Subgroups — #13092970

Order 1472.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#117	C₂	1	1	yes	maximal, characteristic
2	#10625097	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
4	#10625061	C₂	1	2	yes	maximal, complemented
5	#10625059	C₂	1	2	yes	maximal, complemented
6	#10625049	C₂	1	4	yes	maximal
7	C₄×Dic₄₆	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₃	C₂²	1	2	yes	—
10	C₂×Dic₄₆	C₂²	1	4	yes	—
11	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₄²×C₂₃	C₂²	1	2	yes	—
14	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
15	C₄⋊Dic₂₃	C₂²	1	4	yes	—
16	C₄⋊Dic₂₃	C₂²	1	2	yes	—
17	Dic₂₃⋊C₄	C₂²	1	4	yes	—
18	Dic₂₃⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₂₃	C₂²	1	2	yes	—
20	C₄×Dic₂₃	C₂²	1	2	yes	—
21	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
22	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂³×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₃	Q₈	1	4	yes	complemented
25	C₂×C₄×C₂₃	C₂³	1	4	yes	—
26	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₂₃	—	2	4	no	—
29	C₂×Dic₂₃	C₂³	1	2	yes	—
30	C₂×Dic₂₃	C₂³	1	4	yes	—
31	C₂×Dic₂₃	C₂³	1	2	yes	—
32	Dic₄₆	—	2	8	no	—
33	C₂²×C₂₃	—	2	2	no	—
34	C₂²×C₂₃	C₂×Q₈	1	2	yes	—
35	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
36	C₄×C₂₃	—	2	4	no	—
37	C₄×C₂₃	C₂×Q₈	1	2	yes	—
38	C₄×C₂₃	C₂×Q₈	1	2	yes	—
39	Dic₂₃	—	2	4	no	—
40	Dic₂₃	—	2	2	no	—
41	Dic₂₃	#41	1	4	yes	—
42	#533	—	23	1	no	maximal, Sylow
43	C₂×C₂₃	—	2	2	no	—
44	C₂×C₂₃	C₂²×Q₈	1	1	yes	characteristic
45	C₂×C₂₃	C₂×#41	1	2	yes	—
46	C₄⋊Q₈	—	23	2	no	—
47	#125	—	23	2	no	—
48	C₂²⋊Q₈	—	23	4	no	—
49	C₄×Q₈	—	23	4	no	—
50	#117	—	23	1	no	—
51	#117	D₂₃	1	1	yes	characteristic
52	C₂×C₄²	—	23	1	no	—
53	C₂₃	#533	1	1	yes	characteristic, complemented, Sylow
54	C₂×Q₈	—	23	4	no	—
55	C₂²×C₄	—	23	2	no	—
56	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
57	C₄⋊C₄	—	23	2	no	—
58	C₄⋊C₄	—	23	4	no	—
59	C₄⋊C₄	—	23	4	no	—
60	C₄⋊C₄	—	23	4	no	—
61	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
62	C₂²⋊C₄	—	23	2	no	—
63	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
64	C₄²	—	23	1	no	—
65	C₄²	—	23	1	no	—
66	C₄²	—	23	2	no	—
67	C₄²	—	23	2	no	—
68	C₄²	C₂×D₂₃	1	2	yes	—
69	C₂³	C₂²×D₂₃	1	1	yes	characteristic
70	Q₈	—	46	8	no	—
71	C₂×C₄	—	23	2	no	—
72	C₂×C₄	—	23	2	no	—
73	C₂×C₄	—	23	4	no	—
74	C₂×C₄	—	46	4	no	—
75	C₂×C₄	Dic₄₆	1	4	yes	complemented
76	C₂×C₄	C₂²×D₂₃	1	4	yes	—
77	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	C₂×Dic₄₆	1	2	yes	—
81	C₂²	C₂³×D₂₃	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	23	4	no	—
84	C₄	—	46	2	no	—
85	C₄	—	46	4	no	—
86	C₄	C₂×Dic₄₆	1	2	yes	—
87	C₄	C₂×Dic₄₆	1	2	yes	—
88	C₂	—	2	2	no	—
89	C₂	C₂²×Dic₄₆	1	1	yes	characteristic
90	C₂	#41×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Frattini	C₂²	C₂³×D₂₃
Fitting	C₂₃×#117	C₂
prime radical	C₂³×C₂₃	C₂³
hypercenter	#117	D₂₃
hypocenter	C₂₃	#533
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #10625061	C₂	semidirect
5. #10625059	C₂	semidirect
7. C₄×Dic₄₆	C₂	semidirect
24. C₂×C₄×C₂₃	Q₈	semidirect
53. C₂₃	#533	semidirect
75. C₂×C₄	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#117	2
#10625097	2
C₂×C₄×Dic₂₃	2
#10625061	2
#10625059	2
#10625049	2
C₄×Dic₄₆	2
#533	23