Group #15683920 — subgroups

Subgroups — #15683920

Order 2944.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#13092970	C₂	1	2	yes	maximal
2	#13092825	C₂	1	4	yes	maximal, complemented
3	#13092824	C₂	1	4	yes	maximal, complemented
4	#13092455	C₂	1	1	yes	maximal, characteristic
5	#13092453	C₂	1	2	yes	maximal
6	C₂×#10624997	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#117	C₂²	1	2	yes	—
10	#10625097	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	#10625061	C₂²	1	4	yes	—
14	#10625059	C₂²	1	4	yes	—
15	#10625049	—	2	4	no	—
16	C₄×Dic₄₆	—	2	4	no	—
17	C₂₃×#107	C₂²	1	2	yes	—
18	C₂₃×#106	C₂²	1	2	yes	—
19	#10625000	C₂²	1	2	yes	—
20	#10624999	C₂²	1	2	yes	—
21	#10624997	C₂²	1	2	yes	—
22	#10624997	C₂²	1	2	yes	—
23	#10624990	C₂²	1	4	yes	—
24	#10624989	C₂²	1	4	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₃	D₄	1	2	yes	—
27	C₂×Dic₄₆	—	2	4	no	—
28	C₂×C₈×C₂₃	Q₈	1	4	yes	complemented
29	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₄²×C₂₃	—	2	2	no	—
34	C₂²⋊Dic₂₃	—	2	2	no	—
35	C₄⋊Dic₂₃	—	2	4	no	—
36	C₄⋊Dic₂₃	C₂³	1	4	yes	—
37	Dic₂₃⋊C₄	—	2	4	no	—
38	Dic₂₃⋊C₄	—	2	4	no	—
39	C₄×Dic₂₃	—	2	2	no	—
40	C₄×Dic₂₃	D₄	1	2	yes	—
41	C₄×Dic₂₃	C₂³	1	2	yes	—
42	C₂×C₂₃⋊C₈	Q₈	1	4	yes	complemented
43	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
44	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₂₃	—	2	4	no	—
51	C₂×Dic₂₃	—	4	2	no	—
52	C₂×Dic₂₃	C₂×D₄	1	4	yes	—
53	Dic₄₆	—	4	8	no	—
54	C₈×C₂₃	C₂×Q₈	1	2	yes	—
55	C₈×C₂₃	C₂×Q₈	1	2	yes	—
56	C₂₃⋊C₈	C₂×Q₈	1	4	yes	—
57	#3162	—	23	1	no	maximal, Sylow
58	C₂²×C₂₃	—	2	2	no	—
59	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₂₃	—	4	4	no	—
62	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
63	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
64	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
65	Dic₂₃	—	2	4	no	—
66	Dic₂₃	—	4	4	no	—
67	#533	—	23	2	no	—
68	#501	—	23	4	no	—
69	#501	—	23	4	no	—
70	#427	—	23	1	no	—
71	#427	—	23	2	no	—
72	#427	D₂₃	1	1	yes	characteristic
73	C₂×#97	—	23	1	no	—
74	C₂×C₂₃	—	2	2	no	—
75	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₂₃	#575	1	2	yes	—
77	C₂²×C₈	—	23	1	no	—
78	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
79	C₄⋊Q₈	—	23	4	no	—
80	#125	—	23	4	no	—
81	C₂²⋊Q₈	—	46	4	no	—
82	C₄×Q₈	—	46	4	no	—
83	#117	—	23	2	no	—
84	#117	C₂×D₂₃	1	2	yes	—
85	C₂×C₄²	—	23	1	no	—
86	#107	—	23	2	no	—
87	#107	—	23	4	no	—
88	#107	C₂×D₂₃	1	2	yes	—
89	#106	—	23	2	no	—
90	#106	—	23	4	no	—
91	#106	C₂×D₂₃	1	2	yes	—
92	#97	—	23	2	no	—
93	#97	—	23	2	no	—
94	C₂₃	#3162	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₈	—	46	4	no	—
96	C₂²×C₄	—	23	2	no	—
97	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
98	C₂×C₈	—	23	4	no	—
99	C₂×C₈	—	23	2	no	—
100	C₂×C₈	Dic₄₆	1	4	yes	complemented
101	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
103	C₄⋊C₄	—	23	4	no	—
104	C₄⋊C₄	—	46	4	no	—
105	C₄⋊C₄	—	46	4	no	—
106	C₄⋊C₄	—	46	4	no	—
107	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	46	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	23	2	no	—
112	C₄²	—	23	2	no	—
113	C₄²	—	46	2	no	—
114	C₂³	D₄×D₂₃	1	1	yes	characteristic
115	Q₈	—	92	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	23	4	no	—
118	C₂×C₄	—	46	4	no	—
119	C₂×C₄	—	92	2	no	—
120	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
121	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₂₃	1	2	yes	—
123	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
124	C₈	—	23	4	no	—
125	C₈	C₂×Dic₄₆	1	2	yes	—
126	C₈	C₂×Dic₄₆	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#10625059	1	2	yes	—
129	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	46	4	no	—
132	C₄	—	92	4	no	—
133	C₄	#10625059	1	2	yes	—
134	C₄	C₂²×Dic₄₆	1	1	yes	characteristic
135	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#10625059	1	1	yes	characteristic
138	C₂	#13093214	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092970

