Subgroups — #15675677

Order 2944.

128 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	#13092244	C₂	1	4	yes	maximal
3	C₄×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂₃×C₄⋊Q₈	—	2	1	no	—
6	#10625040	—	2	1	no	—
7	C₂₃×#107	—	2	1	no	—
8	C₂₃×#107	C₂²	1	2	yes	—
9	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	#10625000	—	2	1	no	—
11	#10625000	C₂²	1	2	yes	—
12	C₂×C₂₃⋊C₁₆	C₂²	1	2	yes	—
13	C₂×Q₈×C₂₃	—	4	1	no	—
14	C₂×Dic₄₆	—	4	1	no	—
15	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₂₃	—	2	2	no	—
18	C₄⋊C₄×C₂₃	—	4	1	no	—
19	C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₄⋊Dic₂₃	—	2	2	no	—
21	C₄⋊Dic₂₃	—	4	1	no	—
22	C₂₃⋊C₁₆	Q₈	1	4	yes	complemented
23	Q₈×C₂₃	—	8	2	no	—
24	C₂×C₄×C₂₃	—	4	2	no	—
25	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
26	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₂₃	—	4	2	no	—
29	Dic₄₆	—	8	2	no	—
30	C₈×C₂₃	Q₁₆	1	2	yes	—
31	C₈×C₂₃	C₂×Q₈	1	2	yes	—
32	#2254	—	23	1	no	maximal, Sylow
33	C₂²×C₂₃	C₂×D₈	1	1	yes	characteristic
34	C₄×C₂₃	—	8	2	no	—
35	C₄×C₂₃	SD₃₂	1	2	yes	—
36	C₄×C₂₃	SD₃₂	1	2	yes	—
37	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
38	C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
39	Dic₂₃	—	8	2	no	—
40	#500	—	23	1	no	—
41	#500	D₂₃	1	1	yes	characteristic
42	#367	—	23	4	no	—
43	C₄×C₁₆	—	23	1	no	—
44	C₂×C₂₃	#500	1	1	yes	characteristic
45	C₂×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
46	C₂×C₂₃	C₂×SD₃₂	1	1	yes	characteristic
47	C₄⋊Q₈	—	2	1	no	—
48	C₄⋊Q₈	—	46	1	no	—
49	C₂×C₁₆	—	23	2	no	—
50	#107	—	2	1	no	—
51	#107	—	23	2	no	—
52	#107	—	46	1	no	—
53	#107	C₂×D₂₃	1	2	yes	—
54	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
55	C₂₃	#2254	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	92	1	no	—
58	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	46	2	no	—
63	C₄⋊C₄	—	92	1	no	—
64	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
65	C₁₆	—	23	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	184	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	92	2	no	—
70	C₂×C₄	#67272	1	1	yes	characteristic
71	C₂×C₄	#67272	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
73	C₈	#67275	1	2	yes	—
74	C₈	Q₈×D₂₃	1	2	yes	—
75	C₂²	C₂×#67272	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	184	2	no	—
78	C₄	#10625009	1	2	yes	—
79	C₄	#10625010	1	2	yes	—
80	C₄	C₂×#67275	1	1	yes	characteristic
81	C₄	#10625115	1	1	yes	characteristic
82	C₂	#13092820	1	1	yes	characteristic
83	C₂	C₂×#10625010	1	1	yes	characteristic
84	C₂	C₂×#10625009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67272
commutator, cosocle	C₂×C₈×C₂₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#500	C₂
socle, prime radical	C₂²×C₂₃	C₂×D₈
hypercenter	#500	D₂₃
hypocenter	C₂₃	#2254
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
22. C₂₃⋊C₁₆	Q₈	semidirect
55. C₂₃	#2254	semidirect

Maximal subgroups

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

subgroup	index
#13092822	2
#13092244	2
C₄×C₂₃⋊C₁₆	2
C₂₃×#500	2
#2254	23