Group #13092924 — subgroups

Subgroups — #13092924

Order 1472.

330 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 115 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×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625117	C₂	1	2	yes	maximal, complemented
4	#10625109	C₂	1	2	yes	maximal, complemented
5	#10625108	C₂	1	2	yes	maximal, complemented
6	#10625106	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊D₂₃	C₂	1	4	yes	maximal, complemented
8	#10625097	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₂₃	C₂²	1	1	yes	characteristic
10	#41×C₂₃	—	2	4	no	—
11	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	C₂²	1	2	yes	complemented
16	C₂×C₂²⋊D₂₃	C₂²	1	4	yes	—
17	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₂₃	—	2	4	no	—
19	C₂×C₄×D₂₃	C₂²	1	2	yes	—
20	C₂×C₄×D₂₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₂₃	C₂²	1	4	yes	complemented
23	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
24	C₄⋊Dic₂₃	C₂²	1	2	yes	complemented
25	Dic₂₃⋊C₄	C₂²	1	4	yes	complemented
26	C₄×Dic₂₃	C₂²	1	2	yes	complemented
27	C₂³×C₂₃	C₂³	1	2	yes	—
28	C₂³×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×D₂₃	—	2	2	no	—
30	C₂²×D₂₃	—	2	4	no	—
31	C₂²×D₂₃	C₂³	1	2	yes	—
32	Q₈×C₂₃	—	2	2	no	—
33	D₄×C₂₃	—	2	4	no	—
34	D₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	D₄	1	4	yes	complemented
37	C₂×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²⋊D₂₃	—	2	8	no	—
41	C₂×Dic₂₃	—	2	2	no	—
42	C₂×Dic₂₃	C₂³	1	4	yes	—
43	C₂×Dic₂₃	C₂³	1	2	yes	—
44	C₄×D₂₃	—	2	4	no	—
45	C₄×D₂₃	—	2	4	no	—
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	—	2	2	no	—
48	C₂²×C₂₃	—	2	4	no	—
49	C₂²×C₂₃	C₂×D₄	1	2	yes	—
50	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×D₂₃	—	2	8	no	—
52	C₂×D₂₃	—	2	2	no	—
53	C₂×D₂₃	—	2	2	no	—
54	C₂×D₂₃	#41	1	4	yes	complemented
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₄×C₂₃	C₂×D₄	1	2	yes	—
58	Dic₂₃	—	2	2	no	—
59	Dic₂₃	—	2	4	no	—
60	#525	—	23	1	no	maximal, Sylow
61	C₂×C₂₃	—	2	2	no	—
62	C₂×C₂₃	—	4	2	no	—
63	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
64	C₂×C₂₃	C₂×#41	1	2	yes	—
65	D₂₃	—	2	4	no	—
66	C₂×#41	D₂₃	1	1	yes	characteristic, complemented
67	C₂³×C₄	—	23	1	no	—
68	#123	—	23	2	no	—
69	C₂²⋊Q₈	—	23	2	no	—
70	#121	—	23	2	no	—
71	#120	—	23	2	no	—
72	C₄×D₄	—	23	4	no	—
73	#117	—	23	1	no	—
74	C₂₃	#525	1	1	yes	characteristic, complemented, Sylow
75	C₂⁴	—	23	1	no	—
76	#41	—	2	4	no	—
77	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
78	C₂×D₄	—	23	4	no	—
79	C₂×D₄	C₂×D₂₃	1	2	yes	—
80	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic, complemented
81	C₂²×C₄	—	23	1	no	—
82	C₂²×C₄	—	23	2	no	—
83	C₂²×C₄	—	23	2	no	—
84	C₂²×C₄	—	46	4	no	—
85	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₂₃	1	2	yes	complemented
87	C₄⋊C₄	—	23	2	no	—
88	C₄⋊C₄	—	23	4	no	—
89	C₂²⋊C₄	—	23	2	no	—
90	C₂²⋊C₄	—	23	4	no	—
91	C₂²⋊C₄	—	23	4	no	—
92	C₄²	—	23	2	no	—
93	C₂³	—	23	2	no	—
94	C₂³	—	46	4	no	—
95	C₂³	—	46	2	no	—
96	C₂³	C₂²×D₂₃	1	2	yes	—
97	C₂³	C₂²×D₂₃	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	D₄	—	2	4	no	—
100	D₄	—	2	2	no	—
101	D₄	—	46	8	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	23	4	no	—
104	C₂×C₄	—	23	2	no	—
105	C₂×C₄	—	46	2	no	—
106	C₂×C₄	—	46	4	no	—
107	C₂×C₄	—	46	4	no	—
108	C₂×C₄	C₂²⋊D₂₃	1	4	yes	complemented
109	C₂×C₄	C₂²×D₂₃	1	2	yes	—
110	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	23	4	no	—
116	C₂²	—	46	8	no	—
117	C₂²	—	46	2	no	—
118	C₂²	—	46	2	no	—
119	C₂²	C₂×C₂²⋊D₂₃	1	2	yes	—
120	C₂²	C₂³×D₂₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	46	4	no	—
123	C₄	—	46	2	no	—
124	C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
125	C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	4	2	no	—
128	C₂	—	46	4	no	—
129	C₂	#41×D₂₃	1	2	yes	—
130	C₂	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#41×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₂

