Group #13092985 — subgroups

Subgroups — #13092985

Order 1472.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂×#67292	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625118	C₂	1	1	yes	maximal, characteristic, complemented
5	#10625111	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625055	C₂	1	4	yes	maximal, complemented
7	#10625053	C₂	1	4	yes	maximal, complemented
8	#10625045	C₂	1	1	yes	maximal, characteristic, complemented
9	#10625043	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₂₃	C₂²	1	2	yes	—
11	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	complemented
15	#67292	—	2	4	no	—
16	D₄×D₂₃	—	2	4	no	—
17	C₂×D₉₂	C₂²	1	1	yes	characteristic
18	C₂×D₉₂	C₂²	1	2	yes	—
19	C₂×D₉₂	C₂²	1	1	yes	characteristic, complemented
20	C₂×D₉₂	C₂²	1	4	yes	complemented
21	C₂×C₄×D₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₂₃	C₂²	1	2	yes	complemented
23	C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
24	C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
25	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	complemented
27	Dic₂₃⋊C₄	C₂²	1	2	yes	complemented
28	C₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
29	C₂³×C₂₃	C₂³	1	2	yes	—
30	C₂²×D₂₃	—	2	4	no	—
31	C₂²×D₂₃	—	4	4	no	—
32	C₂²×D₂₃	C₂³	1	2	yes	—
33	C₂²×D₂₃	C₂³	1	2	yes	—
34	C₂²×D₂₃	C₂³	1	1	yes	characteristic
35	Q₈×C₂₃	—	2	2	no	—
36	D₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²⋊D₂₃	—	2	4	no	—
41	C₂²⋊D₂₃	—	4	4	no	—
42	C₂×Dic₂₃	C₂³	1	2	yes	—
43	C₂×Dic₂₃	C₂³	1	1	yes	characteristic
44	D₉₂	—	2	4	no	—
45	D₉₂	—	2	2	no	—
46	D₉₂	—	2	2	no	—
47	D₉₂	—	4	4	no	—
48	C₄×D₂₃	—	2	4	no	—
49	C₄×D₂₃	D₄	1	4	yes	complemented
50	C₂²×C₂₃	—	2	2	no	—
51	C₂²×C₂₃	—	2	4	no	—
52	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₂₃	—	2	2	no	—
54	C₂×D₂₃	—	2	2	no	—
55	C₂×D₂₃	—	2	2	no	—
56	C₂×D₂₃	—	2	4	no	—
57	C₂×D₂₃	—	2	4	no	—
58	C₂×D₂₃	—	4	8	no	—
59	C₂×D₂₃	C₂×D₄	1	2	yes	—
60	C₄×C₂₃	—	2	2	no	—
61	C₄×C₂₃	—	2	2	no	—
62	C₄×C₂₃	C₂×D₄	1	2	yes	—
63	Dic₂₃	—	2	2	no	—
64	Dic₂₃	C₂×D₄	1	2	yes	—
65	#535	—	23	1	no	maximal, Sylow
66	C₂×C₂₃	—	4	2	no	—
67	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₂₃	H(5,𝔽₂)	1	2	yes	—
69	D₂₃	—	2	2	no	—
70	D₂₃	—	4	2	no	—
71	D₂₃	—	4	2	no	—
72	C₂×#41	—	23	1	no	—
73	C₂²×D₄	—	23	1	no	—
74	#127	—	23	1	no	—
75	#127	—	23	1	no	—
76	#124	—	23	1	no	—
77	#124	D₂₃	1	1	yes	characteristic, complemented
78	#121	—	23	4	no	—
79	#120	—	23	4	no	—
80	#117	—	23	1	no	—
81	C₂₃	#535	1	1	yes	characteristic, complemented, Sylow
82	C₂⁴	—	23	2	no	—
83	#41	—	46	4	no	—
84	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
85	C₂×D₄	—	23	2	no	—
86	C₂×D₄	—	23	4	no	—
87	C₂×D₄	—	23	2	no	—
88	C₂×D₄	—	23	1	no	—
89	C₂×D₄	—	23	1	no	—
90	C₂×D₄	—	23	4	no	—
91	C₂×D₄	—	46	4	no	—
92	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
93	C₂²×C₄	—	23	1	no	—
94	C₂²×C₄	—	23	2	no	—
95	C₄⋊C₄	—	23	2	no	—
96	C₂²⋊C₄	—	23	2	no	—
97	C₂²⋊C₄	—	23	4	no	—
98	C₂²⋊C₄	C₂×D₂₃	1	4	yes	complemented
99	C₄²	—	23	1	no	—
100	C₄²	C₂×D₂₃	1	1	yes	characteristic, complemented
101	C₂³	—	23	1	no	—
102	C₂³	—	23	2	no	—
103	C₂³	—	23	2	no	—
104	C₂³	—	46	4	no	—
105	C₂³	—	92	4	no	—
106	C₂³	C₂²×D₂₃	1	2	yes	—
107	Q₈	—	2	2	no	—
108	D₄	—	2	2	no	—
109	D₄	—	46	4	no	—
110	D₄	—	46	4	no	—
111	D₄	—	46	2	no	—
112	D₄	—	46	2	no	—
113	D₄	—	92	4	no	—
114	D₄	—	92	4	no	—
115	C₂×C₄	—	23	1	no	—
116	C₂×C₄	—	23	4	no	—
117	C₂×C₄	—	23	2	no	—
118	C₂×C₄	—	46	4	no	—
119	C₂×C₄	C₂²×D₂₃	1	2	yes	—
120	C₂×C₄	C₂²×D₂₃	1	2	yes	—
121	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	23	2	no	—
125	C₂²	—	46	2	no	—
126	C₂²	—	46	2	no	—
127	C₂²	—	46	2	no	—
128	C₂²	—	46	4	no	—
129	C₂²	—	46	4	no	—
130	C₂²	—	92	8	no	—
131	C₂²	C₂³×D₂₃	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	23	2	no	—
135	C₄	—	46	2	no	—
136	C₄	D₄×D₂₃	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	—	46	2	no	—
139	C₂	—	92	2	no	—
140	C₂	—	92	2	no	—
141	C₂	C₂×D₄×D₂₃	1	1	yes	characteristic
142	C₂	#10625160	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67292

