Subgroups — #15673004

Order 2944.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092794	C₂	1	1	yes	maximal, characteristic
2	#13092765	C₂	1	4	yes	maximal, complemented
3	C₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₂₃	—	2	2	no	—
6	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×D₁₈₄	—	2	2	no	—
8	#10625045	C₂²	1	2	yes	—
9	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	#10624991	—	2	8	no	—
11	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
12	C₂×Q₈×C₂₃	—	2	2	no	—
13	C₂×D₉₂	—	2	2	no	—
14	C₂×D₉₂	—	2	4	no	—
15	Q₁₆×C₂₃	—	4	4	no	—
16	C₂×C₈×C₂₃	D₄	1	2	yes	—
17	C₄⋊C₄×C₂₃	—	2	4	no	—
18	C₄²×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
20	D₁₈₄	—	4	4	no	—
21	C₂²×D₂₃	—	4	2	no	—
22	Q₈×C₂₃	—	4	4	no	—
23	C₂×C₄×C₂₃	—	4	2	no	—
24	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
26	D₉₂	—	4	4	no	—
27	D₉₂	—	4	8	no	—
28	C₈×C₂₃	D₈	1	4	yes	—
29	C₂₃⋊C₈	—	2	4	no	—
30	#1709	—	23	1	no	maximal, Sylow
31	C₂²×C₂₃	#127	1	1	yes	characteristic
32	C₂×D₂₃	—	8	2	no	—
33	C₂×D₂₃	—	8	4	no	—
34	C₄×C₂₃	—	8	2	no	—
35	C₄×C₂₃	C₂×D₈	1	2	yes	—
36	C₄×C₂₃	#135	1	4	yes	—
37	#494	D₂₃	1	1	yes	characteristic, complemented
38	#493	—	23	1	no	—
39	#486	—	23	4	no	—
40	C₈²	—	23	1	no	—
41	C₂×C₂₃	#493	1	1	yes	characteristic
42	C₂×C₂₃	#495	1	2	yes	—
43	D₂₃	—	16	2	no	—
44	C₂×Q₁₆	—	2	2	no	—
45	C₂×D₈	—	46	2	no	—
46	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
47	#127	—	23	2	no	—
48	#102	—	46	8	no	—
49	C₄×C₈	—	23	2	no	—
50	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
51	C₂₃	#1709	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	46	4	no	—
54	C₂×D₄	—	46	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	92	4	no	—
57	C₂×C₈	—	23	4	no	—
58	C₂×C₈	C₂²⋊D₂₃	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₂₃	1	1	yes	characteristic
61	C₂³	—	92	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	92	8	no	—
64	D₄	—	92	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₂₃	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
68	C₈	—	46	4	no	—
69	C₈	#67272	1	4	yes	—
70	C₂²	—	184	2	no	—
71	C₂²	—	184	4	no	—
72	C₂²	#10625111	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	#10625090	1	4	yes	—
75	C₄	C₂×#67272	1	2	yes	—
76	C₂	—	368	2	no	—
77	C₂	#13092793	1	1	yes	characteristic
78	C₂	#13092802	1	2	yes	—

	subgroup	G/H
center	C₂²	#10625111
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#494	C₂
socle	C₂²×C₂₃	#127
prime radical	#13092794	C₂
hypercenter	#494	D₂₃
hypocenter	C₂₃	#1709
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #13092765	C₂	semidirect
3. C₈×C₂₃⋊C₈	C₂	semidirect
4. C₂₃×#494	C₂	semidirect
37. #494	D₂₃	semidirect
51. C₂₃	#1709	semidirect

Maximal subgroups

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

subgroup	index
#13092794	2
#13092765	2
C₈×C₂₃⋊C₈	2
C₂₃×#494	2
#1709	23