yes

maximal, characteristic, complemented

#10625117

C₂

yes

maximal, complemented

#10625109

C₂

yes

maximal, complemented

#10625108

C₂

yes

maximal, complemented

#10625106

C₂

yes

maximal, complemented

C₄×C₂²⋊D₂₃

C₂

yes

maximal, complemented

#10625097

C₂

yes

maximal, characteristic, complemented

C₂³×D₂₃

C₂²

yes

characteristic

#41×C₂₃

—

C₂×Q₈×C₂₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₂₃

C₂²

yes

—

C₂×D₄×C₂₃

C₂²

yes

characteristic, complemented

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

complemented

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

C₂²

yes

—

C₂×C₄×D₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

—

C₂²⋊Dic₂₃

C₂²

yes

complemented

(C₂×D₂₃)⋊C₄

C₂²

yes

—

C₄⋊Dic₂₃

C₂²

yes

complemented

Dic₂₃⋊C₄

C₂²

yes

complemented

C₄×Dic₂₃

C₂²

yes

complemented

C₂³×C₂₃

C₂³

yes

—

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₂³

yes

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

D₄

yes

complemented

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₄×D₂₃

—

C₄×D₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×D₄

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

#41

yes

complemented

C₄×C₂₃

—

C₄×C₂₃

C₂×D₄

yes

—

C₄×C₂₃

C₂×D₄

yes

—

Dic₂₃

—

Dic₂₃

—

#525

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₃

C₂×#41

yes

—

D₂₃

—

C₂×#41

D₂₃

yes

characteristic, complemented

C₂³×C₄

—

#123

—

C₂²⋊Q₈

—

#121

—

#120

—

C₄×D₄

—

#117

—

C₂₃

#525

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

C₂×D₂₃

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

—

C₂×D₄

C₂×D₂₃

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₂²×C₄

C₂×D₂₃

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×D₂₃

yes

—

C₂³

C₂²×D₂₃

yes

characteristic

Q₈

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²⋊D₂₃

yes

complemented

109

C₂×C₄

C₂²×D₂₃

yes

—

110

C₂×C₄

C₂²×D₂₃

yes

characteristic

111

C₂×C₄

C₂²×D₂₃

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂×C₂²⋊D₂₃

yes

—

120

C₂²

C₂³×D₂₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₂×C₂²⋊D₂₃

yes

—

125

C₄

C₂×C₂²⋊D₂₃

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

#41×D₂₃

yes

—

130

C₂

C₂²×C₂²⋊D₂₃

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Frattini	C₂²	C₂³×D₂₃
Fitting	C₂×#41×C₂₃	C₂
hypercenter	C₂×#41	D₂₃
hypocenter	C₂₃	#525
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂×C₄

C₂²×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Frattini

C₂²

C₂³×D₂₃

Fitting

C₂×#41×C₂₃

C₂

hypercenter

C₂×#41

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#41×C₂₃	C₂	semidirect
2. C₂²×C₄×D₂₃	C₂	semidirect
3. #10625117	C₂	semidirect
4. #10625109	C₂	semidirect
5. #10625108	C₂	semidirect
6. #10625106	C₂	semidirect
7. C₄×C₂²⋊D₂₃	C₂	semidirect
8. #10625097	C₂	semidirect
11. C₂×Q₈×C₂₃	C₂²	semidirect
13. C₂×D₄×C₂₃	C₂²	semidirect
15. C₂²×C₄×C₂₃	C₂²	semidirect
22. C₂²⋊Dic₂₃	C₂²	semidirect
24. C₄⋊Dic₂₃	C₂²	semidirect
25. Dic₂₃⋊C₄	C₂²	semidirect
26. C₄×Dic₂₃	C₂²	semidirect
36. C₂×C₄×C₂₃	D₄	semidirect
54. C₂×D₂₃	#41	semidirect
66. C₂×#41	D₂₃	semidirect
74. C₂₃	#525	semidirect
77. C₂×Q₈	C₂×D₂₃	semidirect
80. C₂×D₄	C₂×D₂₃	semidirect
86. C₂²×C₄	C₂×D₂₃	semidirect
108. C₂×C₄	C₂²⋊D₂₃	semidirect

H ≅ G/N

type

1. C₂×#41×C₂₃

C₂

semidirect

2. C₂²×C₄×D₂₃

C₂

semidirect

3. #10625117

C₂

semidirect

4. #10625109

C₂

semidirect

5. #10625108

C₂

semidirect

6. #10625106

C₂

semidirect

7. C₄×C₂²⋊D₂₃

C₂

semidirect

8. #10625097

C₂

semidirect

11. C₂×Q₈×C₂₃

C₂²

semidirect

13. C₂×D₄×C₂₃

C₂²

semidirect

15. C₂²×C₄×C₂₃

C₂²

semidirect

22. C₂²⋊Dic₂₃

C₂²

semidirect

24. C₄⋊Dic₂₃

C₂²

semidirect

25. Dic₂₃⋊C₄

C₂²

semidirect

26. C₄×Dic₂₃

C₂²

semidirect

36. C₂×C₄×C₂₃

D₄

semidirect

54. C₂×D₂₃

#41

semidirect

66. C₂×#41

D₂₃

semidirect

74. C₂₃

#525

semidirect

77. C₂×Q₈

C₂×D₂₃

semidirect

80. C₂×D₄

C₂×D₂₃

semidirect

86. C₂²×C₄

C₂×D₂₃

semidirect

108. C₂×C₄

C₂²⋊D₂₃

semidirect

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

subgroup	index
C₂×#41×C₂₃	2
C₂²×C₄×D₂₃	2
#10625117	2
#10625109	2
#10625108	2
#10625106	2
C₄×C₂²⋊D₂₃	2
#10625097	2
#525	23

subgroup

index

C₂×#41×C₂₃

C₂²×C₄×D₂₃

C₄×C₂²⋊D₂₃

Subgroups — #13092924

Conjugacy classes of subgroups

Complements

Maximal subgroups