Subgroups — #15673014

Order 2944.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092822	C₂	1	1	yes	maximal, characteristic
2	#13092820	C₂	1	2	yes	maximal
3	#13092796	C₂	1	2	yes	maximal
4	C₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#500	C₂	1	1	yes	maximal, characteristic
6	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#67275	—	2	4	no	—
8	#10625040	C₂²	1	2	yes	—
9	C₂₃×#107	—	2	2	no	—
10	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	#10625000	—	2	2	no	—
12	#10624989	—	2	4	no	—
13	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₂₃	—	2	2	no	—
16	C₂×Dic₄₆	—	2	2	no	—
17	C₂×C₈×C₂₃	D₄	1	2	yes	—
18	C₄⋊C₄×C₂₃	—	2	2	no	—
19	C₄⋊C₄×C₂₃	—	2	2	no	—
20	C₄²×C₂₃	C₂³	1	1	yes	characteristic
21	#67275	—	4	8	no	—
22	C₄⋊Dic₂₃	—	2	2	no	—
23	C₄⋊Dic₂₃	—	2	2	no	—
24	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
25	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
26	Q₈×C₂₃	—	4	4	no	—
27	C₂×C₄×C₂₃	—	4	2	no	—
28	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₂₃	—	4	2	no	—
32	Dic₄₆	—	4	4	no	—
33	C₈×C₂₃	Q₁₆	1	4	yes	complemented
34	C₂₃⋊C₈	D₈	1	4	yes	—
35	C₂₃⋊C₈	Q₁₆	1	4	yes	complemented
36	#1713	—	23	1	no	maximal, Sylow
37	C₂²×C₂₃	#127	1	1	yes	characteristic
38	C₄×C₂₃	—	8	2	no	—
39	C₄×C₂₃	C₂×D₈	1	2	yes	—
40	C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
41	C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
42	Dic₂₃	—	8	2	no	—
43	#500	—	23	2	no	—
44	#500	—	23	1	no	—
45	#500	D₂₃	1	1	yes	characteristic
46	#494	—	23	2	no	—
47	C₈²	—	23	1	no	—
48	C₂×C₂₃	#493	1	1	yes	characteristic
49	C₂×C₂₃	#494	1	1	yes	characteristic
50	C₂×C₂₃	#494	1	1	yes	characteristic
51	C₂×Q₁₆	—	46	4	no	—
52	C₄⋊Q₈	—	23	2	no	—
53	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
54	#107	—	2	2	no	—
55	#107	—	46	4	no	—
56	#107	—	46	2	no	—
57	C₄×C₈	—	23	1	no	—
58	C₄×C₈	—	23	1	no	—
59	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
60	C₂₃	#1713	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	46	2	no	—
63	Q₁₆	—	92	8	no	—
64	C₂×C₈	—	23	2	no	—
65	C₂×C₈	—	23	2	no	—
66	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	46	2	no	—
70	C₄⋊C₄	—	46	2	no	—
71	C₄²	C₂²×D₂₃	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	92	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	92	2	no	—
76	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
77	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
79	C₈	—	23	4	no	—
80	C₈	—	23	4	no	—
81	C₈	#67275	1	4	yes	complemented
82	C₂²	#10625111	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	184	2	no	—
85	C₄	D₈×D₂₃	1	2	yes	—
86	C₄	Q₁₆×D₂₃	1	2	yes	—
87	C₄	C₂×#67275	1	2	yes	—
88	C₂	#13092792	1	1	yes	characteristic
89	C₂	#13092795	1	1	yes	characteristic
90	C₂	#13092796	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625111
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#500	C₂
socle, prime radical	C₂²×C₂₃	#127
hypercenter	#500	D₂₃
hypocenter	C₂₃	#1713
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
33. C₈×C₂₃	Q₁₆	semidirect
35. C₂₃⋊C₈	Q₁₆	semidirect
60. C₂₃	#1713	semidirect
81. C₈	#67275	semidirect

Maximal subgroups

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

subgroup	index
#13092822	2
#13092820	2
#13092796	2
C₈×C₂₃⋊C₈	2
C₂₃×#500	2
#1713	23