Group #13093151 — subgroups

Subgroups — #13093151

Order 1472.

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

normal subgroups only

This group has 236 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×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625111	C₂	1	1	yes	maximal, characteristic
3	#10625106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
5	#10625065	C₂	1	2	yes	maximal, complemented
6	#10625058	C₂	1	2	yes	maximal, complemented
7	#10625056	C₂	1	2	yes	maximal, complemented
8	#10625056	C₂	1	2	yes	maximal, complemented
9	(C₂²⋊D₂₃)⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₂×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₂₃	C₂²	1	2	yes	—
14	C₂²×Dic₂₃	C₂²	1	1	yes	characteristic
15	C₂×D₉₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₂₃	C₂²	1	2	yes	—
17	C₂×Dic₄₆	C₂²	1	2	yes	—
18	C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₂₃	C₂²	1	2	yes	—
23	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₂₃)⋊C₄	C₂²	1	2	yes	—
26	Dic₂₃⋊C₄	C₂²	1	2	yes	—
27	Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₂₃	C₂²	1	2	yes	—
29	C₄×Dic₂₃	C₂²	1	2	yes	—
30	C₄×Dic₂₃	C₂²	1	1	yes	characteristic
31	C₂³×C₂₃	C₂³	1	1	yes	characteristic
32	C₂³×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×D₂₃	C₂³	1	2	yes	—
34	D₄×C₂₃	—	4	1	no	—
35	C₂×C₄×C₂₃	—	4	1	no	—
36	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²⋊D₂₃	—	2	4	no	—
40	C₂²⋊D₂₃	—	2	4	no	—
41	C₂×Dic₂₃	—	4	1	no	—
42	C₂×Dic₂₃	C₂³	1	2	yes	—
43	C₂×Dic₂₃	C₂³	1	2	yes	—
44	C₂×Dic₂₃	C₂³	1	2	yes	—
45	D₉₂	—	4	1	no	—
46	C₄×D₂₃	—	4	2	no	—
47	Dic₄₆	—	2	4	no	—
48	C₂²×C₂₃	—	2	2	no	—
49	C₂²×C₂₃	—	2	2	no	—
50	C₂²×C₂₃	—	2	1	no	—
51	C₂²×C₂₃	—	2	1	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	C₄×C₂₃	—	2	2	no	—
57	C₄×C₂₃	—	2	2	no	—
58	C₄×C₂₃	—	2	1	no	—
59	Dic₂₃	—	2	2	no	—
60	Dic₂₃	—	2	2	no	—
61	Dic₂₃	#41	1	4	yes	—
62	#559	—	23	1	no	maximal, Sylow
63	C₂×C₂₃	—	4	1	no	—
64	C₂×C₂₃	—	4	1	no	—
65	C₂×C₂₃	C₂×#41	1	2	yes	—
66	C₂×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
67	D₂₃	—	4	2	no	—
68	#127	—	23	1	no	—
69	#124	—	23	2	no	—
70	#124	—	23	2	no	—
71	#123	—	23	1	no	—
72	#123	—	23	2	no	—
73	#123	D₂₃	1	1	yes	characteristic, complemented
74	C₄×Q₈	—	23	2	no	—
75	C₄×D₄	—	23	2	no	—
76	C₄×D₄	—	23	2	no	—
77	C₂₃	#559	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₈	—	23	2	no	—
79	C₂×D₄	—	23	2	no	—
80	C₂×D₄	—	23	2	no	—
81	C₂×D₄	—	23	1	no	—
82	C₂×D₄	C₂×D₂₃	1	1	yes	characteristic
83	C₂²×C₄	—	23	1	no	—
84	C₂²×C₄	—	23	2	no	—
85	C₂²×C₄	C₂×D₂₃	1	1	yes	characteristic
86	C₄⋊C₄	—	23	2	no	—
87	C₄⋊C₄	—	23	2	no	—
88	C₄⋊C₄	C₂×D₂₃	1	2	yes	—
89	C₂²⋊C₄	—	23	2	no	—
90	C₂²⋊C₄	—	23	1	no	—
91	C₂²⋊C₄	—	23	2	no	—
92	C₂²⋊C₄	—	23	2	no	—
93	C₂²⋊C₄	—	23	2	no	—
94	C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
95	C₂²⋊C₄	C₂×D₂₃	1	1	yes	characteristic
96	C₄²	—	23	2	no	—
97	C₄²	—	23	2	no	—
98	C₄²	—	23	1	no	—
99	C₂³	—	23	2	no	—
100	C₂³	C₂²×D₂₃	1	1	yes	characteristic
101	C₂³	C₂²×D₂₃	1	1	yes	characteristic
102	Q₈	—	46	4	no	—
103	D₄	—	4	1	no	—
104	D₄	—	46	4	no	—
105	D₄	—	46	4	no	—
106	D₄	—	92	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	23	2	no	—
109	C₂×C₄	—	23	2	no	—
110	C₂×C₄	—	23	2	no	—
111	C₂×C₄	—	92	2	no	—
112	C₂×C₄	—	92	1	no	—
113	C₂×C₄	C₂²×D₂₃	1	2	yes	—
114	C₂×C₄	C₂²×D₂₃	1	2	yes	—
115	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	46	2	no	—
121	C₂²	—	46	2	no	—
122	C₂²	—	46	2	no	—
123	C₂²	C₂³×D₂₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	2	no	—
127	C₄	—	23	4	no	—
128	C₄	—	46	2	no	—
129	C₄	—	46	2	no	—
130	C₂	—	4	1	no	—
131	C₂	—	4	1	no	—
132	C₂	—	92	2	no	—
133	C₂	#10625155	1	1	yes	characteristic
134	C₂	#41×D₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#123

