Subgroups — #13092475

Order 1472.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10625097	C₂	1	1	yes	maximal, characteristic
2	C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#10624997	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₂₃	C₄	1	1	yes	characteristic
12	C₂²⋊Dic₂₃	C₄	1	1	yes	characteristic
13	C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
14	C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
15	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂³×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	—	2	1	no	—
21	C₂×C₄×C₂₃	—	2	1	no	—
22	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₂₃	C₂×C₄	1	1	yes	characteristic
28	C₈×C₂₃	—	2	1	no	—
29	C₈×C₂₃	—	2	1	no	—
30	C₂₃⋊C₈	—	2	1	no	—
31	C₂₃⋊C₈	—	2	1	no	—
32	C₂²×C₂₃	—	2	1	no	—
33	C₂²×C₂₃	—	2	1	no	—
34	C₂²×C₂₃	—	2	1	no	—
35	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₂₃	—	2	1	no	—
37	C₄×C₂₃	#41	1	1	yes	characteristic
38	C₄×C₂₃	#41	1	1	yes	characteristic
39	Dic₂₃	—	2	1	no	—
40	Dic₂₃	—	2	1	no	—
41	Dic₂₃	—	2	1	no	—
42	Dic₂₃	—	2	1	no	—
43	#433	—	23	1	no	maximal, Sylow
44	C₂×C₂₃	—	4	1	no	—
45	C₂×C₂₃	#117	1	1	yes	characteristic
46	C₂×C₂₃	#131	1	1	yes	characteristic
47	C₂×C₂₃	#131	1	1	yes	characteristic
48	#117	—	23	1	no	—
49	C₄⋊C₈	—	23	1	no	—
50	C₄⋊C₈	—	23	1	no	—
51	C₂²⋊C₈	—	23	1	no	—
52	C₂²⋊C₈	D₂₃	1	1	yes	characteristic
53	#97	—	23	1	no	—
54	C₄×C₈	—	23	1	no	—
55	C₂₃	#433	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
57	C₂×C₈	—	23	1	no	—
58	C₂×C₈	—	23	1	no	—
59	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
60	C₂×C₈	C₂×D₂₃	1	1	yes	characteristic
61	C₄⋊C₄	—	23	1	no	—
62	C₄⋊C₄	—	23	1	no	—
63	C₂²⋊C₄	—	23	1	no	—
64	C₂²⋊C₄	—	23	1	no	—
65	C₄²	—	23	1	no	—
66	C₄²	—	23	1	no	—
67	C₂³	C₄×D₂₃	1	1	yes	characteristic
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	23	1	no	—
71	C₂×C₄	—	23	1	no	—
72	C₂×C₄	—	23	1	no	—
73	C₂×C₄	—	23	1	no	—
74	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₈	—	46	1	no	—
79	C₈	—	46	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	C₂×C₄×D₂₃	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	46	1	no	—
86	C₄	—	46	1	no	—
87	C₄	—	46	1	no	—
88	C₄	—	46	1	no	—
89	C₄	#67290	1	1	yes	characteristic
90	C₄	#67290	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	#10625048	1	1	yes	characteristic
93	C₂	#10625072	1	1	yes	characteristic
94	C₂	#10625078	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting	C₂₃×C₂²⋊C₈	C₂
prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₂²⋊C₈	D₂₃
hypocenter	C₂₃	#433
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. #10624997	C₂	semidirect
5. Dic₂₃⋊C₈	C₂	semidirect
6. Dic₂₃⋊C₈	C₂	semidirect
7. C₈×Dic₂₃	C₂	semidirect
55. C₂₃	#433	semidirect

Maximal subgroups

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

subgroup	index
#10625097	2
C₂₃×C₂²⋊C₈	2
C₂²⋊(C₂₃⋊C₈)	2
#10624997	2
Dic₂₃⋊C₈	2
C₈×Dic₂₃	2
#433	23