Subgroups — #15671446

Order 2944.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#13092170	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#494	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₂₃	—	2	1	no	—
5	C₂×Q₁₆×C₂₃	C₄	1	2	yes	—
6	C₂₃×C₄⋊Q₈	—	2	1	no	—
7	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃⋊C₁₆	—	2	1	no	—
9	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
10	C₂×Q₈×C₂₃	—	2	2	no	—
11	Q₁₆×C₂₃	—	4	2	no	—
12	Q₁₆×C₂₃	—	4	1	no	—
13	Q₁₆×C₂₃	—	4	1	no	—
14	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
16	C₄⋊C₄×C₂₃	—	4	1	no	—
17	C₄⋊C₄×C₂₃	—	4	1	no	—
18	C₄²×C₂₃	D₄	1	1	yes	characteristic
19	C₂×C₂₃⋊C₈	—	4	1	no	—
20	C₂₃⋊C₁₆	—	4	1	no	—
21	Q₈×C₂₃	—	4	2	no	—
22	Q₈×C₂₃	—	4	2	no	—
23	C₂×C₄×C₂₃	—	4	2	no	—
24	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
26	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
27	C₈×C₂₃	—	2	1	no	—
28	C₈×C₂₃	D₈	1	1	yes	characteristic
29	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
30	C₂₃⋊C₈	—	8	1	no	—
31	#1362	—	23	1	no	maximal, Sylow
32	C₂²×C₂₃	#102	1	1	yes	characteristic
33	C₄×C₂₃	—	2	1	no	—
34	C₄×C₂₃	—	8	2	no	—
35	C₄×C₂₃	#100	1	1	yes	characteristic
36	C₄×C₂₃	#102	1	1	yes	characteristic
37	C₄×C₂₃	SD₃₂	1	1	yes	characteristic
38	C₄×C₂₃	Q₃₂	1	1	yes	characteristic
39	#494	D₂₃	1	1	yes	characteristic
40	C₄⋊C₁₆	—	23	1	no	—
41	#335	—	23	1	no	—
42	C₂×C₂₃	#331	1	1	yes	characteristic
43	C₂×C₂₃	#358	1	1	yes	characteristic
44	C₂×C₂₃	#362	1	1	yes	characteristic
45	C₂×Q₁₆	—	2	1	no	—
46	C₂×Q₁₆	Dic₂₃	1	2	yes	—
47	C₄⋊Q₈	—	2	1	no	—
48	C₂×C₁₆	—	46	1	no	—
49	C₄⋊C₈	—	46	1	no	—
50	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
51	C₂₃	#1362	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₈	—	2	2	no	—
53	Q₁₆	—	4	2	no	—
54	Q₁₆	—	4	1	no	—
55	Q₁₆	—	4	1	no	—
56	C₂×C₈	—	92	1	no	—
57	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
62	C₁₆	—	92	1	no	—
63	Q₈	—	4	2	no	—
64	Q₈	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	#67272	1	1	yes	characteristic
67	C₂×C₄	#67273	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	184	1	no	—
71	C₈	#67272	1	1	yes	characteristic
72	C₈	#67273	1	1	yes	characteristic
73	C₂²	#10625014	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	8	2	no	—
76	C₄	#10625010	1	1	yes	characteristic
77	C₄	#10625011	1	1	yes	characteristic
78	C₄	#10625014	1	1	yes	characteristic
79	C₄	#10625015	1	1	yes	characteristic
80	C₂	#13092161	1	1	yes	characteristic
81	C₂	#13092221	1	1	yes	characteristic
82	C₂	#13092233	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625014
commutator	C₂×C₈×C₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#494	C₂
socle, prime radical	C₂²×C₂₃	#102
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#494	D₂₃
hypocenter	C₂₃	#1362
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
51. C₂₃	#1362	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₂₃⋊C₁₆)	2
#13092170	2
C₂₃×#494	2
#1362	23