Subgroups — #15673511

Order 2944.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625018	C₂	1	2	yes	maximal
2	#13092187	C₂	1	4	yes	maximal
3	C₂₃×#519	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
5	C₄×Q₈×C₂₃	—	2	4	no	—
6	C₂₃×#117	C₄	1	4	yes	—
7	C₂₃×#117	C₂²	1	2	yes	—
8	C₂×#67267	C₂²	1	4	yes	—
9	#10625018	—	2	4	no	—
10	C₂×Q₈×C₂₃	—	2	4	no	—
11	C₂×Q₈×C₂₃	D₄	1	4	yes	—
12	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₂₃	—	2	4	no	—
15	C₄⋊C₄×C₂₃	—	2	2	no	—
16	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
17	C₄²×C₂₃	—	2	4	no	—
18	C₄²×C₂₃	—	2	2	no	—
19	#67267	—	2	4	no	—
20	#67267	—	2	4	no	—
21	C₂×C₂₃⋊C₈	—	2	4	no	—
22	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
23	Q₈×C₂₃	—	2	8	no	—
24	C₂×C₄×C₂₃	—	2	4	no	—
25	C₂×C₄×C₂₃	—	2	4	no	—
26	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
27	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₂₃	#41	1	4	yes	—
29	C₂₃⋊C₈	—	4	4	no	—
30	#1830	—	23	1	no	maximal, Sylow
31	C₂²×C₂₃	—	4	1	no	—
32	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×C₂₃	#117	1	2	yes	—
34	C₄×C₂₃	—	2	4	no	—
35	C₄×C₂₃	—	4	4	no	—
36	C₄×C₂₃	#123	1	4	yes	—
37	#519	D₂₃	1	1	yes	characteristic
38	C₂×#101	—	23	2	no	—
39	#344	—	23	4	no	—
40	C₂×C₂₃	—	2	1	no	—
41	C₂×C₂₃	—	2	2	no	—
42	C₂×C₂₃	#381	1	1	yes	characteristic
43	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
44	C₂×M₁₆	—	23	4	no	—
45	C₄×Q₈	—	2	4	no	—
46	#117	Dic₂₃	1	4	yes	—
47	#117	C₂×D₂₃	1	2	yes	—
48	#101	—	46	4	no	—
49	C₂₃	#1830	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	C₂²⋊D₂₃	1	4	yes	—
52	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
53	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
54	M₁₆	—	46	4	no	—
55	M₁₆	—	46	4	no	—
56	C₂×C₈	—	46	4	no	—
57	C₄⋊C₄	—	2	4	no	—
58	C₄⋊C₄	—	2	2	no	—
59	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
60	C₄²	—	2	4	no	—
61	C₄²	—	2	2	no	—
62	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
63	Q₈	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
67	C₂×C₄	#67288	1	4	yes	—
68	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
69	C₈	—	92	4	no	—
70	C₂²	—	4	1	no	—
71	C₂²	#10625097	1	2	yes	—
72	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
73	C₄	—	2	4	no	—
74	C₄	—	4	4	no	—
75	C₄	#10625100	1	4	yes	—
76	C₂	—	2	1	no	—
77	C₂	—	2	2	no	—
78	C₂	#13092280	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092280
commutator, prime radical	C₂³×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#519	C₂
socle	C₂×C₂₃	#381
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#519	D₂₃
hypocenter	C₂₃	#1830
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #519

Complements

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

N	H ≅ G/N	type
49. C₂₃	#1830	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625018	2
#13092187	2
C₂₃×#519	2
#1830	23