C₂²	—	446	1	no	—
47	C₂²	—	446	1	no	—
48	C₂²	C₂²×D₂₂₃	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	446	1	no	—
53	C₄	—	446	1	no	—
54	C₄	—	446	1	no	—
55	C₂	—	892	1	no	—
56	C₂	#16813169	1	1	yes	characteristic
57	C₂	#16813170	1	1	yes	characteristic
58	C₂	#16813171	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂³
Fitting	C₄⋊C₄×C₂₂₃	C₂
hypercenter	C₄⋊C₄	D₂₂₃
hypocenter	C₂₂₃	#126
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊Dic₂₂₃	C₂	semidirect
2. Dic₂₂₃⋊C₄	C₂	semidirect
6. C₄×Dic₂₂₃	C₂	semidirect
7. C₄⋊C₄×C₂₂₃	C₂	semidirect
29. C₂₂₃	#126	semidirect
33. C₄⋊C₄	D₂₂₃	semidirect