Subgroups — #15671361

Order 2944.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#13092170	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
5	C₄×Q₈×C₂₃	—	2	1	no	—
6	C₂₃×#107	C₄	1	1	yes	characteristic
7	C₂₃×#103	—	2	1	no	—
8	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃⋊C₁₆	—	2	1	no	—
10	C₄⋊(C₂₃⋊C₈)	—	2	1	no	—
11	C₂×Q₈×C₂₃	—	2	1	no	—
12	Q₁₆×C₂₃	—	2	1	no	—
13	Q₁₆×C₂₃	C₈	1	2	yes	complemented
14	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
16	C₄⋊C₄×C₂₃	—	2	1	no	—
17	C₄⋊C₄×C₂₃	—	4	1	no	—
18	C₄²×C₂₃	—	4	1	no	—
19	C₄²×C₂₃	D₄	1	1	yes	characteristic
20	C₂×C₂₃⋊C₈	—	4	1	no	—
21	C₂₃⋊C₁₆	—	2	2	no	—
22	Q₈×C₂₃	—	2	2	no	—
23	Q₈×C₂₃	—	4	1	no	—
24	C₂×C₄×C₂₃	—	4	1	no	—
25	C₂×C₄×C₂₃	—	4	1	no	—
26	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
28	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
29	C₈×C₂₃	—	2	1	no	—
30	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
31	C₈×C₂₃	M₁₆	1	1	yes	characteristic
32	C₂₃⋊C₈	—	8	1	no	—
33	#1332	—	23	1	no	maximal, Sylow
34	C₂²×C₂₃	#102	1	1	yes	characteristic
35	C₄×C₂₃	—	2	1	no	—
36	C₄×C₂₃	—	4	2	no	—
37	C₄×C₂₃	—	8	1	no	—
38	C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₂₃	#104	1	1	yes	characteristic
40	C₄×C₂₃	SD₃₂	1	1	yes	characteristic
41	C₄×C₂₃	Q₃₂	1	1	yes	characteristic
42	C₄×Q₁₆	D₂₃	1	1	yes	characteristic
43	C₄×C₁₆	—	23	1	no	—
44	#335	—	23	1	no	—
45	C₂×C₂₃	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₂₃	#358	1	1	yes	characteristic
47	C₂×C₂₃	#359	1	1	yes	characteristic
48	C₂×Q₁₆	Dic₂₃	1	1	yes	characteristic
49	C₄×Q₈	—	2	1	no	—
50	C₂×C₁₆	—	46	1	no	—
51	#107	Dic₂₃	1	1	yes	characteristic
52	C₄⋊C₈	—	46	1	no	—
53	#103	—	2	1	no	—
54	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
55	C₂₃	#1332	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₈	—	2	1	no	—
57	Q₁₆	—	2	1	no	—
58	Q₁₆	C₂₃⋊C₈	1	2	yes	complemented
59	C₂×C₈	—	92	1	no	—
60	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	4	1	no	—
64	C₄²	—	4	1	no	—
65	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
66	C₁₆	—	46	2	no	—
67	Q₈	—	2	2	no	—
68	Q₈	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	#67272	1	1	yes	characteristic
72	C₂×C₄	#67273	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	184	1	no	—
76	C₈	C₂×C₂₃⋊C₈	1	1	yes	characteristic
77	C₈	#67267	1	1	yes	characteristic
78	C₂²	#10625014	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	4	2	no	—
81	C₄	—	8	1	no	—
82	C₄	#10625010	1	1	yes	characteristic
83	C₄	#10625011	1	1	yes	characteristic
84	C₄	C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
85	C₄	#10625019	1	1	yes	characteristic
86	C₂	D₄⋊(C₂₃⋊C₈)	1	1	yes	characteristic
87	C₂	#13092221	1	1	yes	characteristic
88	C₂	#13092224	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#67272
commutator	C₈×C₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₄×Q₁₆×C₂₃	C₂
socle, prime radical	C₂²×C₂₃	#102
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	C₄×Q₁₆	D₂₃
hypocenter	C₂₃	#1332
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
13. Q₁₆×C₂₃	C₈	semidirect
55. C₂₃	#1332	semidirect
58. Q₁₆	C₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂₃⋊C₁₆	2
#13092170	2
C₄×Q₁₆×C₂₃	2
#1332	23