Group #13093138 — subgroups

Subgroups — #13093138

Order 1472.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625117	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₂₃	C₂	1	1	yes	maximal, characteristic
5	#10625097	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625069	C₂	1	2	yes	maximal, complemented
7	#10625063	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625059	C₂	1	1	yes	maximal, characteristic, complemented
9	#10625058	C₂	1	2	yes	maximal, complemented
10	#10625056	C₂	1	2	yes	maximal
11	#10625050	C₂	1	2	yes	maximal, complemented
12	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₆	C₂²	1	2	yes	—
17	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
22	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₂₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₂₃	C₂²	1	2	yes	—
26	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
28	Dic₂₃⋊C₄	C₂²	1	2	yes	—
29	Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₄×Dic₂₃	C₂²	1	2	yes	—
32	C₄×Dic₂₃	C₂²	1	2	yes	—
33	C₄×Dic₂₃	C₂²	1	2	yes	—
34	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
35	C₂³×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×D₂₃	C₂³	1	1	yes	characteristic
37	Q₈×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	2	2	no	—
39	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²⋊D₂₃	—	2	2	no	—
44	C₂×Dic₂₃	C₂³	1	2	yes	—
45	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₃	C₂³	1	2	yes	—
48	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
49	C₄×D₂₃	—	2	2	no	—
50	Dic₄₆	—	2	4	no	—
51	C₂²×C₂₃	—	2	1	no	—
52	C₂²×C₂₃	—	2	1	no	—
53	C₂²×C₂₃	—	2	1	no	—
54	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
55	C₂×D₂₃	—	2	1	no	—
56	C₂×D₂₃	—	2	1	no	—
57	C₂×D₂₃	—	2	1	no	—
58	C₄×C₂₃	—	2	2	no	—
59	C₄×C₂₃	—	2	1	no	—
60	C₄×C₂₃	—	2	2	no	—
61	C₄×C₂₃	#41	1	2	yes	—
62	Dic₂₃	—	2	2	no	—
63	Dic₂₃	—	2	1	no	—
64	Dic₂₃	—	2	2	no	—
65	Dic₂₃	—	2	1	no	—
66	Dic₂₃	#41	1	2	yes	—
67	#556	—	23	1	no	maximal, Sylow
68	C₂×C₂₃	—	4	1	no	—
69	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
71	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
72	D₂₃	—	4	1	no	—
73	C₄⋊Q₈	—	23	1	no	—
74	#126	—	23	2	no	—
75	#126	—	23	2	no	—
76	#124	—	23	2	no	—
77	C₂²⋊Q₈	—	23	1	no	—
78	C₂²⋊Q₈	D₂₃	1	1	yes	characteristic, complemented
79	C₄×Q₈	—	23	1	no	—
80	C₄×Q₈	—	23	2	no	—
81	C₄×D₄	—	23	1	no	—
82	#117	—	23	1	no	—
83	#117	—	23	1	no	—
84	C₂₃	#556	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	—	23	2	no	—
86	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
87	C₂×D₄	—	23	1	no	—
88	C₂²×C₄	—	23	1	no	—
89	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
90	C₄⋊C₄	—	23	1	no	—
91	C₄⋊C₄	—	23	1	no	—
92	C₄⋊C₄	—	23	1	no	—
93	C₄⋊C₄	—	23	2	no	—
94	C₄⋊C₄	—	23	2	no	—
95	C₄⋊C₄	—	23	2	no	—
96	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
98	C₂²⋊C₄	—	23	2	no	—
99	C₂²⋊C₄	—	23	1	no	—
100	C₂²⋊C₄	—	23	2	no	—
101	C₂²⋊C₄	—	23	1	no	—
102	C₂²⋊C₄	—	23	2	no	—
103	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
104	C₄²	—	23	2	no	—
105	C₄²	—	23	2	no	—
106	C₄²	—	23	1	no	—
107	C₄²	—	23	2	no	—
108	C₂³	—	23	1	no	—
109	C₂³	C₂²×D₂₃	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	46	4	no	—
112	D₄	—	46	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	23	1	no	—
115	C₂×C₄	—	23	2	no	—
116	C₂×C₄	—	23	1	no	—
117	C₂×C₄	—	23	1	no	—
118	C₂×C₄	—	23	2	no	—
119	C₂×C₄	—	46	2	no	—
120	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₂₃	1	2	yes	—
122	C₂×C₄	C₂²×D₂₃	1	2	yes	—
123	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	46	1	no	—
128	C₂²	—	46	1	no	—
129	C₂²	—	46	1	no	—
130	C₂²	C₂³×D₂₃	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	23	2	no	—
135	C₄	—	46	1	no	—
136	C₄	—	46	2	no	—
137	C₄	—	46	1	no	—
138	C₄	—	46	2	no	—
139	C₄	#67290	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	92	1	no	—
142	C₂	C₂×#67290	1	1	yes	characteristic
143	C₂	#10625158	1	1	yes	characteristic
144	C₂	#41×D₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×C₂²⋊Q₈

