Subgroups — #13092176

Order 1472.

110 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	C₂×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#67267	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×Dic₂₃	—	2	1	no	—
6	C₄⋊C₄×C₂₃	C₄	1	2	yes	complemented
7	C₂²⋊C₄×C₂₃	—	2	1	no	—
8	C₄²×C₂₃	—	2	1	no	—
9	C₄×Dic₂₃	—	2	1	no	—
10	C₄×Dic₂₃	C₄	1	2	yes	complemented
11	#67267	—	2	1	no	—
12	#67267	C₄	1	2	yes	complemented
13	C₂×C₂₃⋊C₈	—	2	1	no	—
14	C₂³×C₂₃	D₄	1	1	yes	characteristic
15	C₂×C₄×C₂₃	—	2	2	no	—
16	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	Q₈	1	1	yes	characteristic
22	C₂×Dic₂₃	—	2	4	no	—
23	C₂×Dic₂₃	—	2	2	no	—
24	C₂₃⋊C₈	—	2	2	no	—
25	C₂²×C₂₃	—	2	2	no	—
26	C₂²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₂₃	—	4	2	no	—
30	C₄×C₂₃	C₄²	1	1	yes	characteristic
31	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
34	Dic₂₃	—	2	4	no	—
35	#339	—	23	1	no	maximal, Sylow
36	C₂×C₂₃	—	2	1	no	—
37	C₂×C₂₃	—	2	1	no	—
38	C₂×C₂₃	#95	1	1	yes	characteristic
39	C₂×C₂₃	#104	1	2	yes	—
40	C₂×M₁₆	—	23	1	no	—
41	#117	D₂₃	1	1	yes	characteristic
42	C₂×C₄²	—	23	1	no	—
43	C₂₃	#339	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₄	—	46	1	no	—
45	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
46	M₁₆	—	23	2	no	—
47	M₁₆	—	46	1	no	—
48	C₂×C₈	—	46	1	no	—
49	C₄⋊C₄	Dic₂₃	1	2	yes	complemented
50	C₂²⋊C₄	—	2	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	23	2	no	—
53	C₄²	—	46	1	no	—
54	C₂³	C₂²⋊D₂₃	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	46	4	no	—
57	C₂×C₄	—	46	2	no	—
58	C₂×C₄	Dic₄₆	1	1	yes	characteristic
59	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
60	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
61	C₂×C₄	D₉₂	1	1	yes	characteristic
62	C₂×C₄	C₂×Dic₂₃	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
64	C₈	—	46	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	Dic₂₃⋊C₄	1	1	yes	characteristic
67	C₂²	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
68	C₂²	C₂²⋊Dic₂₃	1	1	yes	characteristic
69	C₄	—	4	2	no	—
70	C₄	—	46	4	no	—
71	C₄	C₄×Dic₂₃	1	1	yes	characteristic
72	C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
73	C₄	C₄⋊Dic₂₃	1	1	yes	characteristic
74	C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	C₂₃⋊#95	1	1	yes	characteristic
78	C₂	#10625019	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₃
commutator	C₄×C₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₃
Fitting	C₂₃×#117	C₂
socle	C₂²×C₂₃	C₂²⋊C₄
prime radical	C₂³×C₂₃	D₄
cosocle	C₂²×C₄×C₂₃	C₂²
hypercenter	#117	D₂₃
hypocenter	C₂₃	#339
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
6. C₄⋊C₄×C₂₃	C₄	semidirect
10. C₄×Dic₂₃	C₄	semidirect
12. #67267	C₄	semidirect
43. C₂₃	#339	semidirect
49. C₄⋊C₄	Dic₂₃	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
C₂×C₄×Dic₂₃	2
C₂×#67267	2
#339	23