Group #14521743 — subgroups

Subgroups — #14521743

Order 1984.

238 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758507	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂²⋊D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758472	C₂	1	1	yes	maximal, characteristic, complemented
5	#11758470	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊C₄×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#11758467	C₂	1	1	yes	maximal, characteristic, complemented
8	#11758461	C₂	1	2	yes	maximal, complemented
9	#11758460	C₂	1	2	yes	maximal, complemented
10	#11758453	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₂₄	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
15	C₂×D₁₂₄	C₂²	1	2	yes	—
16	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₃₁	C₂²	1	2	yes	—
19	C₂×C₄×D₃₁	C₂²	1	2	yes	—
20	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₄²×C₃₁	C₂²	1	2	yes	—
24	C₂²⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	(C₂×D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
26	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
28	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
29	C₄⋊Dic₃₁	C₂²	1	2	yes	—
30	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
31	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
32	Dic₃₁⋊C₄	C₂²	1	2	yes	—
33	Dic₃₁⋊C₄	C₂²	1	2	yes	—
34	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
35	C₂³×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×D₃₁	C₂³	1	2	yes	—
37	C₂²×D₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²⋊D₃₁	—	2	2	no	—
44	C₂²⋊D₃₁	—	4	2	no	—
45	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
46	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
47	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₃₁	C₂³	1	2	yes	—
49	D₁₂₄	—	2	4	no	—
50	D₁₂₄	—	2	2	no	—
51	C₄×D₃₁	—	2	2	no	—
52	C₄×D₃₁	—	4	2	no	—
53	C₄×D₃₁	—	4	2	no	—
54	C₂²×C₃₁	—	2	1	no	—
55	C₂²×C₃₁	—	2	1	no	—
56	C₂²×C₃₁	—	2	1	no	—
57	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
58	C₂×D₃₁	—	2	1	no	—
59	C₂×D₃₁	—	2	1	no	—
60	C₂×D₃₁	—	2	2	no	—
61	C₂×D₃₁	—	2	2	no	—
62	C₂×D₃₁	—	2	2	no	—
63	C₂×D₃₁	#41	1	2	yes	—
64	C₄×C₃₁	—	2	2	no	—
65	C₄×C₃₁	—	2	1	no	—
66	C₄×C₃₁	—	2	2	no	—
67	C₄×C₃₁	#41	1	2	yes	—
68	Dic₃₁	—	2	1	no	—
69	Dic₃₁	—	2	1	no	—
70	Dic₃₁	—	2	1	no	—
71	Dic₃₁	—	2	2	no	—
72	#553	—	31	1	no	maximal, Sylow
73	C₂×C₃₁	—	4	1	no	—
74	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
75	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
76	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
77	D₃₁	—	2	2	no	—
78	D₃₁	—	4	2	no	—
79	#126	—	31	2	no	—
80	#125	—	31	1	no	—
81	#123	—	31	2	no	—
82	#121	—	31	1	no	—
83	#121	—	31	1	no	—
84	#121	—	31	2	no	—
85	C₄×D₄	—	31	1	no	—
86	C₄×D₄	—	31	1	no	—
87	C₄×D₄	—	31	2	no	—
88	#117	D₃₁	1	1	yes	characteristic, complemented
89	C₂×C₄⋊C₄	—	31	1	no	—
90	C₃₁	#553	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₄	—	31	1	no	—
92	C₂×D₄	—	31	2	no	—
93	C₂×D₄	—	31	2	no	—
94	C₂×D₄	—	31	1	no	—
95	C₂²×C₄	—	31	1	no	—
96	C₂²×C₄	—	31	2	no	—
97	C₂²×C₄	—	31	2	no	—
98	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
99	C₄⋊C₄	—	31	2	no	—
100	C₄⋊C₄	—	31	1	no	—
101	C₄⋊C₄	—	31	2	no	—
102	C₄⋊C₄	—	31	1	no	—
103	C₄⋊C₄	—	31	2	no	—
104	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
105	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
106	C₂²⋊C₄	—	31	1	no	—
107	C₂²⋊C₄	—	31	1	no	—
108	C₂²⋊C₄	—	31	2	no	—
109	C₂²⋊C₄	—	31	2	no	—
110	C₂²⋊C₄	—	31	2	no	—
111	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
112	C₄²	—	31	1	no	—
113	C₄²	C₂×D₃₁	1	2	yes	—
114	C₂³	—	31	1	no	—
115	C₂³	—	31	2	no	—
116	C₂³	C₂²×D₃₁	1	1	yes	characteristic
117	D₄	—	62	4	no	—
118	D₄	—	62	2	no	—
119	D₄	—	62	2	no	—
120	D₄	—	124	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	31	1	no	—
123	C₂×C₄	—	31	1	no	—
124	C₂×C₄	—	31	1	no	—
125	C₂×C₄	—	31	2	no	—
126	C₂×C₄	—	62	2	no	—
127	C₂×C₄	—	124	2	no	—
128	C₂×C₄	—	124	2	no	—
129	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
130	C₂×C₄	C₂²×D₃₁	1	2	yes	—
131	C₂×C₄	C₂²×D₃₁	1	2	yes	—
132	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	31	2	no	—
137	C₂²	—	62	1	no	—
138	C₂²	—	62	1	no	—
139	C₂²	—	62	2	no	—
140	C₂²	—	62	2	no	—
141	C₂²	—	62	2	no	—
142	C₂²	C₂³×D₃₁	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	62	1	no	—
147	C₄	—	62	1	no	—
148	C₄	—	62	1	no	—
149	C₄	—	62	2	no	—
150	C₄	#92495	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	—	62	2	no	—
153	C₂	—	124	2	no	—
154	C₂	C₂×#92495	1	1	yes	characteristic
155	C₂	#41×D₃₁	1	1	yes	characteristic
156	C₂	#11758566	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#117

