Subgroups — #28384347

Order 14272.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297311	C₂	1	8	yes	maximal, complemented
2	C₂×#16813154	C₂	1	6	yes	maximal, complemented
3	C₂×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#16813154	C₂²	1	4	yes	—
5	#16813154	C₂²	1	12	yes	complemented
6	C₂×C₂₂₃⋊C₈	C₂²	1	12	yes	complemented
7	#41×C₂₂₃	C₄	1	4	yes	—
8	#41×C₂₂₃	C₂²	1	4	yes	—
9	C₂×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
10	C₂×D₄×C₂₂₃	C₄	1	3	yes	—
11	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
12	C₂₂₃⋊C₈	C₂³	1	8	yes	—
13	C₂³×C₂₂₃	C₂×C₄	1	3	yes	—
14	Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
15	D₄×C₂₂₃	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₂₂₃	C₂×C₄	1	6	yes	—
18	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
20	C₂²×C₂₂₃	—	4	3	no	—
21	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₂₂₃	C₂²×C₄	1	6	yes	—
23	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
25	C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
26	C₂×C₂₂₃	—	2	1	no	—
27	C₂×C₂₂₃	—	2	6	no	—
28	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
29	C₂₂₃	#568	1	1	yes	characteristic, complemented, Sylow
30	#568	—	223	1	no	maximal, Sylow
31	C₂×#41	D₂₂₃	1	1	yes	characteristic
32	#131	—	223	8	no	—
33	C₂×M₁₆	—	223	6	no	—
34	#41	Dic₂₂₃	1	4	yes	—
35	#41	C₂×D₂₂₃	1	4	yes	—
36	C₂×Q₈	Dic₂₂₃	1	1	yes	characteristic
37	C₂×D₄	Dic₂₂₃	1	3	yes	—
38	C₂²×C₄	C₂×D₂₂₃	1	3	yes	—
39	M₁₆	—	223	4	no	—
40	M₁₆	—	223	12	no	—
41	C₂×C₈	—	223	12	no	—
42	C₂³	C₂×Dic₂₂₃	1	3	yes	—
43	Q₈	C₂×Dic₂₂₃	1	4	yes	—
44	D₄	C₂×Dic₂₂₃	1	12	yes	—
45	C₂×C₄	C₂×Dic₂₂₃	1	3	yes	—
46	C₂×C₄	C₂×Dic₂₂₃	1	6	yes	—
47	C₂×C₄	C₂²×D₂₂₃	1	6	yes	—
48	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
49	C₈	—	223	8	no	—
50	C₂²	—	4	3	no	—
51	C₂²	C₂²×Dic₂₂₃	1	1	yes	characteristic
52	C₂²	C₂²×Dic₂₂₃	1	6	yes	—
53	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
54	C₄	C₂²×Dic₂₂₃	1	6	yes	—
55	C₄	C₂³×D₂₂₃	1	1	yes	characteristic
56	C₂	—	2	1	no	—
57	C₂	—	2	6	no	—
58	C₂	C₂³×Dic₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₂₂₃
commutator, socle	C₂×C₂₂₃	C₂³×C₄
Fitting, prime radical	C₂×#41×C₂₂₃	C₂
cosocle	C₄×C₂₂₃	C₂⁴
hypercenter	C₂×#41	D₂₂₃
hypocenter	C₂₂₃	#568
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20297311	C₂	semidirect
2. C₂×#16813154	C₂	semidirect
5. #16813154	C₂²	semidirect
6. C₂×C₂₂₃⋊C₈	C₂²	semidirect
29. C₂₂₃	#568	semidirect

Maximal subgroups

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

subgroup	index
#20297311	2
C₂×#16813154	2
C₂×#41×C₂₂₃	2
#568	223