Group #13093038 — subgroups

Subgroups — #13093038

Order 1472.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625101	C₂	1	1	yes	maximal, characteristic
3	#10625095	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625069	C₂	1	2	yes	maximal, complemented
5	#10625067	C₂	1	2	yes	maximal, complemented
6	#10625056	C₂	1	2	yes	maximal, complemented
7	#10625054	C₂	1	2	yes	maximal, complemented
8	#10625046	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₉₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#10625040	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
14	C₂×D₉₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₃	C₂²	1	2	yes	—
16	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₆	C₂²	1	2	yes	—
18	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
25	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
28	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
29	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
31	C₄⋊Dic₂₃	C₂²	1	2	yes	—
32	Dic₂₃⋊C₄	C₂²	1	2	yes	—
33	Dic₂₃⋊C₄	C₂²	1	2	yes	—
34	C₄×Dic₂₃	C₂²	1	2	yes	—
35	C₂³×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²×D₂₃	C₂³	1	2	yes	—
37	C₂×C₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²⋊D₂₃	—	4	2	no	—
45	C₂×Dic₂₃	C₂³	1	2	yes	—
46	C₂×Dic₂₃	C₂³	1	2	yes	—
47	C₂×Dic₂₃	C₂³	1	2	yes	—
48	D₉₂	—	2	2	no	—
49	C₄×D₂₃	—	4	2	no	—
50	Dic₄₆	—	2	2	no	—
51	Dic₄₆	—	4	2	no	—
52	C₂²×C₂₃	—	2	1	no	—
53	C₂²×C₂₃	—	2	1	no	—
54	C₂²×C₂₃	—	2	1	no	—
55	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
56	C₂×D₂₃	—	2	2	no	—
57	C₂×D₂₃	—	2	2	no	—
58	C₂×D₂₃	—	2	2	no	—
59	C₄×C₂₃	—	2	1	no	—
60	C₄×C₂₃	—	2	1	no	—
61	C₄×C₂₃	—	2	1	no	—
62	C₄×C₂₃	—	2	1	no	—
63	C₄×C₂₃	—	2	1	no	—
64	C₄×C₂₃	#41	1	2	yes	—
65	Dic₂₃	—	2	2	no	—
66	Dic₂₃	—	2	2	no	—
67	Dic₂₃	—	2	2	no	—
68	#542	—	23	1	no	maximal, Sylow
69	C₂×C₂₃	—	4	1	no	—
70	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
71	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
73	D₂₃	—	4	2	no	—
74	C₄⋊Q₈	—	23	1	no	—
75	#126	—	23	2	no	—
76	#124	—	23	2	no	—
77	#124	—	23	1	no	—
78	#123	—	23	2	no	—
79	C₂²⋊Q₈	—	23	1	no	—
80	C₂²⋊Q₈	—	23	2	no	—
81	#121	—	23	1	no	—
82	C₄×Q₈	—	23	1	no	—
83	C₄×D₄	—	23	1	no	—
84	#117	D₂₃	1	1	yes	characteristic, complemented
85	C₂₃	#542	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₈	—	23	1	no	—
87	C₂×Q₈	—	23	2	no	—
88	C₂×D₄	—	23	2	no	—
89	C₂×D₄	—	23	1	no	—
90	C₂²×C₄	—	23	2	no	—
91	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
92	C₄⋊C₄	—	23	1	no	—
93	C₄⋊C₄	—	23	1	no	—
94	C₄⋊C₄	—	23	2	no	—
95	C₄⋊C₄	—	23	2	no	—
96	C₄⋊C₄	—	23	2	no	—
97	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
99	C₂²⋊C₄	—	23	2	no	—
100	C₂²⋊C₄	—	23	2	no	—
101	C₂²⋊C₄	—	23	2	no	—
102	C₂²⋊C₄	—	23	2	no	—
103	C₂²⋊C₄	—	23	2	no	—
104	C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
105	C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
106	C₄²	—	23	2	no	—
107	C₄²	C₂×D₂₃	1	1	yes	characteristic
108	C₄²	C₂×D₂₃	1	1	yes	characteristic
109	C₂³	—	23	2	no	—
110	C₂³	C₂²×D₂₃	1	1	yes	characteristic
111	Q₈	—	46	2	no	—
112	Q₈	—	92	2	no	—
113	D₄	—	46	2	no	—
114	D₄	—	92	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	23	2	no	—
117	C₂×C₄	—	23	2	no	—
118	C₂×C₄	—	23	2	no	—
119	C₂×C₄	—	92	2	no	—
120	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	46	2	no	—
130	C₂²	—	46	2	no	—
131	C₂²	—	46	2	no	—
132	C₂²	C₂³×D₂₃	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	46	2	no	—
139	C₄	—	46	2	no	—
140	C₄	—	46	2	no	—
141	C₄	#67288	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	92	2	no	—
144	C₂	C₂×#67288	1	1	yes	characteristic
145	C₂	#10625160	1	1	yes	characteristic
146	C₂	#10625161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#117

