Group #15682338 — subgroups

Subgroups — #15682338

Order 2944.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 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	#13092548	C₂	1	8	yes	maximal, complemented
2	#13092444	C₂	1	2	yes	maximal
3	#13092441	C₂	1	1	yes	maximal, characteristic
4	#13092373	C₂	1	2	yes	maximal
5	C₂×#10624985	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#533	C₂	1	1	yes	maximal, characteristic
7	C₂₃×C₄⋊Q₈	C₄	1	2	yes	—
8	C₂₃×#125	C₄	1	2	yes	—
9	C₂₃×C₂²⋊Q₈	C₄	1	4	yes	—
10	C₄×Q₈×C₂₃	C₂²	1	4	yes	—
11	C₂₃×#117	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67267	C₂²	1	2	yes	—
14	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
17	C₄⋊(C₂₃⋊C₈)	C₂²	1	8	yes	—
18	#10624985	C₂²	1	4	yes	—
19	#10624985	C₂²	1	4	yes	—
20	C₄×C₂₃⋊C₈	C₂²	1	4	yes	—
21	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
25	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
26	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
28	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₃	C₂³	1	2	yes	—
31	C₄²×C₂₃	C₂³	1	4	yes	—
32	#67267	—	2	4	no	—
33	C₂×C₂₃⋊C₈	—	2	2	no	—
34	C₂×C₂₃⋊C₈	Q₈	1	4	yes	complemented
35	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
36	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
37	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
38	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₂₃	—	4	4	no	—
40	C₂×C₄×C₂₃	—	4	2	no	—
41	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₂₃	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₂₃	#41	1	2	yes	—
47	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
48	C₂₃⋊C₈	—	2	2	no	—
49	C₂₃⋊C₈	—	2	4	no	—
50	C₂₃⋊C₈	C₂×Q₈	1	4	yes	—
51	#2961	—	23	1	no	maximal, Sylow
52	C₂²×C₂₃	—	2	2	no	—
53	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
54	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	2	8	no	—
58	C₄×C₂₃	C₄×Q₈	1	2	yes	—
59	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
60	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
61	#533	D₂₃	1	1	yes	characteristic
62	#446	—	23	8	no	—
63	#424	—	23	2	no	—
64	#423	—	23	1	no	—
65	#405	—	23	2	no	—
66	C₂×#97	—	23	1	no	—
67	C₂×C₂₃	—	2	2	no	—
68	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
69	C₂×C₂₃	#568	1	2	yes	—
70	C₂×M₁₆	—	23	2	no	—
71	C₂²×C₈	—	23	1	no	—
72	C₂²×C₈	—	23	1	no	—
73	C₄⋊Q₈	Dic₂₃	1	2	yes	—
74	#125	Dic₂₃	1	2	yes	—
75	C₂²⋊Q₈	Dic₂₃	1	4	yes	—
76	C₄×Q₈	C₂×D₂₃	1	4	yes	—
77	#117	C₂×D₂₃	1	2	yes	—
78	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
79	C₄⋊C₈	—	23	4	no	—
80	C₄⋊C₈	—	23	8	no	—
81	#97	—	23	4	no	—
82	#97	—	23	4	no	—
83	C₄×C₈	—	23	4	no	—
84	C₂₃	#2961	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	C₂×Dic₂₃	1	4	yes	—
86	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
88	M₁₆	—	46	4	no	—
89	C₂×C₈	—	23	4	no	—
90	C₂×C₈	—	23	2	no	—
91	C₂×C₈	—	23	4	no	—
92	C₂×C₈	—	23	2	no	—
93	C₂×C₈	—	46	2	no	—
94	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
95	C₄⋊C₄	C₂×Dic₂₃	1	8	yes	—
96	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
98	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
99	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
100	C₄²	C₂²×D₂₃	1	4	yes	—
101	C₄²	C₂²×D₂₃	1	2	yes	—
102	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	Q₈×D₂₃	1	2	yes	—
106	C₂×C₄	#67292	1	2	yes	—
107	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
108	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
109	C₂×C₄	C₂²×Dic₂₃	1	8	yes	—
110	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
112	C₈	—	23	4	no	—
113	C₈	—	46	2	no	—
114	C₈	—	46	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	Q₈×Dic₂₃	1	2	yes	—
117	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	Q₈×Dic₂₃	1	2	yes	—
122	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
123	C₄	C₂×#67292	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₂₃	1	1	yes	characteristic
126	C₂	#13093184	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092548

