Subgroups — #13092528

Order 1472.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₈₄	C₂	1	1	yes	maximal, characteristic
2	#10625064	C₂	1	2	yes	maximal, complemented
3	C₂₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	#10624997	C₂	1	1	yes	maximal, characteristic, complemented
5	#10624991	C₂	1	2	yes	maximal, complemented
6	C₂×D₉₂	C₂²	1	2	yes	—
7	C₂×C₄×D₂₃	—	2	2	no	—
8	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	(C₂×D₂₃)⋊C₄	—	2	2	no	—
11	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	D₁₈₄	C₄	1	4	yes	complemented
14	C₂²×D₂₃	—	2	2	no	—
15	C₂×C₄×C₂₃	—	2	2	no	—
16	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂×Dic₂₃	D₄	1	2	yes	—
18	D₉₂	—	2	2	no	—
19	D₉₂	C₂×C₄	1	4	yes	complemented
20	C₄×D₂₃	—	4	4	no	—
21	C₈×C₂₃	C₂×C₄	1	2	yes	complemented
22	C₂₃⋊C₈	—	2	1	no	—
23	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₃	—	2	4	no	—
25	C₂×D₂₃	—	4	2	no	—
26	C₂×D₂₃	—	4	2	no	—
27	C₄×C₂₃	—	4	2	no	—
28	C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₃	#41	1	1	yes	characteristic
30	Dic₂₃	—	2	2	no	—
31	#442	—	23	1	no	maximal, Sylow
32	C₂×C₂₃	C₄×D₄	1	1	yes	characteristic
33	C₂×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
34	D₂₃	—	4	4	no	—
35	C₂×D₈	—	23	1	no	—
36	C₄×D₄	—	23	2	no	—
37	#106	D₂₃	1	1	yes	characteristic, complemented
38	#102	—	23	2	no	—
39	#97	—	23	1	no	—
40	C₂₃	#442	1	1	yes	characteristic, complemented, Sylow
41	C₂×D₄	—	23	2	no	—
42	C₂²×C₄	—	46	2	no	—
43	D₈	—	23	4	no	—
44	C₂×C₈	—	23	1	no	—
45	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
46	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
47	C₂²⋊C₄	—	46	2	no	—
48	C₄²	—	23	1	no	—
49	C₂³	—	46	2	no	—
50	D₄	—	23	4	no	—
51	D₄	—	46	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	23	2	no	—
54	C₂×C₄	—	92	4	no	—
55	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
56	C₈	—	46	1	no	—
57	C₈	C₄×D₂₃	1	2	yes	complemented
58	C₂²	—	46	4	no	—
59	C₂²	—	92	2	no	—
60	C₂²	—	92	2	no	—
61	C₂²	D₄×D₂₃	1	1	yes	characteristic
62	C₄	—	4	2	no	—
63	C₄	—	46	2	no	—
64	C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
65	C₄	#67292	1	1	yes	characteristic
66	C₂	—	92	4	no	—
67	C₂	#10625064	1	1	yes	characteristic
68	C₂	#10625085	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₃
commutator	C₄×C₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#106	C₂
socle	C₂²×C₂₃	C₂×D₄
prime radical	C₂×D₁₈₄	C₂
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	#106	D₂₃
hypocenter	C₂₃	#442
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
2. #10625064	C₂	semidirect
3. C₂₃×#106	C₂	semidirect
4. #10624997	C₂	semidirect
5. #10624991	C₂	semidirect
13. D₁₈₄	C₄	semidirect
19. D₉₂	C₂×C₄	semidirect
21. C₈×C₂₃	C₂×C₄	semidirect
37. #106	D₂₃	semidirect
40. C₂₃	#442	semidirect
57. C₈	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₈₄	2
#10625064	2
C₂₃×#106	2
#10624997	2
#10624991	2
#442	23