C₂

yes

maximal, characteristic, complemented

#10625101

C₂

yes

maximal, characteristic

#10625095

C₂

yes

maximal, characteristic, complemented

#10625069

C₂

yes

maximal, complemented

#10625067

C₂

yes

maximal, complemented

#10625056

C₂

yes

maximal, complemented

#10625054

C₂

yes

maximal, complemented

#10625046

C₂

yes

maximal, characteristic, complemented

C₄×D₉₂

C₂

yes

maximal, characteristic, complemented

#10625040

C₂

yes

maximal, characteristic, complemented

C₄×Dic₄₆

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×D₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

characteristic

C₄²×C₂₃

C₂²

yes

characteristic

C₄²×C₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₃

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

C₂³

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

—

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

—

C₄×C₂₃

—

C₄×C₂₃

#41

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

#542

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

H(5,𝔽₂)

yes

characteristic

C₂×C₂₃

2₋¹⁺⁴

yes

characteristic

D₂₃

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

D₂₃

yes

characteristic, complemented

C₂₃

#542

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃

yes

characteristic

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

C₂×D₂₃

yes

characteristic

105

C₂²⋊C₄

C₂×D₂₃

yes

characteristic

106

C₄²

—

107

C₄²

C₂×D₂₃

yes

characteristic

108

C₄²

C₂×D₂₃

yes

characteristic

109

C₂³

—

110

C₂³

C₂²×D₂₃

yes

characteristic

111

Q₈

—

112

Q₈

—

113

D₄

—

114

D₄

—

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

characteristic

122

C₂×C₄

C₂²×D₂₃

yes

characteristic

123

C₂×C₄

C₂²×D₂₃

yes

characteristic

124

C₂×C₄

C₂²×D₂₃

yes

characteristic

125

C₂×C₄

C₂²×D₂₃

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×D₂₃

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#67288

yes

—

142

C₂

—

143

C₂

—

144

C₂

C₂×#67288

yes

characteristic

145

C₂

#10625160

yes

characteristic

146

C₂

#10625161

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#117	C₂
prime radical	#10625101	C₂
hypercenter	#117	D₂₃
hypocenter	C₂₃	#542
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#117

C₂

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂₃×#117	C₂	semidirect
3. #10625095	C₂	semidirect
4. #10625069	C₂	semidirect
5. #10625067	C₂	semidirect
6. #10625056	C₂	semidirect
7. #10625054	C₂	semidirect
8. #10625046	C₂	semidirect
9. C₄×D₉₂	C₂	semidirect
10. #10625040	C₂	semidirect
11. C₄×Dic₄₆	C₂	semidirect
84. #117	D₂₃	semidirect
85. C₂₃	#542	semidirect

H ≅ G/N

type

1. C₂₃×#117

C₂

semidirect

3. #10625095

C₂

semidirect

4. #10625069

C₂

semidirect

5. #10625067

C₂

semidirect

6. #10625056

C₂

semidirect

7. #10625054

C₂

semidirect

8. #10625046

C₂

semidirect

9. C₄×D₉₂

C₂

semidirect

10. #10625040

C₂

semidirect

11. C₄×Dic₄₆

C₂

semidirect

84. #117

D₂₃

semidirect

85. C₂₃

#542

semidirect

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

subgroup	index
C₂₃×#117	2
#10625101	2
#10625095	2
#10625069	2
#10625067	2
#10625056	2
#10625054	2
#10625046	2
C₄×D₉₂	2
#10625040	2
C₄×Dic₄₆	2
#542	23

subgroup

index

C₂₃×#117

C₄×D₉₂

C₄×Dic₄₆

Subgroups — #13093038

Conjugacy classes of subgroups

Complements

Maximal subgroups