Group #13093150 — subgroups

Subgroups — #13093150

Order 1472.

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

normal subgroups only

This group has 236 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	C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625111	C₂	1	1	yes	maximal, characteristic
3	#10625107	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625100	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
6	#10625064	C₂	1	1	yes	maximal, characteristic, complemented
7	#10625058	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625056	C₂	1	2	yes	maximal, complemented
9	#10625054	C₂	1	2	yes	maximal, complemented
10	#10625046	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
16	C₂×D₉₂	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₂₃	C₂²	1	2	yes	—
18	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
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	C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
29	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
30	Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₂₃⋊C₄	C₂²	1	2	yes	—
32	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₃	C₂²	1	2	yes	—
35	C₂³×C₂₃	C₂³	1	2	yes	—
36	C₂²×D₂₃	C₂³	1	2	yes	—
37	D₄×C₂₃	—	2	2	no	—
38	C₂×C₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	C₂³	1	2	yes	—
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	C₂²⋊D₂₃	—	2	4	no	—
44	C₂²⋊D₂₃	—	4	2	no	—
45	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₃	C₂³	1	2	yes	—
49	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
50	D₉₂	—	2	2	no	—
51	C₄×D₂₃	—	4	2	no	—
52	Dic₄₆	—	2	2	no	—
53	Dic₄₆	—	2	2	no	—
54	C₂²×C₂₃	—	2	2	no	—
55	C₂²×C₂₃	—	2	2	no	—
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
58	C₂×D₂₃	—	2	2	no	—
59	C₂×D₂₃	—	2	2	no	—
60	C₂×D₂₃	—	2	2	no	—
61	C₄×C₂₃	—	2	1	no	—
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	2	1	no	—
64	C₄×C₂₃	#41	1	2	yes	—
65	Dic₂₃	—	2	2	no	—
66	Dic₂₃	—	2	1	no	—
67	Dic₂₃	—	2	1	no	—
68	Dic₂₃	—	2	1	no	—
69	Dic₂₃	#41	1	2	yes	—
70	#559	—	23	1	no	maximal, Sylow
71	C₂×C₂₃	—	4	2	no	—
72	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
73	C₂×C₂₃	C₂×#41	1	1	yes	characteristic
74	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
75	D₂₃	—	4	2	no	—
76	#127	—	23	1	no	—
77	#124	—	23	1	no	—
78	#124	—	23	2	no	—
79	#124	—	23	1	no	—
80	#123	—	23	2	no	—
81	#123	—	23	2	no	—
82	C₄×Q₈	—	23	1	no	—
83	C₄×Q₈	—	23	1	no	—
84	C₄×D₄	—	23	2	no	—
85	C₄×D₄	—	23	1	no	—
86	C₄×D₄	D₂₃	1	1	yes	characteristic, complemented
87	C₂₃	#559	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₈	—	23	1	no	—
89	C₂×Q₈	—	23	1	no	—
90	C₂×D₄	—	23	2	no	—
91	C₂×D₄	—	23	2	no	—
92	C₂×D₄	—	23	1	no	—
93	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
94	C₂²×C₄	—	23	2	no	—
95	C₂²×C₄	C₂×D₂₃	1	2	yes	—
96	C₄⋊C₄	—	23	1	no	—
97	C₄⋊C₄	—	23	2	no	—
98	C₄⋊C₄	—	23	1	no	—
99	C₄⋊C₄	—	23	1	no	—
100	C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
101	C₂²⋊C₄	—	23	2	no	—
102	C₂²⋊C₄	—	23	2	no	—
103	C₂²⋊C₄	—	23	2	no	—
104	C₂²⋊C₄	—	23	2	no	—
105	C₂²⋊C₄	—	23	2	no	—
106	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
107	C₄²	—	23	2	no	—
108	C₄²	—	23	1	no	—
109	C₄²	—	23	1	no	—
110	C₄²	C₂×D₂₃	1	1	yes	characteristic
111	C₂³	—	23	2	no	—
112	C₂³	C₂²×D₂₃	1	2	yes	—
113	Q₈	—	46	2	no	—
114	Q₈	—	46	2	no	—
115	D₄	—	2	2	no	—
116	D₄	—	46	4	no	—
117	D₄	—	46	2	no	—
118	D₄	—	92	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	23	1	no	—
121	C₂×C₄	—	23	2	no	—
122	C₂×C₄	—	23	1	no	—
123	C₂×C₄	—	23	1	no	—
124	C₂×C₄	—	23	1	no	—
125	C₂×C₄	—	92	2	no	—
126	C₂×C₄	C₂²×D₂₃	1	2	yes	—
127	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
128	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	46	2	no	—
134	C₂²	—	46	2	no	—
135	C₂²	—	46	2	no	—
136	C₂²	C₂³×D₂₃	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	23	2	no	—
141	C₄	—	46	2	no	—
142	C₄	—	46	1	no	—
143	C₄	—	46	1	no	—
144	C₄	—	46	1	no	—
145	C₄	#67288	1	2	yes	—
146	C₂	—	4	2	no	—
147	C₂	—	92	2	no	—
148	C₂	C₂×#67288	1	1	yes	characteristic
149	C₂	#10625155	1	1	yes	characteristic
150	C₂	#41×D₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄×C₂₃