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₂₃

C₂

yes

maximal, characteristic, complemented

C₂₃×#124

C₂

yes

maximal, characteristic, complemented

#10625118

C₂

yes

maximal, characteristic, complemented

#10625111

C₂

yes

maximal, characteristic, complemented

#10625055

C₂

yes

maximal, complemented

#10625053

C₂

yes

maximal, complemented

#10625045

C₂

yes

maximal, characteristic, complemented

#10625043

C₂

yes

maximal, characteristic, complemented

C₂³×D₂₃

C₂²

yes

—

C₂×Q₈×C₂₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

complemented

#67292

—

D₄×D₂₃

—

C₂×D₉₂

C₂²

yes

characteristic

C₂×D₉₂

C₂²

yes

—

C₂×D₉₂

C₂²

yes

characteristic, complemented

C₂×D₉₂

C₂²

yes

complemented

C₂×C₄×D₂₃

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

complemented

C₂²⋊C₄×C₂₃

C₂²

yes

complemented

C₄²×C₂₃

C₂²

yes

characteristic, complemented

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

complemented

Dic₂₃⋊C₄

C₂²

yes

complemented

C₄×Dic₂₃

C₂²

yes

characteristic, complemented

C₂³×C₂₃

C₂³

yes

—

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₂³

yes

—

C₂²×D₂₃

C₂³

yes

—

C₂²×D₂₃

C₂³

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

characteristic

D₉₂

—

D₉₂

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

D₄

yes

complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

C₂×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×D₄

yes

—

Dic₂₃

—

Dic₂₃

C₂×D₄

yes

—

#535

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₃

H(5,𝔽₂)

