Group #15682803 — subgroups

Subgroups — #15682803

Order 2944.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#13092988	C₂	1	3	yes	maximal, complemented
2	#13092607	C₂	1	8	yes	maximal, complemented
3	#13092391	C₂	1	3	yes	maximal, complemented
4	C₂₃×#585	C₂	1	1	yes	maximal, characteristic
5	C₂₃×2₋¹⁺⁴	—	2	3	no	—
6	C₂₃×H(5,𝔽₂)	—	2	3	no	—
7	C₂₃×H(5,𝔽₂)	C₂²	1	4	yes	—
8	C₂×#41×C₂₃	—	2	6	no	—
9	C₂×#41×C₂₃	C₂²	1	3	yes	—
10	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
11	#10625115	—	2	3	no	—
12	#10625107	—	2	3	no	—
13	#10625106	C₂²	1	12	yes	complemented
14	#10625097	C₂²	1	3	yes	complemented
15	#10625095	—	2	6	no	—
16	#10625016	C₂²	1	12	yes	complemented
17	#41×C₂₃	—	2	4	no	—
18	#41×C₂₃	—	2	12	no	—
19	#41×C₂₃	—	2	6	no	—
20	#41×C₂₃	—	2	12	no	—
21	#41×C₂₃	—	2	6	no	—
22	C₂×Q₈×C₂₃	—	2	6	no	—
23	C₂×Q₈×C₂₃	D₄	1	3	yes	—
24	C₂×D₄×C₂₃	—	2	6	no	—
25	C₂×D₄×C₂₃	—	2	12	no	—
26	C₂×D₄×C₂₃	D₄	1	3	yes	—
27	C₂×D₄×C₂₃	C₂³	1	6	yes	—
28	C₂²×C₄×C₂₃	—	2	4	no	—
29	C₂²×C₄×C₂₃	D₄	1	6	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₂₃	C₂³	1	2	yes	—
32	C₂×Dic₄₆	—	2	3	no	—
33	C₂×Dic₄₆	—	2	3	no	—
34	C₂²⋊Dic₂₃	—	2	12	no	—
35	C₂²⋊Dic₂₃	C₂³	1	6	yes	—
36	C₄⋊Dic₂₃	—	2	3	no	—
37	Dic₂₃⋊C₄	—	2	12	no	—
38	C₄×Dic₂₃	—	2	3	no	—
39	C₂³×C₂₃	—	2	4	no	—
40	C₂³×C₂₃	C₂×D₄	1	6	yes	—
41	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
42	Q₈×C₂₃	—	2	4	no	—
43	Q₈×C₂₃	—	2	6	no	—
44	D₄×C₂₃	—	2	12	no	—
45	D₄×C₂₃	—	2	12	no	—
46	D₄×C₂₃	—	2	6	no	—
47	C₂×C₄×C₂₃	—	2	12	no	—
48	C₂×C₄×C₂₃	—	2	6	no	—
49	C₂×C₄×C₂₃	—	2	6	no	—
50	C₂×C₄×C₂₃	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₂₃	C₂×D₄	1	3	yes	—
53	C₂×Dic₂₃	—	2	6	no	—
54	C₂×Dic₂₃	—	2	6	no	—
55	Dic₄₆	—	4	1	no	—
56	Dic₄₆	—	4	3	no	—
57	#3028	—	23	1	no	maximal, Sylow
58	C₂²×C₂₃	—	2	6	no	—
59	C₂²×C₂₃	—	4	1	no	—
60	C₂²×C₂₃	—	4	6	no	—
61	C₂²×C₂₃	—	8	4	no	—
62	C₂²×C₂₃	C₂²×D₄	1	3	yes	—
63	C₄×C₂₃	—	2	6	no	—
64	C₄×C₂₃	#120	1	1	yes	characteristic
65	C₄×C₂₃	#120	1	3	yes	—
66	Dic₂₃	—	4	6	no	—
67	Dic₂₃	—	4	2	no	—
68	#585	D₂₃	1	1	yes	characteristic
69	#536	—	23	3	no	—
70	#458	—	23	8	no	—
71	#410	—	23	3	no	—
72	C₂×C₂₃	—	2	3	no	—
73	C₂×C₂₃	—	4	6	no	—
74	C₂×C₂₃	C₂×#120	1	1	yes	characteristic
75	2₋¹⁺⁴	—	2	3	no	—
76	H(5,𝔽₂)	—	2	3	no	—
77	H(5,𝔽₂)	C₂×D₂₃	1	4	yes	—
78	C₂×#41	—	2	6	no	—
79	C₂×#41	C₂×D₂₃	1	3	yes	—
80	C₂²×Q₈	—	23	1	no	—
81	C₄⋊Q₈	—	46	3	no	—
82	#124	—	46	3	no	—
83	#123	—	23	12	no	—
84	C₂²⋊Q₈	—	46	6	no	—
85	#117	—	23	3	no	—
86	#99	—	23	12	no	—
87	C₂₃	#3028	1	1	yes	characteristic, complemented, Sylow
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	46	3	no	—
95	C₂×Q₈	—	46	3	no	—
96	C₂×Q₈	C₂²⋊D₂₃	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₂₃	1	3	yes	—
100	C₂×D₄	C₂²×D₂₃	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	23	2	no	—
103	C₂²×C₄	C₂²⋊D₂₃	1	6	yes	—
104	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
105	C₄⋊C₄	—	46	3	no	—
106	C₄⋊C₄	—	46	12	no	—
107	C₂²⋊C₄	—	23	6	no	—
108	C₂²⋊C₄	—	46	12	no	—
109	C₄²	—	46	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₂₃	1	6	yes	—
112	C₂³	C₂³×D₂₃	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	92	1	no	—
116	Q₈	—	92	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	46	6	no	—
124	C₂×C₄	—	46	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₂₃	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₂₃	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₂₃	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	92	6	no	—
135	C₄	—	92	2	no	—
136	C₄	#10625124	1	1	yes	characteristic
137	C₄	#10625124	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#10625124	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092988