C₂

yes

maximal, characteristic, complemented

#11758507

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₃₁

C₂

yes

maximal, characteristic, complemented

#11758472

C₂

yes

maximal, characteristic, complemented

#11758470

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₃₁

C₂

yes

maximal, characteristic, complemented

#11758467

C₂

yes

maximal, characteristic, complemented

#11758461

C₂

yes

maximal, complemented

#11758460

C₂

yes

maximal, complemented

#11758453

C₂

yes

maximal, complemented

C₄×D₁₂₄

C₂

yes

maximal, complemented

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂×D₁₂₄

C₂²

yes

—

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₂³×C₃₁

C₂³

yes

characteristic

C₂²×D₃₁

C₂³

yes

—

C₂²×D₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊D₃₁

—

C₂²⋊D₃₁

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

D₁₂₄

—

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

#41

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#41

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

#553

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

characteristic

D₃₁

—

D₃₁

—

—

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

—

#117

D₃₁

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₃₁

#553

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

C₂×D₃₁

yes

characteristic

105

C₄⋊C₄

C₂×D₃₁

yes

characteristic

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

C₂×D₃₁

yes

—

112

C₄²

—

113

C₄²

C₂×D₃₁

yes

—

114

C₂³

—

115

C₂³

—

116

C₂³

C₂²×D₃₁

yes

characteristic

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

124

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

124

—

128

C₂×C₄

—

124

—

129

C₂×C₄

C₂²×D₃₁

yes

characteristic

130

C₂×C₄

C₂²×D₃₁

yes

—

131

C₂×C₄

C₂²×D₃₁

yes

—

132

C₂×C₄

C₂²×D₃₁

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₃₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#92495

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

124

—

154

C₂

C₂×#92495

yes

characteristic

155

C₂

#41×D₃₁

yes

characteristic

156

C₂

#11758566

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#117	C₂
hypercenter	#117	D₃₁
hypocenter	C₃₁	#553
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×#117

C₂

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₃₁×#117	C₂	semidirect
2. #11758507	C₂	semidirect
3. C₄×C₂²⋊D₃₁	C₂	semidirect
4. #11758472	C₂	semidirect
5. #11758470	C₂	semidirect
6. C₄⋊C₄×D₃₁	C₂	semidirect
7. #11758467	C₂	semidirect
8. #11758461	C₂	semidirect
9. #11758460	C₂	semidirect
10. #11758453	C₂	semidirect
11. C₄×D₁₂₄	C₂	semidirect
88. #117	D₃₁	semidirect
90. C₃₁	#553	semidirect

H ≅ G/N

type

1. C₃₁×#117

C₂

semidirect

2. #11758507

C₂

semidirect

3. C₄×C₂²⋊D₃₁

C₂

semidirect

4. #11758472

C₂

semidirect

5. #11758470

C₂

semidirect

6. C₄⋊C₄×D₃₁

C₂

semidirect

7. #11758467

C₂

semidirect

8. #11758461

C₂

semidirect

9. #11758460

C₂

semidirect

10. #11758453

C₂

semidirect

11. C₄×D₁₂₄

C₂

semidirect

88. #117

D₃₁

semidirect

90. C₃₁

#553

semidirect

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

subgroup	index
C₃₁×#117	2
#11758507	2
C₄×C₂²⋊D₃₁	2
#11758472	2
#11758470	2
C₄⋊C₄×D₃₁	2
#11758467	2
#11758461	2
#11758460	2
#11758453	2
C₄×D₁₂₄	2
#553	31

subgroup

index

C₃₁×#117

C₄×C₂²⋊D₃₁

C₄⋊C₄×D₃₁

C₄×D₁₂₄

Subgroups — #14521743

Conjugacy classes of subgroups

Complements

Maximal subgroups