yes

—

D₂₃

—

D₂₃

—

D₂₃

—

C₂×#41

—

C₂²×D₄

—

—

—

—

D₂₃

yes

characteristic, complemented

#121

—

#120

—

#117

—

C₂₃

#535

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

C₂×D₂₃

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

complemented

C₄²

—

100

C₄²

C₂×D₂₃

yes

characteristic, complemented

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

C₂²×D₂₃

yes

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×D₂₃

yes

—

120

C₂×C₄

C₂²×D₂₃

yes

—

121

C₂×C₄

C₂²×D₂₃

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂³×D₂₃

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

D₄×D₂₃

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

C₂×D₄×D₂₃

yes

characteristic

142

C₂

#10625160

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#124	C₂
hypercenter	#124	D₂₃
hypocenter	C₂₃	#535
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#124

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₂×#67292	C₂	semidirect
2. C₂×D₄×D₂₃	C₂	semidirect
3. C₂₃×#124	C₂	semidirect
4. #10625118	C₂	semidirect
5. #10625111	C₂	semidirect
6. #10625055	C₂	semidirect
7. #10625053	C₂	semidirect
8. #10625045	C₂	semidirect
9. #10625043	C₂	semidirect
11. C₂×Q₈×C₂₃	C₂²	semidirect
14. C₂×C₂²⋊D₂₃	C₂²	semidirect
19. C₂×D₉₂	C₂²	semidirect
20. C₂×D₉₂	C₂²	semidirect
22. C₂×C₄×D₂₃	C₂²	semidirect
23. C₂²⋊C₄×C₂₃	C₂²	semidirect
24. C₄²×C₂₃	C₂²	semidirect
26. (C₂×D₂₃)⋊C₄	C₂²	semidirect
27. Dic₂₃⋊C₄	C₂²	semidirect
28. C₄×Dic₂₃	C₂²	semidirect
49. C₄×D₂₃	D₄	semidirect
77. #124	D₂₃	semidirect
81. C₂₃	#535	semidirect
84. C₂×Q₈	C₂×D₂₃	semidirect
98. C₂²⋊C₄	C₂×D₂₃	semidirect
100. C₄²	C₂×D₂₃	semidirect

H ≅ G/N

type

1. C₂×#67292

C₂

semidirect

2. C₂×D₄×D₂₃

C₂

semidirect

3. C₂₃×#124

C₂

semidirect

4. #10625118

C₂

semidirect

5. #10625111

C₂

semidirect

6. #10625055

C₂

semidirect

7. #10625053

C₂

semidirect

8. #10625045

C₂

semidirect

9. #10625043

C₂

semidirect

11. C₂×Q₈×C₂₃

C₂²

semidirect

14. C₂×C₂²⋊D₂₃

C₂²

semidirect

19. C₂×D₉₂

C₂²

semidirect

20. C₂×D₉₂

C₂²

semidirect

22. C₂×C₄×D₂₃

C₂²

semidirect

23. C₂²⋊C₄×C₂₃

C₂²

semidirect

24. C₄²×C₂₃

C₂²

semidirect

26. (C₂×D₂₃)⋊C₄

C₂²

semidirect

27. Dic₂₃⋊C₄

C₂²

semidirect

28. C₄×Dic₂₃

C₂²

semidirect

49. C₄×D₂₃

D₄

semidirect

77. #124

D₂₃

semidirect

81. C₂₃

#535

semidirect

84. C₂×Q₈

C₂×D₂₃

semidirect

98. C₂²⋊C₄

C₂×D₂₃

semidirect

100. C₄²

C₂×D₂₃

semidirect

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

subgroup	index
C₂×#67292	2
C₂×D₄×D₂₃	2
C₂₃×#124	2
#10625118	2
#10625111	2
#10625055	2
#10625053	2
#10625045	2
#10625043	2
#535	23

subgroup

index

C₂×#67292

C₂×D₄×D₂₃

C₂₃×#124

Subgroups — #13092985

Conjugacy classes of subgroups

Complements

Maximal subgroups