Subgroups — #13092670

Order 1472.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625101	C₂	1	2	yes	maximal, complemented
3	C₂×D₁₈₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625003	C₂	1	2	yes	maximal, complemented
5	#10624999	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₂₃	—	2	2	no	—
8	C₂×D₉₂	C₂²	1	2	yes	—
9	M₁₆×C₂₃	—	2	2	no	—
10	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	(C₂×D₂₃)⋊C₄	—	2	2	no	—
13	C₄⋊Dic₂₃	C₂²	1	2	yes	—
14	D₁₈₄	—	2	2	no	—
15	C₂³×C₂₃	D₄	1	1	yes	characteristic
16	C₂²×D₂₃	—	2	2	no	—
17	C₂×C₄×C₂₃	—	2	1	no	—
18	C₂×C₄×C₂₃	—	2	1	no	—
19	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂²⋊D₂₃	—	4	4	no	—
22	C₂×Dic₂₃	—	2	2	no	—
23	D₉₂	—	2	2	no	—
24	D₉₂	—	2	2	no	—
25	C₈×C₂₃	—	2	1	no	—
26	C₈×C₂₃	D₄	1	2	yes	complemented
27	C₂²×C₂₃	—	2	1	no	—
28	C₂²×C₂₃	—	2	2	no	—
29	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂×D₂₃	—	4	2	no	—
31	C₂×D₂₃	—	4	2	no	—
32	C₂×D₂₃	—	4	2	no	—
33	C₄×C₂₃	—	2	1	no	—
34	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₃	#41	1	1	yes	characteristic
36	Dic₂₃	—	4	2	no	—
37	#469	—	23	1	no	maximal, Sylow
38	C₂×C₂₃	—	4	1	no	—
39	C₂×C₂₃	#121	1	1	yes	characteristic
40	C₂×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
41	D₂₃	—	8	2	no	—
42	C₂×D₈	—	23	1	no	—
43	C₂×M₁₆	D₂₃	1	1	yes	characteristic, complemented
44	#121	—	23	2	no	—
45	#106	—	23	1	no	—
46	#102	—	23	2	no	—
47	C₂₃	#469	1	1	yes	characteristic, complemented, Sylow
48	C₂×D₄	—	23	2	no	—
49	C₂×D₄	—	46	2	no	—
50	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
51	D₈	—	46	2	no	—
52	M₁₆	—	2	2	no	—
53	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
54	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
55	C₄⋊C₄	—	23	2	no	—
56	C₂²⋊C₄	—	46	2	no	—
57	C₂³	—	46	2	no	—
58	C₂³	D₉₂	1	1	yes	characteristic
59	D₄	—	46	2	no	—
60	D₄	—	46	2	no	—
61	D₄	—	92	4	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	46	2	no	—
65	C₂×C₄	D₉₂	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	C₂²⋊D₂₃	1	2	yes	complemented
69	C₂²	—	2	2	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	92	2	no	—
72	C₂²	—	92	2	no	—
73	C₂²	—	92	2	no	—
74	C₂²	C₂×D₉₂	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	92	2	no	—
77	C₄	#67288	1	1	yes	characteristic
78	C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
79	C₂	—	4	1	no	—
80	C₂	—	184	2	no	—
81	C₂	#10625079	1	2	yes	—
82	C₂	#10625101	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂×D₄
hypercenter	C₂×M₁₆	D₂₃
hypocenter	C₂₃	#469
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×C₂₃	C₂	semidirect
2. #10625101	C₂	semidirect
3. C₂×D₁₈₄	C₂	semidirect
4. #10625003	C₂	semidirect
5. #10624999	C₂	semidirect
26. C₈×C₂₃	D₄	semidirect
43. C₂×M₁₆	D₂₃	semidirect
47. C₂₃	#469	semidirect
68. C₈	C₂²⋊D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₂₃	2
#10625101	2
C₂×D₁₈₄	2
#10625003	2
#10624999	2
#469	23