Group #13092929 — subgroups

Subgroups — #13092929

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×#67288	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625068	C₂	1	2	yes	maximal, complemented
5	#10625066	C₂	1	2	yes	maximal, complemented
6	#10625057	C₂	1	2	yes	maximal, complemented
7	#10625053	C₂	1	2	yes	maximal, complemented
8	C₄×D₉₂	C₂	1	4	yes	maximal, complemented
9	C₂³×D₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
12	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
13	#67288	—	2	4	no	—
14	C₂×D₉₂	C₂²	1	4	yes	—
15	C₂×D₉₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₄×D₂₃	—	2	4	no	—
17	C₂×C₄×D₂₃	C₂²	1	2	yes	—
18	C₂×C₄×D₂₃	C₂²	1	2	yes	—
19	C₂×C₄×D₂₃	C₂²	1	2	yes	complemented
20	C₂×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
21	C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
22	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₄²×C₂₃	C₂²	1	2	yes	complemented
24	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	—
25	(C₂×D₂₃)⋊C₄	C₂²	1	4	yes	complemented
26	C₄⋊Dic₂₃	C₂²	1	4	yes	complemented
27	C₂³×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×D₂₃	—	2	2	no	—
29	C₂²×D₂₃	—	2	4	no	—
30	C₂²×D₂₃	C₂³	1	2	yes	—
31	C₂²×D₂₃	C₂³	1	2	yes	—
32	C₂×C₄×C₂₃	D₄	1	4	yes	complemented
33	C₂×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂²⋊D₂₃	—	2	4	no	—
37	C₂×Dic₂₃	—	2	2	no	—
38	C₂×Dic₂₃	C₂³	1	2	yes	—
39	C₂×Dic₂₃	C₂³	1	2	yes	—
40	D₉₂	—	2	8	no	—
41	D₉₂	—	2	2	no	—
42	C₄×D₂₃	—	2	4	no	—
43	C₄×D₂₃	—	2	4	no	—
44	C₄×D₂₃	—	2	4	no	—
45	Dic₄₆	—	2	2	no	—
46	C₂²×C₂₃	—	2	2	no	—
47	C₂²×C₂₃	C₂×D₄	1	2	yes	—
48	C₂²×C₂₃	C₂⁴	1	1	yes	characteristic
49	C₂×D₂₃	—	2	8	no	—
50	C₂×D₂₃	—	2	2	no	—
51	C₂×D₂₃	—	2	2	no	—
52	C₂×D₂₃	—	2	2	no	—
53	C₂×D₂₃	—	2	4	no	—
54	C₂×D₂₃	#41	1	4	yes	complemented
55	C₄×C₂₃	—	2	4	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	2	no	—
60	#525	—	23	1	no	maximal, Sylow
61	C₂×C₂₃	—	2	2	no	—
62	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₂₃	C₂×#41	1	2	yes	—
64	D₂₃	—	2	4	no	—
65	D₂₃	—	4	2	no	—
66	C₂×#41	—	23	1	no	—
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	D₂₃	1	1	yes	characteristic, complemented
74	C₂₃	#525	1	1	yes	characteristic, complemented, Sylow
75	C₂⁴	—	23	1	no	—
76	#41	—	46	4	no	—
77	C₂×Q₈	—	23	1	no	—
78	C₂×D₄	—	23	2	no	—
79	C₂×D₄	—	23	4	no	—
80	C₂×D₄	—	23	1	no	—
81	C₂²×C₄	—	23	1	no	—
82	C₂²×C₄	—	23	2	no	—
83	C₂²×C₄	—	23	2	no	—
84	C₂²×C₄	—	23	2	no	—
85	C₂²×C₄	—	46	4	no	—
86	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
87	C₄⋊C₄	—	23	4	no	—
88	C₄⋊C₄	C₂×D₂₃	1	2	yes	complemented
89	C₂²⋊C₄	—	23	4	no	—
90	C₂²⋊C₄	—	23	4	no	—
91	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
92	C₄²	C₂×D₂₃	1	2	yes	complemented
93	C₂³	—	23	2	no	—
94	C₂³	—	23	2	no	—
95	C₂³	—	46	4	no	—
96	C₂³	—	46	2	no	—
97	C₂³	C₂²×D₂₃	1	1	yes	characteristic
98	Q₈	—	46	2	no	—
99	D₄	—	46	8	no	—
100	D₄	—	46	4	no	—
101	D₄	—	46	2	no	—
102	C₂×C₄	—	23	2	no	—
103	C₂×C₄	—	23	2	no	—
104	C₂×C₄	—	46	2	no	—
105	C₂×C₄	—	46	4	no	—
106	C₂×C₄	—	46	4	no	—
107	C₂×C₄	—	46	4	no	—
108	C₂×C₄	D₉₂	1	4	yes	complemented
109	C₂×C₄	C₂²×D₂₃	1	4	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	—	23	4	no	—
114	C₂²	—	46	8	no	—
115	C₂²	—	46	2	no	—
116	C₂²	—	46	2	no	—
117	C₂²	—	46	2	no	—
118	C₂²	—	46	4	no	—
119	C₂²	C₂×D₉₂	1	2	yes	—
120	C₂²	C₂³×D₂₃	1	1	yes	characteristic
121	C₄	—	2	4	no	—
122	C₄	—	46	2	no	—
123	C₄	—	46	2	no	—
124	C₄	C₂×D₉₂	1	2	yes	—
125	C₄	C₂×D₉₂	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	46	4	no	—
128	C₂	—	92	2	no	—
129	C₂	C₂²×D₉₂	1	1	yes	characteristic
130	C₂	#41×D₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#67288