C₂

yes

maximal, characteristic, complemented

#10625117

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₂₃

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₂₃

C₂

yes

maximal, characteristic

#10625097

C₂

yes

maximal, characteristic, complemented

#10625069

C₂

yes

maximal, complemented

#10625063

C₂

yes

maximal, characteristic, complemented

#10625059

C₂

yes

maximal, characteristic, complemented

#10625058

C₂

yes

maximal, complemented

#10625056

C₂

yes

maximal

#10625050

C₂

yes

maximal, complemented

C₂×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

C₂³

yes

characteristic

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₄×D₂₃

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#41

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

#41

yes

—

#556

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

D₂₃

—

C₄⋊Q₈

—

#126

—

#126

—

#124

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₂₃

yes

characteristic, complemented

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

#117

—

C₂₃

#556

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₂₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₂₃

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

—

109

C₂³

C₂²×D₂₃

yes

characteristic

110

Q₈

—

111

Q₈

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²×D₂₃

yes

characteristic

121

C₂×C₄

C₂²×D₂₃

yes

—

122

C₂×C₄

C₂²×D₂₃

yes

—

123

C₂×C₄

C₂²×D₂₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂³×D₂₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#67290

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₂×#67290

yes

characteristic

143

C₂

#10625158

yes

characteristic

144

C₂

#41×D₂₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×C₂²⋊Q₈	C₂
prime radical	C₂×C₂²⋊D₂₃	C₂²
hypercenter	C₂²⋊Q₈	D₂₃
hypocenter	C₂₃	#556
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×C₂²⋊Q₈

C₂

prime radical

C₂×C₂²⋊D₂₃

C₂²

hypercenter

C₂²⋊Q₈

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. C₂₃×C₂²⋊Q₈	C₂	semidirect
2. #10625117	C₂	semidirect
3. Q₈×Dic₂₃	C₂	semidirect
5. #10625097	C₂	semidirect
6. #10625069	C₂	semidirect
7. #10625063	C₂	semidirect
8. #10625059	C₂	semidirect
9. #10625058	C₂	semidirect
11. #10625050	C₂	semidirect
78. C₂²⋊Q₈	D₂₃	semidirect
84. C₂₃	#556	semidirect

H ≅ G/N

type

1. C₂₃×C₂²⋊Q₈

C₂

semidirect

2. #10625117

C₂

semidirect

3. Q₈×Dic₂₃

C₂

semidirect

5. #10625097

C₂

semidirect

6. #10625069

C₂

semidirect

7. #10625063

C₂

semidirect

8. #10625059

C₂

semidirect

9. #10625058

C₂

semidirect

11. #10625050

C₂

semidirect

78. C₂²⋊Q₈

D₂₃

semidirect

84. C₂₃

#556

semidirect

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

subgroup	index
C₂₃×C₂²⋊Q₈	2
#10625117	2
Q₈×Dic₂₃	2
C₄×C₂²⋊D₂₃	2
#10625097	2
#10625069	2
#10625063	2
#10625059	2
#10625058	2
#10625056	2
#10625050	2
#556	23

subgroup

index

C₂₃×C₂²⋊Q₈

#10625117

Q₈×Dic₂₃

C₄×C₂²⋊D₂₃

Subgroups — #13093138

Conjugacy classes of subgroups

Complements

Maximal subgroups