Subgroups — #13092187

Order 1472.

90 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×#67267	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	C₄⋊C₄×C₂₃	—	2	1	no	—
6	C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
7	C₄²×C₂₃	—	2	1	no	—
8	#67267	—	2	1	no	—
9	#67267	—	2	1	no	—
10	#67267	—	2	1	no	—
11	#67267	C₄	1	2	yes	complemented
12	C₂×C₂₃⋊C₈	—	2	1	no	—
13	C₂×C₂₃⋊C₈	C₄	1	2	yes	complemented
14	C₂³×C₂₃	C₂×C₄	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	D₄	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	C₂₃⋊C₈	—	2	2	no	—
23	C₂₃⋊C₈	—	2	2	no	—
24	C₂²×C₂₃	—	4	1	no	—
25	C₂²×C₂₃	C₄²	1	1	yes	characteristic
26	C₂²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₂₃	—	4	2	no	—
29	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
33	#344	—	23	1	no	maximal, Sylow
34	C₂×C₂₃	—	2	1	no	—
35	C₂×C₂₃	—	2	1	no	—
36	C₂×C₂₃	—	2	1	no	—
37	C₂×C₂₃	#95	1	1	yes	characteristic
38	C₂×M₁₆	—	23	1	no	—
39	C₂×M₁₆	—	23	1	no	—
40	#117	D₂₃	1	1	yes	characteristic
41	C₂₃	#344	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
43	M₁₆	—	23	2	no	—
44	M₁₆	—	46	1	no	—
45	M₁₆	—	46	1	no	—
46	M₁₆	—	46	1	no	—
47	C₂×C₈	—	23	2	no	—
48	C₂×C₈	—	46	1	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	Dic₂₃	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₂³	C₂×Dic₂₃	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	Dic₄₆	1	1	yes	characteristic
55	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
56	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
57	C₂×C₄	D₉₂	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
60	C₈	—	46	2	no	—
61	C₈	—	46	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₄×Dic₂₃	1	1	yes	characteristic
64	C₂²	C₄⋊Dic₂₃	1	1	yes	characteristic
65	C₂²	C₂²⋊Dic₂₃	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
68	C₄	Dic₂₃⋊C₄	1	1	yes	characteristic
69	C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
70	C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₂₃)⋊C₄
commutator	C₂²×C₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₃
Fitting	C₂₃×#117	C₂
socle	C₂×C₂₃	#95
prime radical	C₂³×C₂₃	C₂×C₄
cosocle	C₂²×C₄×C₂₃	C₂²
hypercenter	#117	D₂₃
hypocenter	C₂₃	#344
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
11. #67267	C₄	semidirect
13. C₂×C₂₃⋊C₈	C₄	semidirect
41. C₂₃	#344	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×#67267	2
#344	23