Subgroups — #15671540

Order 2944.

72 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	C₂×C₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625072	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	#10624994	C₄	1	1	yes	characteristic
7	#10624980	—	2	1	no	—
8	C₁₆×D₂₃	—	2	1	no	—
9	C₂₃×C₃₂	—	2	1	no	—
10	C₂₃⋊C₃₂	—	2	1	no	—
11	#67288	C₈	1	1	yes	characteristic
12	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	#67267	C₈	1	1	yes	characteristic
14	#67262	—	2	1	no	—
15	C₈×D₂₃	—	2	1	no	—
16	C₁₆×C₂₃	D₄	1	1	yes	characteristic
17	C₁₆×C₂₃	D₄	1	1	yes	characteristic
18	C₂₃⋊C₁₆	—	2	1	no	—
19	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
20	C₂²⋊D₂₃	—	2	1	no	—
21	D₉₂	C₁₆	1	1	yes	characteristic
22	C₄×D₂₃	—	2	1	no	—
23	Dic₄₆	C₁₆	1	1	yes	characteristic
24	C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₂₃	M₁₆	1	1	yes	characteristic
26	C₂₃⋊C₈	—	2	1	no	—
27	#1401	—	23	1	no	maximal, Sylow
28	C₂²×C₂₃	M₃₂	1	1	yes	characteristic
29	C₂×D₂₃	—	2	1	no	—
30	C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
32	Dic₂₃	—	2	1	no	—
33	#504	—	23	1	no	—
34	M₆₄	D₂₃	1	1	yes	characteristic, complemented
35	C₂×C₃₂	—	23	1	no	—
36	C₂×C₂₃	—	2	1	no	—
37	C₂×C₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
38	D₂₃	—	4	1	no	—
39	#131	—	23	1	no	—
40	M₃₂	—	23	1	no	—
41	M₃₂	—	46	1	no	—
42	C₂×C₁₆	—	46	1	no	—
43	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	—	46	1	no	—
46	C₂₃	#1401	1	1	yes	characteristic, complemented, Sylow
47	#41	—	23	1	no	—
48	M₁₆	—	23	1	no	—
49	M₁₆	—	46	1	no	—
50	C₂×C₈	—	46	1	no	—
51	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
52	C₁₆	—	46	1	no	—
53	C₁₆	D₉₂	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
55	Q₈	—	23	1	no	—
56	D₄	—	23	1	no	—
57	D₄	—	46	1	no	—
58	C₂×C₄	—	46	1	no	—
59	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
60	C₈	—	46	1	no	—
61	C₈	#67262	1	1	yes	characteristic
62	C₈	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
63	C₂²	—	46	1	no	—
64	C₂²	#10624980	1	1	yes	characteristic
65	C₄	—	46	1	no	—
66	C₄	C₁₆×D₂₃	1	1	yes	characteristic
67	C₄	(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	92	1	no	—
70	C₂	(C₂×D₂₃)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₂₃
Fitting	C₂₃×M₆₄	C₂
socle	C₂×C₂₃	C₂²⋊C₁₆
prime radical	#67288	C₈
cosocle	C₂×C₁₆×C₂₃	C₂²
hypercenter	M₆₄	D₂₃
hypocenter	C₂₃	#1401
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃⋊C₃₂	C₂	semidirect
3. C₂₃×M₆₄	C₂	semidirect
34. M₆₄	D₂₃	semidirect
46. C₂₃	#1401	semidirect

Maximal subgroups

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

subgroup	index
#13092833	2
C₂×C₂₃⋊C₃₂	2
C₂₃×M₆₄	2
#1401	23