C₂

yes

maximal, characteristic, complemented

#10625111

C₂

yes

maximal, characteristic

#10625107

C₂

yes

maximal, characteristic, complemented

#10625100

C₂

yes

maximal, complemented

C₄×C₂²⋊D₂₃

C₂

yes

maximal, complemented

#10625064

C₂

yes

maximal, characteristic, complemented

#10625058

C₂

yes

maximal, characteristic, complemented

#10625056

C₂

yes

maximal, complemented

#10625054

C₂

yes

maximal, complemented

#10625046

C₂

yes

maximal, characteristic, complemented

C₄×Dic₄₆

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×D₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

characteristic

C₂×Dic₄₆

C₂²

yes

characteristic

C₄⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₄²×C₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

characteristic

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

characteristic

C₄×Dic₂₃

C₂²

yes

—

C₂³×C₂₃

C₂³

yes

—

C₂²×D₂₃

C₂³

yes

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

characteristic

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

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

#41

yes

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

#41

yes

—

#559

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

C₂×#41

yes

characteristic

C₂×C₂₃

H(5,𝔽₂)

yes

characteristic

D₂₃

—

—

—

—

—

—

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

D₂₃

yes

characteristic, complemented

C₂₃

#559

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

C₂×D₂₃

yes

characteristic

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

C₂×D₂₃

yes

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂×D₂₃

yes

characteristic

111

C₂³

—

112

C₂³

C₂²×D₂₃

yes

—

113

Q₈

—

114

Q₈

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂²×D₂₃

yes

—

127

C₂×C₄

C₂²×D₂₃

yes

characteristic

128

C₂×C₄

C₂²×D₂₃

yes

characteristic

129

C₂×C₄

C₂²×D₂₃

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂³×D₂₃

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#67288

yes

—

146

C₂

—

147

C₂

—

148

C₂

C₂×#67288

yes

characteristic

149

C₂

#10625155

yes

characteristic

150

C₂

#41×D₂₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₄×D₄×C₂₃	C₂
prime radical	#10625111	C₂
hypercenter	C₄×D₄	D₂₃
hypocenter	C₂₃	#559
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₄×D₄×C₂₃

C₂

prime radical

#10625111

C₂

hypercenter

C₄×D₄

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₄×D₄×C₂₃	C₂	semidirect
3. #10625107	C₂	semidirect
4. #10625100	C₂	semidirect
5. C₄×C₂²⋊D₂₃	C₂	semidirect
6. #10625064	C₂	semidirect
7. #10625058	C₂	semidirect
8. #10625056	C₂	semidirect
9. #10625054	C₂	semidirect
10. #10625046	C₂	semidirect
11. C₄×Dic₄₆	C₂	semidirect
86. C₄×D₄	D₂₃	semidirect
87. C₂₃	#559	semidirect

H ≅ G/N

type

1. C₄×D₄×C₂₃

C₂

semidirect

3. #10625107

C₂

semidirect

4. #10625100

C₂

semidirect

5. C₄×C₂²⋊D₂₃

C₂

semidirect

6. #10625064

C₂

semidirect

7. #10625058

C₂

semidirect

8. #10625056

C₂

semidirect

9. #10625054

C₂

semidirect

10. #10625046

C₂

semidirect

11. C₄×Dic₄₆

C₂

semidirect

86. C₄×D₄

D₂₃

semidirect

87. C₂₃

#559

semidirect

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

subgroup	index
C₄×D₄×C₂₃	2
#10625111	2
#10625107	2
#10625100	2
C₄×C₂²⋊D₂₃	2
#10625064	2
#10625058	2
#10625056	2
#10625054	2
#10625046	2
C₄×Dic₄₆	2
#559	23

subgroup

index

C₄×D₄×C₂₃

C₄×C₂²⋊D₂₃

C₄×Dic₄₆

Subgroups — #13093150

Conjugacy classes of subgroups

Complements

Maximal subgroups