C₂

yes

maximal, complemented

#13092607

C₂

yes

maximal, complemented

#13092391

C₂

yes

maximal, complemented

C₂₃×#585

C₂

yes

maximal, characteristic

C₂₃×2₋¹⁺⁴

—

C₂₃×H(5,𝔽₂)

—

C₂₃×H(5,𝔽₂)

C₂²

yes

—

C₂×#41×C₂₃

—

C₂×#41×C₂₃

C₂²

yes

—

C₂²×Dic₄₆

C₂²

yes

characteristic, complemented

#10625115

—

#10625107

—

#10625106

C₂²

yes

complemented

#10625097

C₂²

yes

complemented

#10625095

—

#10625016

C₂²

yes

complemented

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

D₄

yes

—

C₂×D₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂³

yes

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂²⋊Dic₂₃

—

C₂²⋊Dic₂₃

C₂³

yes

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

—

C₂³×C₂₃

C₂⁴

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

#3028

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

#120

yes

—

Dic₂₃

—

Dic₂₃

—

#585

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#120

yes

characteristic

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₂₃

yes

—

C₂×#41

—

C₂×#41

C₂×D₂₃

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#99

—

C₂₃

#3028

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₃

yes

—

100

C₂×D₄

C₂²×D₂₃

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₂₃

yes

—

104

C₂²×C₄

C₂²×D₂₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₂₃

yes

—

112

C₂³

C₂³×D₂₃

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

126

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

127

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₂₃

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#10625124

yes

characteristic

137

C₄

#10625124

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#10625124

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#10625124
commutator, cosocle	C₂³×C₂₃	C₂⁴
Frattini	C₂³	C₂³×D₂₃
Fitting, prime radical	C₂₃×#585	C₂
socle	C₂×C₂₃	C₂×#120
hypercenter	#585	D₂₃
hypocenter	C₂₃	#3028
LCS	G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#10625124

commutator, cosocle

C₂³×C₂₃

C₂⁴

Frattini

C₂³

C₂³×D₂₃

Fitting, prime radical

C₂₃×#585

C₂

C₂×C₂₃

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #13092988	C₂	semidirect
2. #13092607	C₂	semidirect
3. #13092391	C₂	semidirect
10. C₂²×Dic₄₆	C₂²	semidirect
13. #10625106	C₂²	semidirect
14. #10625097	C₂²	semidirect
16. #10625016	C₂²	semidirect
87. C₂₃	#3028	semidirect

H ≅ G/N

type

1. #13092988

C₂

semidirect

2. #13092607

C₂

semidirect

3. #13092391

C₂

semidirect

10. C₂²×Dic₄₆

C₂²

semidirect

13. #10625106

C₂²

semidirect

14. #10625097

C₂²

semidirect

16. #10625016

C₂²

semidirect

87. C₂₃

#3028

semidirect

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

subgroup	index
#13092988	2
#13092607	2
#13092391	2
C₂₃×#585	2
#3028	23

subgroup

index

C₂₃×#585

Subgroups — #15682803

Conjugacy classes of subgroups

Complements

Maximal subgroups