C₂

yes

maximal

#13092825

C₂

yes

maximal, complemented

#13092824

C₂

yes

maximal, complemented

#13092455

C₂

yes

maximal, characteristic

#13092453

C₂

yes

maximal

C₂×#10624997

C₂

yes

maximal, characteristic

C₂₃×#427

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

#10625097

C₂²

yes

—

C₂×C₄×Dic₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

C₂²

yes

characteristic

#10625061

C₂²

yes

—

#10625059

C₂²

yes

—

#10625049

—

C₄×Dic₄₆

—

C₂₃×#107

C₂²

yes

—

C₂₃×#106

C₂²

yes

—

#10625000

C₂²

yes

—

#10624999

C₂²

yes

—

#10624997

C₂²

yes

—

#10624997

C₂²

yes

—

#10624990

C₂²

yes

—

#10624989

C₂²

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

D₄

yes

—

C₂×Dic₄₆

—

C₂×C₈×C₂₃

Q₈

yes

complemented

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

C₂³

yes

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

C₂³

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

complemented

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂×D₄

yes

—

Dic₄₆

—

C₈×C₂₃

C₂×Q₈

yes

—

C₈×C₂₃

C₂×Q₈

yes

—

C₂₃⋊C₈

C₂×Q₈

yes

—

#3162

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

C₂²×Q₈

yes

characteristic

C₄×C₂₃

C₂²×Q₈

yes

characteristic

Dic₂₃

—

Dic₂₃

—

—

—

—

—

—

D₂₃

yes

characteristic

C₂×#97

—

C₂×C₂₃

—

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

#575

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₂₃

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₂₃

yes

—

#106

—

#106

—

#106

C₂×D₂₃

yes

—

#97

—

#97

—

C₂₃

#3162

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₄₆

yes

complemented

101

C₂×C₈

C₂²×D₂₃

yes

characteristic

102

C₂×C₈

C₂²×D₂₃

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²×D₂₃

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

D₄×D₂₃

yes

characteristic

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×Dic₄₆

yes

—

121

C₂×C₄

D₄×D₂₃

yes

characteristic

122

C₂×C₄

Q₈×D₂₃

yes

—

123

C₂×C₄

C₂³×D₂₃

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₄₆

yes

—

126

C₈

C₂×Dic₄₆

yes

—

127

C₂²

—

128

C₂²

#10625059

yes

—

129

C₂²

C₂×D₄×D₂₃

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#10625059

yes

—

134

C₄

C₂²×Dic₄₆

yes

characteristic

135

C₄

C₂×Q₈×D₂₃

yes

characteristic

136

C₂

—

137

C₂

C₂×#10625059

yes

characteristic

138

C₂

#13093214

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#427	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	#427	D₂₃
hypocenter	C₂₃	#3162
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂₃×#427

C₂

socle

C₂²×C₂₃

C₂²×D₄

prime radical

C₂³×C₂₃

C₂×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #13092825	C₂	semidirect
3. #13092824	C₂	semidirect
28. C₂×C₈×C₂₃	Q₈	semidirect
42. C₂×C₂₃⋊C₈	Q₈	semidirect
94. C₂₃	#3162	semidirect
100. C₂×C₈	Dic₄₆	semidirect

H ≅ G/N

type

2. #13092825

C₂

semidirect

3. #13092824

C₂

semidirect

28. C₂×C₈×C₂₃

Q₈

semidirect

42. C₂×C₂₃⋊C₈

Q₈

semidirect

94. C₂₃

#3162

semidirect

100. C₂×C₈

Dic₄₆

semidirect

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

subgroup	index
#13092970	2
#13092825	2
#13092824	2
#13092455	2
#13092453	2
C₂×#10624997	2
C₂₃×#427	2
#3162	23

subgroup

index

C₂₃×#427

Subgroups — #15683920

Conjugacy classes of subgroups

Complements

Maximal subgroups