Subgroups — #15675692

Order 2944.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092833	C₂	1	1	yes	maximal, characteristic
2	#13092250	C₂	1	2	yes	maximal, complemented
3	C₂×C₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₂₃	C₂	1	2	yes	maximal, complemented
5	C₂₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625072	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	#10624994	C₄	1	1	yes	characteristic
9	#10624980	C₄	1	2	yes	—
10	C₁₆×D₂₃	C₂²	1	2	yes	—
11	C₂₃×C₃₂	C₂²	1	2	yes	—
12	C₂₃⋊C₃₂	C₂²	1	1	yes	characteristic
13	C₂₃⋊C₃₂	C₂²	1	1	yes	characteristic
14	#67288	C₈	1	1	yes	characteristic
15	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	#67267	C₈	1	1	yes	characteristic
17	#67262	C₈	1	2	yes	—
18	C₈×D₂₃	C₂×C₄	1	2	yes	—
19	C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
23	C₂²⋊D₂₃	C₁₆	1	2	yes	—
24	D₉₂	C₁₆	1	1	yes	characteristic
25	C₄×D₂₃	C₂×C₈	1	2	yes	—
26	Dic₄₆	C₁₆	1	1	yes	characteristic
27	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
28	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂₃⋊C₈	C₂×C₈	1	2	yes	—
30	#2258	—	23	1	no	maximal, Sylow
31	C₂²×C₂₃	C₂×C₁₆	1	1	yes	characteristic
32	C₂×D₂₃	C₂×C₁₆	1	2	yes	—
33	C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
34	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
35	Dic₂₃	C₂×C₁₆	1	2	yes	—
36	#504	—	23	1	no	—
37	M₆₄	—	23	2	no	—
38	M₆₄	D₂₃	1	1	yes	characteristic, complemented
39	C₂×C₃₂	—	23	1	no	—
40	C₂×C₃₂	—	23	2	no	—
41	C₂×C₂₃	—	2	1	no	—
42	C₂×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
43	D₂₃	—	2	2	no	—
44	#131	—	23	1	no	—
45	M₃₂	—	23	1	no	—
46	M₃₂	—	23	2	no	—
47	C₂×C₁₆	—	23	2	no	—
48	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
49	C₃₂	—	23	1	no	—
50	C₃₂	—	23	1	no	—
51	C₃₂	C₂×D₂₃	1	2	yes	—
52	C₂₃	#2258	1	1	yes	characteristic, complemented, Sylow
53	#41	—	23	1	no	—
54	M₁₆	—	23	1	no	—
55	M₁₆	—	23	2	no	—
56	C₂×C₈	—	23	2	no	—
57	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
58	C₁₆	—	23	2	no	—
59	C₁₆	C₄×D₂₃	1	1	yes	characteristic
60	C₁₆	C₂²×D₂₃	1	1	yes	characteristic
61	Q₈	—	23	1	no	—
62	D₄	—	23	2	no	—
63	D₄	—	23	1	no	—
64	C₂×C₄	—	23	2	no	—
65	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
66	C₈	—	23	2	no	—
67	C₈	C₈×D₂₃	1	1	yes	characteristic
68	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
69	C₂²	—	23	2	no	—
70	C₂²	C₁₆×D₂₃	1	1	yes	characteristic
71	C₄	—	23	2	no	—
72	C₄	C₁₆×D₂₃	1	1	yes	characteristic
73	C₄	C₂×C₈×D₂₃	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	46	2	no	—
76	C₂	C₂×C₁₆×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₂₃
commutator, socle	C₂×C₂₃	C₂²×C₁₆
Fitting	C₂₃×M₆₄	C₂
prime radical	#67288	C₈
cosocle	C₁₆×C₂₃	C₂³
hypercenter	M₆₄	D₂₃
hypocenter	C₂₃	#2258
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #13092250	C₂	semidirect
3. C₂×C₂₃⋊C₃₂	C₂	semidirect
4. C₃₂×D₂₃	C₂	semidirect
5. C₂₃×M₆₄	C₂	semidirect
38. M₆₄	D₂₃	semidirect
52. C₂₃	#2258	semidirect

Maximal subgroups

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

subgroup	index
#13092833	2
#13092250	2
C₂×C₂₃⋊C₃₂	2
C₃₂×D₂₃	2
C₂₃×M₆₄	2
#2258	23