Subgroups — #13093028

Order 1472.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625117	C₂	1	2	yes	maximal
3	Q₈×Dic₂₃	C₂	1	2	yes	maximal, complemented
4	#10625069	C₂	1	4	yes	maximal
5	#10625060	C₂	1	4	yes	maximal, complemented
6	#10625043	C₂	1	1	yes	maximal, characteristic
7	#10625041	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×C₂₃	C₂²	1	2	yes	—
9	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
11	C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
13	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊Dic₂₃	C₂²	1	2	yes	—
15	C₄⋊Dic₂₃	C₂²	1	4	yes	—
16	Dic₂₃⋊C₄	C₂²	1	4	yes	—
17	Dic₂₃⋊C₄	C₂²	1	2	yes	—
18	Dic₂₃⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₃	C₂²	1	4	yes	—
21	C₂²×D₂₃	C₂³	1	1	yes	characteristic
22	Q₈×C₂₃	—	2	4	no	—
23	C₂×C₄×C₂₃	C₂³	1	4	yes	—
24	C₂×C₄×C₂₃	C₂³	1	2	yes	—
25	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₂₃	C₂³	1	2	yes	—
27	C₂×Dic₂₃	C₂³	1	4	yes	—
28	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
29	C₄×D₂₃	—	2	2	no	—
30	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
31	C₂×D₂₃	—	2	1	no	—
32	C₂×D₂₃	—	2	2	no	—
33	C₄×C₂₃	—	2	4	no	—
34	C₄×C₂₃	—	2	2	no	—
35	C₄×C₂₃	#41	1	2	yes	—
36	Dic₂₃	—	2	1	no	—
37	Dic₂₃	—	2	2	no	—
38	Dic₂₃	—	2	4	no	—
39	#541	—	23	1	no	maximal, Sylow
40	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
41	C₂×C₂₃	2₋¹⁺⁴	1	2	yes	—
42	D₂₃	—	4	1	no	—
43	C₄⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
44	#126	—	23	4	no	—
45	#125	—	23	4	no	—
46	#125	—	23	1	no	—
47	C₂²⋊Q₈	—	23	2	no	—
48	C₄×Q₈	—	23	2	no	—
49	#117	—	23	1	no	—
50	C₂₃	#541	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₈	C₂×D₂₃	1	2	yes	—
52	C₂²×C₄	—	23	1	no	—
53	C₄⋊C₄	—	23	2	no	—
54	C₄⋊C₄	—	23	2	no	—
55	C₄⋊C₄	—	23	4	no	—
56	C₄⋊C₄	—	23	4	no	—
57	C₄⋊C₄	—	23	4	no	—
58	C₄⋊C₄	C₂×D₂₃	1	4	yes	—
59	C₂²⋊C₄	—	23	2	no	—
60	C₂²⋊C₄	—	23	4	no	—
61	C₄²	—	23	1	no	—
62	C₄²	—	23	4	no	—
63	C₄²	C₂×D₂₃	1	1	yes	characteristic
64	C₂³	—	23	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	23	2	no	—
67	C₂×C₄	—	23	4	no	—
68	C₂×C₄	—	23	1	no	—
69	C₂×C₄	—	46	2	no	—
70	C₂×C₄	C₂²×D₂₃	1	4	yes	—
71	C₂×C₄	C₂²×D₂₃	1	2	yes	—
72	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
73	C₂²	—	46	1	no	—
74	C₂²	—	46	2	no	—
75	C₂²	C₂³×D₂₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	46	1	no	—
79	C₄	—	46	2	no	—
80	C₄	—	46	4	no	—
81	C₄	#67290	1	2	yes	—
82	C₂	—	92	1	no	—
83	C₂	C₂×#67290	1	1	yes	characteristic
84	C₂	#10625158	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×C₄⋊Q₈	C₂
prime radical	C₂²×D₂₃	C₂³
hypercenter	C₄⋊Q₈	D₂₃
hypocenter	C₂₃	#541
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂₃×C₄⋊Q₈	C₂	semidirect
3. Q₈×Dic₂₃	C₂	semidirect
5. #10625060	C₂	semidirect
7. #10625041	C₂	semidirect
43. C₄⋊Q₈	D₂₃	semidirect
50. C₂₃	#541	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×C₄⋊Q₈	2
#10625117	2
Q₈×Dic₂₃	2
#10625069	2
#10625060	2
#10625043	2
#10625041	2
#541	23