Group #15683771 — subgroups

Subgroups — #15683771

Order 2944.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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	#13092974	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092781	C₂	1	4	yes	maximal, complemented
3	#13092763	C₂	1	4	yes	maximal, complemented
4	#13092422	C₂	1	4	yes	maximal, complemented
5	C₄×#67267	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₂₃	—	2	2	no	—
8	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
9	C₂₃×#127	—	2	2	no	—
10	C₂₃×#127	C₂²	1	2	yes	—
11	C₂₃×#127	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	#10625097	C₂²	1	2	yes	—
14	#10625095	—	2	2	no	—
15	C₂×#67267	C₂²	1	2	yes	—
16	#10625041	C₂²	1	2	yes	complemented
17	#10625040	C₂²	1	2	yes	complemented
18	C₄×Dic₄₆	—	2	2	no	—
19	#10625014	C₂²	1	16	yes	complemented
20	#10624985	C₂²	1	2	yes	complemented
21	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
22	C₂⁴×C₂₃	—	2	2	no	—
23	C₂×D₄×C₂₃	—	2	2	no	—
24	C₂×D₄×C₂₃	—	2	4	no	—
25	C₂×D₄×C₂₃	—	2	4	no	—
26	C₂×D₄×C₂₃	—	2	8	no	—
27	C₂×D₄×C₂₃	—	4	8	no	—
28	C₂×D₄×C₂₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₂₃	D₄	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
31	C₂×Dic₄₆	—	2	2	no	—
32	C₄²×C₂₃	D₄	1	2	yes	—
33	C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₂₃	—	2	2	no	—
36	C₄⋊Dic₂₃	—	2	2	no	—
37	C₄⋊Dic₂₃	C₂³	1	4	yes	—
38	Dic₂₃⋊C₄	—	2	4	no	—
39	C₄×Dic₂₃	—	2	2	no	—
40	#67267	—	2	4	no	—
41	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
42	C₂³×C₂₃	—	2	4	no	—
43	C₂³×C₂₃	—	4	8	no	—
44	C₂³×C₂₃	—	4	4	no	—
45	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
46	D₄×C₂₃	—	2	8	no	—
47	D₄×C₂₃	—	4	8	no	—
48	D₄×C₂₃	—	4	4	no	—
49	D₄×C₂₃	—	4	8	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₂₃	#41	1	4	yes	—
55	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₂₃	—	2	4	no	—
57	Dic₄₆	—	4	4	no	—
58	C₂₃⋊C₈	—	2	4	no	—
59	#3138	—	23	1	no	maximal, Sylow
60	C₂²×C₂₃	—	2	2	no	—
61	C₂²×C₂₃	—	4	4	no	—
62	C₂²×C₂₃	—	4	8	no	—
63	C₂²×C₂₃	—	8	8	no	—
64	C₂²×C₂₃	#124	1	2	yes	—
65	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	#124	1	2	yes	—
68	C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₂₃	C₂×#41	1	2	yes	—
70	Dic₂₃	—	4	4	no	—
71	#533	—	23	1	no	—
72	C₂×#127	D₂₃	1	1	yes	characteristic
73	#490	—	23	4	no	—
74	#486	—	23	4	no	—
75	#418	—	23	4	no	—
76	C₄×M₁₆	—	23	1	no	—
77	C₂×C₂₃	—	2	2	no	—
78	C₂×C₂₃	—	8	4	no	—
79	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
80	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
81	C₂²×D₄	—	2	2	no	—
82	C₂²×D₄	C₂×D₂₃	1	2	yes	—
83	C₂×M₁₆	—	23	2	no	—
84	C₄⋊Q₈	—	23	2	no	—
85	#127	—	2	2	no	—
86	#127	C₂×D₂₃	1	2	yes	—
87	#127	C₂×D₂₃	1	2	yes	—
88	#125	—	23	2	no	—
89	C₂²⋊Q₈	—	46	2	no	—
90	C₄×Q₈	—	46	2	no	—
91	#117	—	23	2	no	—
92	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
93	#102	—	23	16	no	—
94	#97	—	23	2	no	—
95	C₄×C₈	—	23	2	no	—
96	C₂₃	#3138	1	1	yes	characteristic, complemented, Sylow
97	C₂⁴	—	2	2	no	—
98	C₂×Q₈	—	46	2	no	—
99	C₂×D₄	—	2	8	no	—
100	C₂×D₄	—	2	2	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	2	4	no	—
103	C₂×D₄	—	4	8	no	—
104	C₂×D₄	C₂²×D₂₃	1	4	yes	—
105	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
107	M₁₆	—	46	4	no	—
108	C₂×C₈	—	23	4	no	—
109	C₄⋊C₄	—	23	4	no	—
110	C₄⋊C₄	—	46	4	no	—
111	C₄⋊C₄	—	46	2	no	—
112	C₂²⋊C₄	—	46	2	no	—
113	C₄²	—	46	2	no	—
114	C₄²	C₂²⋊D₂₃	1	2	yes	—
115	C₄²	C₂²×D₂₃	1	1	yes	characteristic
116	C₄²	C₂²×D₂₃	1	1	yes	characteristic
117	C₂³	—	2	4	no	—
118	C₂³	—	4	8	no	—
119	C₂³	—	4	4	no	—
120	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
121	Q₈	—	92	4	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	8	no	—
124	D₄	—	4	8	no	—
125	D₄	—	4	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	46	4	no	—
128	C₂×C₄	#67290	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
132	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
133	C₈	—	46	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	4	no	—
136	C₂²	—	4	8	no	—
137	C₂²	—	8	8	no	—
138	C₂²	#10625107	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	92	4	no	—
142	C₄	#10625103	1	4	yes	—
143	C₄	#10625107	1	2	yes	—
144	C₄	C₂×#67290	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	8	4	no	—
147	C₂	C₂×#10625107	1	1	yes	characteristic
148	C₂	C₂×#10625103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13092974