C₂

yes

maximal, characteristic, complemented

#10625111

C₂

yes

maximal, characteristic

#10625106

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₂₃

C₂

yes

maximal, complemented

#10625065

C₂

yes

maximal, complemented

#10625058

C₂

yes

maximal, complemented

#10625056

C₂

yes

maximal, complemented

#10625056

C₂

yes

maximal, complemented

(C₂²⋊D₂₃)⋊C₄

C₂

yes

maximal, complemented

C₂×D₄×C₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₃

C₂²

yes

—

C₂²×Dic₂₃

C₂²

yes

characteristic

C₂×D₉₂

C₂²

yes

characteristic

C₂×C₄×D₂₃

C₂²

yes

—

C₂×Dic₄₆

C₂²

yes

—

C₄⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₃)⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

Dic₂₃⋊C₄

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

—

C₄×Dic₂₃

C₂²

yes

characteristic

C₂³×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

C₂³

yes

characteristic

C₂²×D₂₃

C₂³

yes

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₃

C₂³

yes

—

C₂×C₄×C₂₃

C₂³

yes

—

C₂²⋊D₂₃

—

C₂²⋊D₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

C₂×Dic₂₃

C₂³

yes

—

D₉₂

—

C₄×D₂₃

—

Dic₄₆

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂⁴

yes

characteristic

C₂×D₂₃

—

C₂×D₂₃

—

C₂×D₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

Dic₂₃

—

Dic₂₃

—

Dic₂₃

#41

yes

—

#559

—

maximal, Sylow

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#41

yes

—

C₂×C₂₃

H(5,𝔽₂)

yes

characteristic

D₂₃

—

—

—

—

—

—

D₂₃

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂₃

#559

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₂₃

yes

—

C₂²⋊C₄

C₂×D₂₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₂³

—

100

C₂³

C₂²×D₂₃

yes

characteristic

101

C₂³

C₂²×D₂₃

yes

characteristic

102

Q₈

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₂₃

yes

—

114

C₂×C₄

C₂²×D₂₃

yes

—

115

C₂×C₄

C₂²×D₂₃

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₂³×D₂₃

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₂

—

131

C₂

—

132

C₂

—

133

C₂

#10625155

yes

characteristic

134

C₂

#41×D₂₃

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁴
Fitting	C₂₃×#123	C₂
prime radical	#10625111	C₂
hypercenter	#123	D₂₃
hypocenter	C₂₃	#559
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃

commutator, socle, cosocle

C₂²×C₂₃

C₂⁴

Fitting

C₂₃×#123

C₂

C₂

D₂₃

C₂₃

G ▷ C₂²×C₂₃ ▷ C₂₃

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
1. C₂₃×#123	C₂	semidirect
3. #10625106	C₂	semidirect
4. C₄×C₂²⋊D₂₃	C₂	semidirect
5. #10625065	C₂	semidirect
6. #10625058	C₂	semidirect
7. #10625056	C₂	semidirect
8. #10625056	C₂	semidirect
9. (C₂²⋊D₂₃)⋊C₄	C₂	semidirect
73. #123	D₂₃	semidirect
77. C₂₃	#559	semidirect

H ≅ G/N

type

1. C₂₃×#123

C₂

semidirect

3. #10625106

C₂

semidirect

4. C₄×C₂²⋊D₂₃

C₂

semidirect

5. #10625065

C₂

semidirect

6. #10625058

C₂

semidirect

7. #10625056

C₂

semidirect

8. #10625056

C₂

semidirect

9. (C₂²⋊D₂₃)⋊C₄

C₂

semidirect

73. #123

D₂₃

semidirect

77. C₂₃

#559

semidirect

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

subgroup	index
C₂₃×#123	2
#10625111	2
#10625106	2
C₄×C₂²⋊D₂₃	2
#10625065	2
#10625058	2
#10625056	2
(C₂²⋊D₂₃)⋊C₄	2
#559	23

subgroup

index

C₂₃×#123

C₄×C₂²⋊D₂₃

(C₂²⋊D₂₃)⋊C₄

Subgroups — #13093151

Conjugacy classes of subgroups

Complements

Maximal subgroups