Subgroups — #20297275

Order 7136.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₂₂₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₂₂₃	C₂	1	2	yes	maximal, complemented
5	C₂×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₂₃	C₂²	1	1	yes	characteristic
7	C₂²⋊D₂₂₃	—	2	4	no	—
8	C₂×Dic₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₂₃	C₂²	1	2	yes	—
10	C₄×D₂₂₃	—	2	2	no	—
11	C₂³×C₂₂₃	C₂²	1	2	yes	—
12	D₄×C₂₂₃	—	2	2	no	—
13	C₂×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₂₂₃	—	2	2	no	—
15	C₂²×C₂₂₃	—	2	2	no	—
16	C₂²×C₂₂₃	—	2	2	no	—
17	C₂²×C₂₂₃	C₂³	1	1	yes	characteristic
18	C₂×D₂₂₃	—	2	1	no	—
19	C₂×D₂₂₃	—	2	1	no	—
20	C₂×D₂₂₃	D₄	1	2	yes	complemented
21	C₄×C₂₂₃	D₄	1	2	yes	complemented
22	Dic₂₂₃	—	2	1	no	—
23	Dic₂₂₃	—	2	2	no	—
24	C₂×C₂₂₃	—	4	2	no	—
25	C₂×C₂₂₃	C₂×D₄	1	1	yes	characteristic
26	C₂×C₂₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₂₂₃	#41	1	1	yes	characteristic
28	D₂₂₃	—	2	2	no	—
29	C₂₂₃	#121	1	1	yes	characteristic, complemented, Sylow
30	#121	—	223	1	no	maximal, Sylow
31	C₂×D₄	—	223	2	no	—
32	C₂×D₄	D₂₂₃	1	1	yes	characteristic, complemented
33	C₂²×C₄	—	223	1	no	—
34	C₄⋊C₄	—	223	1	no	—
35	C₂²⋊C₄	—	223	2	no	—
36	C₂³	—	223	1	no	—
37	C₂³	C₂×D₂₂₃	1	2	yes	—
38	D₄	—	2	2	no	—
39	D₄	—	446	4	no	—
40	C₂×C₄	—	446	2	no	—
41	C₂×C₄	—	223	1	no	—
42	C₂×C₄	—	223	2	no	—
43	C₂×C₄	C₂×D₂₂₃	1	1	yes	characteristic
44	C₂²	—	2	2	no	—
45	C₂²	—	2	2	no	—
46	C₂²	—	2	2	no	—
47	C₂²	—	446	1	no	—
48	C₂²	—	446	1	no	—
49	C₂²	—	223	2	no	—
50	C₂²	C₂²×D₂₂₃	1	1	yes	characteristic
51	C₄	—	446	2	no	—
52	C₄	—	446	1	no	—
53	C₄	C₂²⋊D₂₂₃	1	2	yes	complemented
54	C₂	—	4	2	no	—
55	C₂	—	446	2	no	—
56	C₂	D₄×D₂₂₃	1	1	yes	characteristic
57	C₂	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
58	C₂	#16813169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₂₂₃
commutator, socle	C₂²×C₂₂₃	C₂³
Fitting	C₂×D₄×C₂₂₃	C₂
hypercenter	C₂×D₄	D₂₂₃
hypocenter	C₂₂₃	#121
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊D₂₂₃	C₂	semidirect
2. C₂×C₄×D₂₂₃	C₂	semidirect
3. C₄⋊Dic₂₂₃	C₂	semidirect
4. C₂²⋊Dic₂₂₃	C₂	semidirect
5. C₂×D₄×C₂₂₃	C₂	semidirect
20. C₂×D₂₂₃	D₄	semidirect
21. C₄×C₂₂₃	D₄	semidirect
29. C₂₂₃	#121	semidirect
32. C₂×D₄	D₂₂₃	semidirect
53. C₄	C₂²⋊D₂₂₃	semidirect