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₃

C₂

yes

maximal, characteristic, complemented

C₂₃×#117

C₂

yes

maximal, characteristic, complemented

#10625068

C₂

yes

maximal, complemented

#10625066

C₂

yes

maximal, complemented

#10625057

C₂

yes

maximal, complemented

#10625053

C₂

yes

maximal, complemented

C₄×D₉₂

C₂

yes

maximal, complemented

C₂³×D₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

characteristic

#67288

—

C₂×D₉₂

C₂²

yes

—

C₂×D₉₂

C₂²

yes

characteristic, complemented

C₂×C₄×D₂₃

—

C₂×C₄×D₂₃

C₂²

yes

—

C₂×C₄×D₂₃

C₂²

yes

—

C₂×C₄×D₂₃

C₂²

yes

complemented

C₂×Dic₄₆

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₂₃

C₂²

yes

complemented

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₄²×C₂₃

C₂²

yes

complemented

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

complemented

C₄⋊Dic₂₃

C₂²

yes

complemented

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

—

C₂²×D₂₃

—

C₂²×D₂₃

C₂³

yes

—

C₂²×D₂₃

C₂³

yes

—

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

—

D₉₂

—

D₉₂

—

C₄×D₂₃

—

C₄×D₂₃

—

C₄×D₂₃

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×D₄

yes

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

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

C₂²×D₄

yes

characteristic

C₂×C₂₃

C₂×#41

yes

—

D₂₃

—

D₂₃

—

C₂×#41

—

C₂³×C₄

—

#123

—

C₂²⋊Q₈

—

#121

—

#120

—

C₄×D₄

—

#117

D₂₃

yes

characteristic, complemented

C₂₃

#525

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₄²

C₂×D₂₃

yes

complemented

C₂³

—

C₂³

—

C₂³

—

C₂³

—

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

D₉₂

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×D₉₂

yes

—

120

C₂²

C₂³×D₂₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₂×D₉₂

yes

—

125

C₄

C₂×D₉₂

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

C₂²×D₉₂

yes

characteristic

130

C₂

#41×D₂₃

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Frattini	C₂²	C₂³×D₂₃
Fitting	C₂₃×#117	C₂
hypercenter	#117	D₂₃
hypocenter	C₂₃	#525
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center

C₂×C₄

C₂²×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Frattini

C₂²

C₂³×D₂₃

Fitting

C₂₃×#117

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
1. C₂×#67288	C₂	semidirect
2. C₂²×C₄×D₂₃	C₂	semidirect
3. C₂₃×#117	C₂	semidirect
4. #10625068	C₂	semidirect
5. #10625066	C₂	semidirect
6. #10625057	C₂	semidirect
7. #10625053	C₂	semidirect
8. C₄×D₉₂	C₂	semidirect
15. C₂×D₉₂	C₂²	semidirect
19. C₂×C₄×D₂₃	C₂²	semidirect
20. C₂×Dic₄₆	C₂²	semidirect
21. C₄⋊C₄×C₂₃	C₂²	semidirect
23. C₄²×C₂₃	C₂²	semidirect
25. (C₂×D₂₃)⋊C₄	C₂²	semidirect
26. C₄⋊Dic₂₃	C₂²	semidirect
32. C₂×C₄×C₂₃	D₄	semidirect
54. C₂×D₂₃	#41	semidirect
73. #117	D₂₃	semidirect
74. C₂₃	#525	semidirect
88. C₄⋊C₄	C₂×D₂₃	semidirect
92. C₄²	C₂×D₂₃	semidirect
108. C₂×C₄	D₉₂	semidirect

H ≅ G/N

type

1. C₂×#67288

C₂

semidirect

2. C₂²×C₄×D₂₃

C₂

semidirect

3. C₂₃×#117

C₂

semidirect

4. #10625068

C₂

semidirect

5. #10625066

C₂

semidirect

6. #10625057

C₂

semidirect

7. #10625053

C₂

semidirect

8. C₄×D₉₂

C₂

semidirect

15. C₂×D₉₂

C₂²

semidirect

19. C₂×C₄×D₂₃

C₂²

semidirect

20. C₂×Dic₄₆

C₂²

semidirect

21. C₄⋊C₄×C₂₃

C₂²

semidirect

23. C₄²×C₂₃

C₂²

semidirect

25. (C₂×D₂₃)⋊C₄

C₂²

semidirect

26. C₄⋊Dic₂₃

C₂²

semidirect

32. C₂×C₄×C₂₃

D₄

semidirect

54. C₂×D₂₃

#41

semidirect

73. #117

D₂₃

semidirect

74. C₂₃

#525

semidirect

88. C₄⋊C₄

C₂×D₂₃

semidirect

92. C₄²

C₂×D₂₃

semidirect

108. C₂×C₄

D₉₂

semidirect

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

subgroup	index
C₂×#67288	2
C₂²×C₄×D₂₃	2
C₂₃×#117	2
#10625068	2
#10625066	2
#10625057	2
#10625053	2
C₄×D₉₂	2
#525	23

subgroup

index

C₂×#67288

C₂²×C₄×D₂₃

C₂₃×#117

C₄×D₉₂

Subgroups — #13092929

Conjugacy classes of subgroups

Complements

Maximal subgroups