Subgroups — #15671380

Order 2944.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092547	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#346	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	#10624997	—	2	1	no	—
8	Dic₂₃⋊C₈	—	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	—	4	1	no	—
27	C₈×C₂₃	—	2	1	no	—
28	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
29	C₈×C₂₃	Q₁₆	1	1	yes	characteristic
30	C₂₃⋊C₈	—	4	1	no	—
31	#1338	—	23	1	no	maximal, Sylow
32	C₂²×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₂₃	—	2	1	no	—
34	C₄×C₂₃	—	2	1	no	—
35	C₄×C₂₃	#103	1	1	yes	characteristic
36	C₄×C₂₃	#104	1	1	yes	characteristic
37	Dic₂₃	—	4	1	no	—
38	Dic₂₃	—	4	1	no	—
39	#446	—	23	1	no	—
40	C₄⋊C₁₆	—	23	1	no	—
41	#346	D₂₃	1	1	yes	characteristic
42	C₂×C₂₃	Q₈⋊C₈	1	1	yes	characteristic
43	C₂×C₂₃	#349	1	1	yes	characteristic
44	C₂×C₂₃	#350	1	1	yes	characteristic
45	C₄×Q₈	—	23	1	no	—
46	C₂×C₁₆	—	2	1	no	—
47	C₂×C₁₆	—	46	1	no	—
48	C₄⋊C₈	—	23	1	no	—
49	C₄⋊C₈	—	46	1	no	—
50	#97	—	46	1	no	—
51	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
52	C₂₃	#1338	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₈	—	23	1	no	—
54	C₂×C₈	—	46	1	no	—
55	C₂×C₈	D₉₂	1	1	yes	characteristic
56	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
57	C₄⋊C₄	—	23	1	no	—
58	C₄⋊C₄	—	46	1	no	—
59	C₄²	—	46	1	no	—
60	C₄²	C₄×D₂₃	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₁₆	—	92	1	no	—
63	Q₈	—	92	1	no	—
64	C₂×C₄	—	46	1	no	—
65	C₂×C₄	—	46	1	no	—
66	C₂×C₄	C₈×D₂₃	1	1	yes	characteristic
67	C₂×C₄	#67262	1	1	yes	characteristic
68	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	92	1	no	—
71	C₈	#67263	1	1	yes	characteristic
72	C₈	Dic₉₂	1	1	yes	characteristic
73	C₂²	(C₂×D₂₃)⋊C₈	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	92	1	no	—
77	C₄	—	92	1	no	—
78	C₄	#10624987	1	1	yes	characteristic
79	C₄	#10624998	1	1	yes	characteristic
80	C₂	#13092149	1	1	yes	characteristic
81	C₂	#13092198	1	1	yes	characteristic
82	C₂	#13092202	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₂₃)⋊C₄
commutator	C₂×C₄×C₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#346	C₂
socle, prime radical	C₂²×C₂₃	C₂²⋊C₈
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#346	D₂₃
hypocenter	C₂₃	#1338
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
52. C₂₃	#1338	semidirect

Maximal subgroups

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

subgroup	index
#13092547	2
C₄⋊(C₂₃⋊C₁₆)	2
C₂₃×#346	2
#1338	23