Subgroups — #15673143

Order 2944.

296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₃⋊C₈	C₂	1	2	yes	maximal
2	#13092172	C₂	1	4	yes	maximal
3	C₄³×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄²×C₂₃	C₄	1	2	yes	—
5	C₂×C₄²×C₂₃	C₄	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₂₃⋊C₈	C₂²	1	4	yes	—
9	C₄×C₂₃⋊C₈	C₄	1	8	yes	complemented
10	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
11	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
14	C₄²×C₂₃	—	2	4	no	—
15	C₄²×C₂₃	—	2	4	no	—
16	C₄²×C₂₃	C₈	1	8	yes	complemented
17	C₄²×C₂₃	C₂×C₄	1	4	yes	—
18	C₂×C₂₃⋊C₈	—	2	8	no	—
19	C₂×C₂₃⋊C₈	C₂×C₄	1	8	yes	—
20	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	—	2	4	no	—
22	C₂×C₄×C₂₃	—	2	4	no	—
23	C₂×C₄×C₂₃	C₄²	1	2	yes	—
24	C₂×C₄×C₂₃	C₄²	1	2	yes	—
25	C₂×C₄×C₂₃	C₂×C₈	1	8	yes	—
26	C₂×C₄×C₂₃	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₂₃	M₁₆	1	4	yes	—
28	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
29	C₂×C₄×C₂₃	#41	1	4	yes	—
30	C₂₃⋊C₈	—	2	8	no	—
31	#1744	—	23	1	no	maximal, Sylow
32	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₂₃	#117	1	2	yes	—
34	C₂²×C₂₃	C₂²×C₈	1	2	yes	—
35	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
36	C₄×C₂₃	—	2	8	no	—
37	C₄×C₂₃	C₄×C₈	1	4	yes	—
38	C₄×C₂₃	#97	1	4	yes	—
39	C₄×C₂₃	#117	1	4	yes	—
40	C₂×C₄×C₈	—	23	2	no	—
41	C₄³	D₂₃	1	1	yes	characteristic
42	#336	—	23	4	no	—
43	C₂×C₂₃	#376	1	1	yes	characteristic
44	C₂×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
45	C₂×C₂₃	C₂×#97	1	1	yes	characteristic
46	C₂×C₂₃	#431	1	4	yes	—
47	C₂²×C₈	—	23	4	no	—
48	C₂×C₄²	Dic₂₃	1	2	yes	—
49	C₂×C₄²	Dic₂₃	1	2	yes	—
50	C₂×C₄²	C₂×D₂₃	1	2	yes	—
51	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
52	C₄×C₈	—	23	8	no	—
53	C₂₃	#1744	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
55	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
57	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
58	C₂×C₈	—	23	8	no	—
59	C₂×C₈	—	46	8	no	—
60	C₄²	—	2	4	no	—
61	C₄²	—	2	4	no	—
62	C₄²	C₂₃⋊C₈	1	8	yes	complemented
63	C₄²	C₄×D₂₃	1	4	yes	—
64	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₂×C₂₃⋊C₈	1	8	yes	—
68	C₂×C₄	C₂×C₂₃⋊C₈	1	4	yes	—
69	C₂×C₄	#67267	1	4	yes	—
70	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
71	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
72	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
73	C₂×C₄	#67288	1	4	yes	—
74	C₈	—	46	8	no	—
75	C₂²	C₂²×C₂₃⋊C₈	1	2	yes	—
76	C₂²	C₂×#67267	1	2	yes	—
77	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
78	C₂²	#10625097	1	2	yes	—
79	C₄	—	2	8	no	—
80	C₄	C₄×C₂₃⋊C₈	1	4	yes	—
81	C₄	#10624985	1	4	yes	—
82	C₄	#10625043	1	4	yes	—
83	C₂	#13092263	1	1	yes	characteristic
84	C₂	C₂×C₄×C₂₃⋊C₈	1	1	yes	characteristic
85	C₂	C₂×#10624985	1	1	yes	characteristic
86	C₂	#13092470	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₂₃
commutator	C₂×C₂₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₄³×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₄³	D₂₃
hypocenter	C₂₃	#1744
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
9. C₄×C₂₃⋊C₈	C₄	semidirect
16. C₄²×C₂₃	C₈	semidirect
53. C₂₃	#1744	semidirect
62. C₄²	C₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₂₃⋊C₈	2
#13092172	2
C₄³×C₂₃	2
#1744	23