C₂

yes

maximal, characteristic, complemented

#13092781

C₂

yes

maximal, complemented

#13092763

C₂

yes

maximal, complemented

#13092422

C₂

yes

maximal, complemented

C₄×#67267

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#127

C₂

yes

maximal, characteristic

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

C₂²

yes

—

C₂₃×#127

—

C₂₃×#127

C₂²

yes

—

C₂₃×#127

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

#10625097

C₂²

yes

—

#10625095

—

C₂×#67267

C₂²

yes

—

#10625041

C₂²

yes

complemented

#10625040

C₂²

yes

complemented

C₄×Dic₄₆

—

#10625014

C₂²

yes

complemented

#10624985

C₂²

yes

complemented

C₄×C₂₃⋊C₈

C₂²

yes

complemented

C₂⁴×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×Dic₄₆

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

#67267

—

C₂×C₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

Dic₄₆

—

C₂₃⋊C₈

—

#3138

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#124

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

C₂×#41

yes

—

Dic₂₃

—

#533

—

C₂×#127

D₂₃

yes

—

—

—

C₄×M₁₆

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

C₂×#124

yes

characteristic

C₂×C₂₃

C₂×AGL(1,ℤ/8)

yes

—

C₂²×D₄

—

C₂²×D₄

C₂×D₂₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#127

C₂×D₂₃

yes

—

#127

C₂×D₂₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂₃

#3138

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

C₂²×D₂₃

yes

—

105

C₂²×C₄

C₂²⋊D₂₃

yes

—

106

C₂²×C₄

C₂²×D₂₃

yes

characteristic

107

M₁₆

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₂₃

yes

—

115

C₄²

C₂²×D₂₃

yes

characteristic

116

C₄²

C₂²×D₂₃

yes

characteristic

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

121

Q₈

—

122

D₄

—

123

D₄

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#67290

yes

—

129

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₂₃

yes

—

131

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

132

C₂×C₄

C₂³×D₂₃

yes

characteristic

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#10625107

yes

—

139

C₂²

C₂²×C₂²⋊D₂₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#10625103

yes

—

143

C₄

#10625107

yes

—

144

C₄

C₂×#67290

yes

—

145

C₂

—

146

C₂

—

147

C₂

C₂×#10625107

yes

characteristic

148

C₂

C₂×#10625103

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting, prime radical	C₂×C₂₃×#127	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	C₂×#127	D₂₃
hypocenter	C₂₃	#3138
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₂₃

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting, prime radical

C₂×C₂₃×#127

C₂

socle

C₂²×C₂₃

C₂²×D₄

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #13092974	C₂	semidirect
2. #13092781	C₂	semidirect
3. #13092763	C₂	semidirect
4. #13092422	C₂	semidirect
5. C₄×#67267	C₂	semidirect
16. #10625041	C₂²	semidirect
17. #10625040	C₂²	semidirect
19. #10625014	C₂²	semidirect
20. #10624985	C₂²	semidirect
21. C₄×C₂₃⋊C₈	C₂²	semidirect
96. C₂₃	#3138	semidirect

H ≅ G/N

type

1. #13092974

C₂

semidirect

2. #13092781

C₂

semidirect

3. #13092763

C₂

semidirect

4. #13092422

C₂

semidirect

5. C₄×#67267

C₂

semidirect

16. #10625041

C₂²

semidirect

17. #10625040

C₂²

semidirect

19. #10625014

C₂²

semidirect

20. #10624985

C₂²

semidirect

21. C₄×C₂₃⋊C₈

C₂²

semidirect

96. C₂₃

#3138

semidirect

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

subgroup	index
#13092974	2
#13092781	2
#13092763	2
#13092422	2
C₄×#67267	2
C₂×C₂₃×#127	2
#3138	23

subgroup

index

C₂×C₂₃×#127

Subgroups — #15683771

Conjugacy classes of subgroups

Complements

Maximal subgroups