Subgroups — #13092852

Order 1472.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₈₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625010	C₂	1	2	yes	maximal, complemented
4	#10625008	C₂	1	2	yes	maximal, complemented
5	#10624994	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₂₃	—	2	1	no	—
7	C₂×D₉₂	—	2	1	no	—
8	Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
9	SD₁₆×C₂₃	—	2	1	no	—
10	D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	D₁₈₄	—	2	1	no	—
13	D₁₈₄	C₂²	1	2	yes	—
14	C₂₃⋊C₁₆	C₂²	1	2	yes	complemented
15	C₂²×D₂₃	—	4	1	no	—
16	Q₈×C₂₃	—	2	1	no	—
17	D₄×C₂₃	—	2	1	no	—
18	D₄×C₂₃	—	4	1	no	—
19	C₂×C₄×C₂₃	—	4	1	no	—
20	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
21	D₉₂	—	2	2	no	—
22	D₉₂	—	4	1	no	—
23	C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₈×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂₃	—	4	1	no	—
26	C₂²×C₂₃	D₈	1	1	yes	characteristic
27	C₂×D₂₃	—	4	2	no	—
28	C₂×D₂₃	—	8	2	no	—
29	C₄×C₂₃	—	4	1	no	—
30	C₄×C₂₃	D₈	1	1	yes	characteristic
31	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
32	#509	—	23	1	no	maximal, Sylow
33	C₂×C₂₃	—	2	1	no	—
34	C₂×C₂₃	—	8	1	no	—
35	C₂×C₂₃	C₂×D₈	1	1	yes	characteristic
36	D₂₃	—	8	2	no	—
37	#135	D₂₃	1	1	yes	characteristic, complemented
38	C₂×D₈	—	23	1	no	—
39	SD₃₂	—	23	2	no	—
40	D₁₆	—	23	2	no	—
41	M₃₂	—	23	1	no	—
42	C₂₃	#509	1	1	yes	characteristic, complemented, Sylow
43	#41	—	2	1	no	—
44	C₂×D₄	—	46	1	no	—
45	Q₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
46	SD₁₆	—	2	1	no	—
47	D₈	—	23	2	no	—
48	D₈	—	46	1	no	—
49	D₈	C₂×D₂₃	1	1	yes	characteristic, complemented
50	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
51	C₁₆	—	23	2	no	—
52	C₂³	—	92	1	no	—
53	Q₈	—	2	1	no	—
54	D₄	—	2	1	no	—
55	D₄	—	4	1	no	—
56	D₄	—	46	2	no	—
57	D₄	—	92	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
60	C₈	C₂²⋊D₂₃	1	1	yes	characteristic
61	C₈	C₂²×D₂₃	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	—	92	2	no	—
64	C₂²	—	184	2	no	—
65	C₂²	#67272	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#67272	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	1	no	—
71	C₂	—	184	2	no	—
72	C₂	C₂×#67272	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#67272
commutator, cosocle	C₈×C₂₃	C₂³
Frattini	C₈	C₂²×D₂₃
Fitting	C₂₃×#135	C₂
socle	C₂×C₂₃	C₂×D₈
hypercenter	#135	D₂₃
hypocenter	C₂₃	#509
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂₃×#135	C₂	semidirect
2. C₂×D₁₈₄	C₂	semidirect
3. #10625010	C₂	semidirect
4. #10625008	C₂	semidirect
5. #10624994	C₂	semidirect
8. Q₁₆×C₂₃	C₂²	semidirect
10. D₈×C₂₃	C₂²	semidirect
14. C₂₃⋊C₁₆	C₂²	semidirect
37. #135	D₂₃	semidirect
42. C₂₃	#509	semidirect
45. Q₁₆	C₂×D₂₃	semidirect
49. D₈	C₂×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#135	2
C₂×D₁₈₄	2
#10625010	2
#10625008	2
#10624994	2
#509	23