Subgroups — #15675299

Order 2944.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₄₆	C₂	1	1	yes	maximal, characteristic
2	Dic₂₃⋊C₁₆	C₂	1	2	yes	maximal
3	C₄×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₂₃	C₂	1	2	yes	maximal
5	C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₄₆	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₃	C₂²	1	2	yes	—
8	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	Dic₂₃⋊C₈	C₄	1	2	yes	—
10	C₈×Dic₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₄⋊(C₂₃⋊C₈)	C₄	1	1	yes	characteristic
14	C₂×Dic₄₆	C₈	1	1	yes	characteristic
15	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
17	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₂₃	C₈	1	1	yes	characteristic
19	Dic₂₃⋊C₄	C₈	1	2	yes	—
20	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₂₃⋊C₈	C₂×C₄	1	2	yes	—
22	C₁₆×C₂₃	—	2	2	no	—
23	C₂₃⋊C₁₆	—	2	1	no	—
24	C₂₃⋊C₁₆	Q₈	1	2	yes	complemented
25	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
29	C₂×Dic₂₃	C₂×C₈	1	2	yes	—
30	Dic₄₆	C₁₆	1	4	yes	complemented
31	C₈×C₂₃	—	2	1	no	—
32	C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₂₃	#41	1	1	yes	characteristic
34	C₂₃⋊C₈	—	2	2	no	—
35	Q₈×C₁₆	—	23	1	no	maximal, Sylow
36	C₂²×C₂₃	C₂²×C₈	1	1	yes	characteristic
37	C₄×C₂₃	—	2	1	no	—
38	C₄×C₂₃	C₂×C₁₆	1	2	yes	—
39	C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
40	C₄×C₂₃	#131	1	1	yes	characteristic
41	Dic₂₃	—	2	2	no	—
42	Dic₂₃	C₂×C₁₆	1	4	yes	—
43	C₈×Q₈	—	23	1	no	—
44	C₄⋊C₁₆	—	23	2	no	—
45	C₄⋊C₁₆	D₂₃	1	1	yes	characteristic
46	C₄×C₁₆	—	23	2	no	—
47	C₄×C₁₆	—	23	1	no	—
48	C₂×C₂₃	C₈×Q₈	1	1	yes	characteristic
49	C₂×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
50	C₂×C₂₃	#504	1	1	yes	characteristic
51	C₄×Q₈	—	23	1	no	—
52	C₂×C₁₆	—	23	1	no	—
53	C₂×C₁₆	—	23	1	no	—
54	C₂×C₁₆	C₂×D₂₃	1	2	yes	—
55	C₄⋊C₈	—	23	2	no	—
56	C₄⋊C₈	—	23	1	no	—
57	C₄×C₈	—	23	2	no	—
58	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
59	C₂₃	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	—	23	1	no	—
61	C₂×C₈	—	23	2	no	—
62	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
64	C₄⋊C₄	—	23	2	no	—
65	C₄⋊C₄	—	23	1	no	—
66	C₄²	—	23	2	no	—
67	C₄²	C₄×D₂₃	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	23	2	no	—
70	C₁₆	—	46	1	no	—
71	Q₈	—	23	4	no	—
72	C₂×C₄	—	23	2	no	—
73	C₂×C₄	—	23	2	no	—
74	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
75	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	46	2	no	—
79	C₈	#67290	1	1	yes	characteristic
80	C₈	Q₈×D₂₃	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₂₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	23	4	no	—
84	C₄	—	46	2	no	—
85	C₄	C₁₆×D₂₃	1	2	yes	—
86	C₄	#10625058	1	1	yes	characteristic
87	C₄	#10625078	1	1	yes	characteristic
88	C₂	#13092541	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₂₃	1	1	yes	characteristic
90	C₂	#13092835	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₃
commutator	C₂×C₂₃	C₂²×C₁₆
Fitting	C₂₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₂₃	C₂²×C₈
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	C₄⋊C₁₆	D₂₃
hypocenter	C₂₃	Q₈×C₁₆
LCS	G ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
24. C₂₃⋊C₁₆	Q₈	semidirect
30. Dic₄₆	C₁₆	semidirect
59. C₂₃	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₄₆	2
Dic₂₃⋊C₁₆	2
C₄×C₂₃⋊C₁₆	2
C₁₆×Dic₂₃	2
C₂₃×C₄⋊C₁₆	2
Q₈×C₁₆	23