Group #15672892 — subgroups

Subgroups — #15672892

Order 2944.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092797	C₂	1	1	yes	maximal, characteristic
2	#13092703	C₂	1	2	yes	maximal
3	#13092547	C₂	1	1	yes	maximal, characteristic
4	#13092162	C₂	1	2	yes	maximal
5	C₂₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₉₂	—	2	1	no	—
7	C₂×Dic₉₂	—	2	1	no	—
8	#10625040	C₂²	1	2	yes	—
9	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
10	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	#10624999	—	2	2	no	—
13	#10624998	—	2	2	no	—
14	#10624997	—	2	1	no	—
15	Dic₂₃⋊C₈	—	2	1	no	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₆	—	2	2	no	—
18	C₂×Dic₄₆	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₃	—	2	2	no	—
20	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₄²×C₂₃	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₂₃	—	2	2	no	—
24	C₄⋊Dic₂₃	—	2	2	no	—
25	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
26	Dic₂₃⋊C₄	—	2	1	no	—
27	C₄×Dic₂₃	—	2	1	no	—
28	C₂×C₂₃⋊C₈	—	2	1	no	—
29	Dic₉₂	—	4	2	no	—
30	Dic₉₂	—	4	2	no	—
31	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
34	C₂×Dic₂₃	—	2	1	no	—
35	C₂×Dic₂₃	—	2	1	no	—
36	C₂×Dic₂₃	—	4	2	no	—
37	Dic₄₆	—	2	2	no	—
38	Dic₄₆	—	4	2	no	—
39	Dic₄₆	—	4	2	no	—
40	C₈×C₂₃	—	2	1	no	—
41	C₈×C₂₃	—	4	2	no	—
42	C₈×C₂₃	SD₁₆	1	2	yes	—
43	C₂₃⋊C₈	—	4	1	no	—
44	#1690	—	23	1	no	maximal, Sylow
45	C₂²×C₂₃	#121	1	1	yes	characteristic
46	C₄×C₂₃	—	2	1	no	—
47	C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
48	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₄×C₂₃	#137	1	2	yes	—
50	Dic₂₃	—	4	1	no	—
51	Dic₂₃	—	4	1	no	—
52	Dic₂₃	—	8	2	no	—
53	#494	—	23	1	no	—
54	#475	—	23	2	no	—
55	#446	—	23	1	no	—
56	#334	D₂₃	1	1	yes	characteristic
57	#332	—	23	2	no	—
58	C₂×C₂₃	#460	1	1	yes	characteristic
59	C₂×C₂₃	#470	1	1	yes	characteristic
60	C₂×C₂₃	#473	1	1	yes	characteristic
61	C₂×Q₁₆	—	46	1	no	—
62	C₂×Q₁₆	—	46	1	no	—
63	C₄⋊Q₈	—	23	2	no	—
64	C₄×Q₈	—	23	1	no	—
65	#106	—	46	2	no	—
66	C₄⋊C₈	—	23	1	no	—
67	C₄⋊C₈	—	46	1	no	—
68	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
69	#103	—	46	2	no	—
70	#97	—	46	1	no	—
71	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
72	C₂₃	#1690	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₈	—	23	1	no	—
74	C₂×Q₈	—	46	2	no	—
75	Q₁₆	—	92	2	no	—
76	Q₁₆	—	92	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	46	1	no	—
79	C₂×C₈	D₉₂	1	1	yes	characteristic
80	C₂×C₈	D₉₂	1	1	yes	characteristic
81	C₄⋊C₄	—	23	1	no	—
82	C₄⋊C₄	—	46	1	no	—
83	C₄⋊C₄	—	46	2	no	—
84	C₄⋊C₄	—	46	2	no	—
85	C₄²	—	46	1	no	—
86	C₄²	C₂²×D₂₃	1	1	yes	characteristic
87	Q₈	—	46	2	no	—
88	Q₈	—	92	2	no	—
89	Q₈	—	92	2	no	—
90	C₂×C₄	—	46	1	no	—
91	C₂×C₄	—	46	1	no	—
92	C₂×C₄	—	92	2	no	—
93	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
94	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
95	C₂×C₄	#67292	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	4	2	no	—
98	C₈	—	92	1	no	—
99	C₈	#67263	1	2	yes	—
100	C₂²	#10625066	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	92	1	no	—
103	C₄	—	92	1	no	—
104	C₄	—	184	2	no	—
105	C₄	C₂×#67263	1	1	yes	characteristic
106	C₄	#10625079	1	1	yes	characteristic
107	C₄	#10625086	1	2	yes	—
108	C₂	#13092618	1	1	yes	characteristic
109	C₂	#13092676	1	1	yes	characteristic
110	C₂	#13092693	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092797

C₂

yes

maximal, characteristic

#13092703

C₂

yes

maximal

#13092547

C₂

yes

maximal, characteristic

#13092162

C₂

yes

maximal

C₂₃×#334

C₂

yes

maximal, characteristic

C₂×Dic₉₂

—

C₂×Dic₉₂

—

#10625040

C₂²

yes

—

C₄×Dic₄₆

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

—

—

Dic₂₃⋊C₈

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

characteristic

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

D₄

yes

characteristic

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

Dic₉₂

—

Dic₉₂

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

SD₁₆

yes

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

—

—

—

D₂₃

yes

characteristic

#332

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

—

#103

—

#97

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1690

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₉₂

yes

characteristic

C₂×C₈

D₉₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

D₄×D₂₃

yes

characteristic

C₂×C₄

#67292

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#67263

yes

—

100

C₂²

#10625066

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

184

—

105

C₄

C₂×#67263

yes

characteristic

106

C₄

#10625079

yes

characteristic

107

C₄

#10625086

yes

—

108

C₂

#13092618

yes

characteristic

109

C₂

#13092676

yes

characteristic

110

C₂

#13092693

yes

characteristic

	subgroup	G/H
center	C₂²	#10625066
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#334	C₂
socle, prime radical	C₂²×C₂₃	#121
hypercenter	#334	D₂₃
hypocenter	C₂₃	#1690
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#10625066

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#334

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
72. C₂₃	#1690	semidirect

H ≅ G/N

type

72. C₂₃

#1690

semidirect

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

subgroup	index
#13092797	2
#13092703	2
#13092547	2
#13092162	2
C₂₃×#334	2
#1690	23

subgroup

index

C₂₃×#334

Subgroups — #15672892

Conjugacy classes of subgroups

Complements

Maximal subgroups