C₂

yes

maximal, complemented

#13092444

C₂

yes

maximal

#13092441

C₂

yes

maximal, characteristic

#13092373

C₂

yes

maximal

C₂×#10624985

C₂

yes

maximal, characteristic

C₂₃×#533

C₂

yes

maximal, characteristic

C₂₃×C₄⋊Q₈

C₄

yes

—

C₂₃×#125

C₄

yes

—

C₂₃×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₂₃

C₂²

yes

—

C₂₃×#117

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×#67267

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

#10624985

C₂²

yes

—

#10624985

C₂²

yes

—

C₄×C₂₃⋊C₈

C₂²

yes

—

C₂×Q₈×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

—

C₄²×C₂₃

C₂³

yes

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

Q₈

yes

complemented

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂₃⋊C₈

—

C₂₃⋊C₈

—

C₂₃⋊C₈

C₂×Q₈

yes

—

#2961

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄×Q₈

yes

—

C₂²×C₂₃

C₂³×C₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₄×Q₈

yes

—

C₄×C₂₃

C₂²×Q₈

yes

characteristic

C₄×C₂₃

C₂×#41

yes

characteristic

#533

D₂₃

yes

—

—

—

—

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₂₃

#568

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₂₃

yes

—

#125

Dic₂₃

yes

—

C₂²⋊Q₈

Dic₂₃

yes

—

C₄×Q₈

C₂×D₂₃

yes

—

#117

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂₃

#2961

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂×Dic₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₄⋊C₄

C₂×Dic₂₃

yes

—

C₂²⋊C₄

C₂×Dic₂₃

yes

—

C₄²

C₂×Dic₂₃

yes

characteristic

C₄²

C₂×Dic₂₃

yes

characteristic

100

C₄²

C₂²×D₂₃

yes

—

101

C₄²

C₂²×D₂₃

yes

—

102

C₂³

C₂²×Dic₂₃

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

Q₈×D₂₃

yes

—

106

C₂×C₄

#67292

yes

—

107

C₂×C₄

C₂²×Dic₂₃

yes

—

108

C₂×C₄

C₂²×Dic₂₃

yes

—

109

C₂×C₄

C₂²×Dic₂₃

yes

—

110

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

111

C₂×C₄

C₂³×D₂₃

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

Q₈×Dic₂₃

yes

—

117

C₂²

C₂³×Dic₂₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×Dic₂₃

yes

—

122

C₄

C₂×Q₈×D₂₃

yes

characteristic

123

C₄

C₂×#67292

yes

characteristic

124

C₂

—

125

C₂

C₂×Q₈×Dic₂₃

yes

characteristic

126

C₂

#13093184

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#533	C₂
prime radical	C₂³×C₂₃	C₂²×C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#533	D₂₃
hypocenter	C₂₃	#2961
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₃

commutator, socle

C₂²×C₂₃

C₂³×C₄

Fitting

C₂₃×#533

C₂

prime radical

C₂³×C₂₃

C₂²×C₄

cosocle

C₂×C₄×C₂₃

C₂⁴

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #13092548	C₂	semidirect
34. C₂×C₂₃⋊C₈	Q₈	semidirect
84. C₂₃	#2961	semidirect

H ≅ G/N

type

1. #13092548

C₂

semidirect

34. C₂×C₂₃⋊C₈

Q₈

semidirect

84. C₂₃

#2961

semidirect

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

subgroup	index
#13092548	2
#13092444	2
#13092441	2
#13092373	2
C₂×#10624985	2
C₂₃×#533	2
#2961	23

subgroup

index

C₂₃×#533

Subgroups — #15682338

Conjugacy classes of subgroups

Complements

Maximal subgroups