Subgroups — #15671377

Order 2944.

94 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	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₄₆	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₃	—	2	1	no	—
6	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	Dic₂₃⋊C₈	—	2	1	no	—
8	C₈×Dic₂₃	—	2	1	no	—
9	C₂×C₂₃⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₂₃⋊C₈)	C₄	1	1	yes	characteristic
11	C₂×Dic₄₆	C₈	1	1	yes	characteristic
12	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
14	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₄⋊Dic₂₃	C₈	1	1	yes	characteristic
16	Dic₂₃⋊C₄	—	2	1	no	—
17	C₄×Dic₂₃	—	2	1	no	—
18	C₂×C₂₃⋊C₈	—	2	1	no	—
19	C₁₆×C₂₃	—	2	2	no	—
20	C₂₃⋊C₁₆	—	4	1	no	—
21	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₂₃	M₁₆	1	1	yes	characteristic
24	C₂×Dic₂₃	—	2	1	no	—
25	C₂×Dic₂₃	—	2	1	no	—
26	Dic₄₆	—	2	1	no	—
27	Dic₄₆	C₁₆	1	2	yes	complemented
28	C₈×C₂₃	—	2	1	no	—
29	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₂₃	Q₁₆	1	1	yes	characteristic
31	C₂₃⋊C₈	—	4	1	no	—
32	Q₈⋊C₁₆	—	23	1	no	maximal, Sylow
33	C₂²×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₄×C₂₃	—	2	1	no	—
35	C₄×C₂₃	#103	1	1	yes	characteristic
36	C₄×C₂₃	#104	1	1	yes	characteristic
37	C₄×C₂₃	C₂×C₁₆	1	1	yes	characteristic
38	C₄×C₂₃	M₃₂	1	1	yes	characteristic
39	Dic₂₃	—	2	2	no	—
40	Dic₂₃	—	4	1	no	—
41	C₈×Q₈	—	23	1	no	—
42	C₄⋊C₁₆	—	23	1	no	—
43	C₄×C₁₆	D₂₃	1	1	yes	characteristic
44	C₂×C₂₃	Q₈⋊C₈	1	1	yes	characteristic
45	C₂×C₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
46	C₂×C₂₃	#350	1	1	yes	characteristic
47	C₄×Q₈	—	23	1	no	—
48	C₂×C₁₆	—	2	1	no	—
49	C₂×C₁₆	—	46	1	no	—
50	C₄⋊C₈	—	23	1	no	—
51	C₄⋊C₈	—	46	1	no	—
52	C₄×C₈	—	46	1	no	—
53	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
54	C₂₃	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₈	—	23	1	no	—
56	C₂×C₈	—	46	1	no	—
57	C₂×C₈	D₉₂	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
59	C₄⋊C₄	—	23	1	no	—
60	C₄⋊C₄	—	46	1	no	—
61	C₄²	—	46	1	no	—
62	C₄²	C₄×D₂₃	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	92	1	no	—
65	Q₈	—	23	2	no	—
66	Q₈	—	46	1	no	—
67	C₂×C₄	—	46	1	no	—
68	C₂×C₄	—	46	1	no	—
69	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
70	C₂×C₄	#67262	1	1	yes	characteristic
71	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	92	1	no	—
74	C₈	#67263	1	1	yes	characteristic
75	C₈	Dic₉₂	1	1	yes	characteristic
76	C₂²	(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	46	2	no	—
79	C₄	—	92	1	no	—
80	C₄	C₁₆×D₂₃	1	1	yes	characteristic
81	C₄	#10624980	1	1	yes	characteristic
82	C₄	#10624987	1	1	yes	characteristic
83	C₄	#10624998	1	1	yes	characteristic
84	C₂	#13092149	1	1	yes	characteristic
85	C₂	(C₂×D₂₃)⋊C₁₆	1	1	yes	characteristic
86	C₂	#13092200	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₉₂
commutator	C₄×C₂₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₄×C₁₆×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	C₂²⋊C₈
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	C₄×C₁₆	D₂₃
hypocenter	C₂₃	Q₈⋊C₁₆
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
27. Dic₄₆	C₁₆	semidirect
54. 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
C₄⋊(C₂₃⋊C₁₆)	2
C₄×C₁₆×C₂₃	2
Q₈⋊C₁₆	23