Group #14521578 — subgroups

Subgroups — #14521578

Order 1984.

234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758523	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758515	C₂	1	1	yes	maximal, characteristic
5	D₄×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758460	C₂	1	2	yes	maximal, complemented
7	(C₂²⋊D₃₁)⋊C₄	C₂	1	2	yes	maximal, complemented
8	#11758456	C₂	1	2	yes	maximal, complemented
9	#11758454	C₂	1	2	yes	maximal, complemented
10	C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#11758447	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
15	C₂²×Dic₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₄²×C₃₁	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
23	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
24	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
25	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₃₁	C₂²	1	2	yes	—
28	Dic₃₁⋊C₄	C₂²	1	2	yes	—
29	Dic₃₁⋊C₄	C₂²	1	2	yes	—
30	Dic₃₁⋊C₄	C₂²	1	2	yes	—
31	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
32	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
33	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₃₁	C₂²	1	2	yes	—
35	C₂³×C₃₁	C₂³	1	2	yes	—
36	C₂²×D₃₁	C₂³	1	1	yes	characteristic
37	Q₈×C₃₁	—	2	2	no	—
38	D₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
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	4	no	—
44	C₂×Dic₃₁	—	2	4	no	—
45	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
46	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
47	C₂×Dic₃₁	C₂³	1	2	yes	—
48	C₂×Dic₃₁	C₂³	1	2	yes	—
49	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
50	C₄×D₃₁	—	2	2	no	—
51	C₄×D₃₁	—	2	2	no	—
52	C₄×D₃₁	—	2	2	no	—
53	C₂²×C₃₁	—	2	2	no	—
54	C₂²×C₃₁	—	2	2	no	—
55	C₂²×C₃₁	—	2	2	no	—
56	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
57	C₂×D₃₁	—	2	1	no	—
58	C₂×D₃₁	—	2	1	no	—
59	C₂×D₃₁	#41	1	2	yes	—
60	C₄×C₃₁	—	2	1	no	—
61	C₄×C₃₁	—	2	1	no	—
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	#41	1	2	yes	—
64	Dic₃₁	—	2	1	no	—
65	Dic₃₁	—	2	1	no	—
66	Dic₃₁	—	2	2	no	—
67	Dic₃₁	—	2	2	no	—
68	Dic₃₁	#41	1	2	yes	—
69	#529	—	31	1	no	maximal, Sylow
70	C₂×C₃₁	—	4	2	no	—
71	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
72	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
73	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
74	D₃₁	—	2	2	no	—
75	#126	—	31	2	no	—
76	#125	—	31	1	no	—
77	#124	D₃₁	1	1	yes	characteristic, complemented
78	#123	—	31	2	no	—
79	C₂²⋊Q₈	—	31	1	no	—
80	#121	—	31	1	no	—
81	C₄×Q₈	—	31	1	no	—
82	C₄×D₄	—	31	1	no	—
83	C₄×D₄	—	31	2	no	—
84	#117	—	31	2	no	—
85	C₂×C₄²	—	31	1	no	—
86	C₃₁	#529	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
88	C₂×D₄	—	31	2	no	—
89	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
90	C₂²×C₄	—	31	2	no	—
91	C₂²×C₄	—	31	1	no	—
92	C₂²×C₄	—	31	1	no	—
93	C₂²×C₄	—	31	1	no	—
94	C₄⋊C₄	—	31	2	no	—
95	C₄⋊C₄	—	31	2	no	—
96	C₄⋊C₄	—	31	2	no	—
97	C₄⋊C₄	—	31	1	no	—
98	C₄⋊C₄	—	31	1	no	—
99	C₄⋊C₄	—	31	2	no	—
100	C₂²⋊C₄	—	31	2	no	—
101	C₂²⋊C₄	—	31	2	no	—
102	C₂²⋊C₄	—	31	2	no	—
103	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
104	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
105	C₄²	—	31	1	no	—
106	C₄²	—	31	1	no	—
107	C₄²	—	31	1	no	—
108	C₄²	—	31	2	no	—
109	C₄²	C₂×D₃₁	1	1	yes	characteristic
110	C₂³	—	31	1	no	—
111	C₂³	C₂²×D₃₁	1	2	yes	—
112	Q₈	—	2	2	no	—
113	D₄	—	2	2	no	—
114	D₄	—	62	4	no	—
115	C₂×C₄	—	31	1	no	—
116	C₂×C₄	—	31	1	no	—
117	C₂×C₄	—	31	1	no	—
118	C₂×C₄	—	31	2	no	—
119	C₂×C₄	—	31	2	no	—
120	C₂×C₄	—	62	2	no	—
121	C₂×C₄	—	62	2	no	—
122	C₂×C₄	—	62	4	no	—
123	C₂×C₄	—	62	2	no	—
124	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₃₁	1	2	yes	—
127	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	31	2	no	—
132	C₂²	—	62	1	no	—
133	C₂²	—	62	1	no	—
134	C₂²	C₂³×D₃₁	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	31	2	no	—
139	C₄	—	62	1	no	—
140	C₄	—	62	2	no	—
141	C₄	—	62	1	no	—
142	C₄	—	62	2	no	—
143	C₄	#92497	1	2	yes	—
144	C₂	—	4	2	no	—
145	C₂	—	62	2	no	—
146	C₂	C₂×#92497	1	1	yes	characteristic
147	C₂	#41×D₃₁	1	1	yes	characteristic
148	C₂	#41×D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#124

C₂

yes

maximal, characteristic, complemented

#11758523

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₃₁

C₂

yes

maximal, characteristic, complemented

#11758515

C₂

yes

maximal, characteristic

D₄×Dic₃₁

C₂

yes

maximal, characteristic, complemented

#11758460

C₂

yes

maximal, complemented

(C₂²⋊D₃₁)⋊C₄

C₂

yes

maximal, complemented

#11758456

C₂

yes

maximal, complemented

#11758454

C₂

yes

maximal, complemented

C₄²×D₃₁

C₂

yes

maximal, characteristic, complemented

#11758447

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₁

C₂²

yes

characteristic

C₂×D₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂²×Dic₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊Dic₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

—

C₂³×C₃₁

C₂³

yes

—

C₂²×D₃₁

C₂³

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊D₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

#41

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#41

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

#41

yes

—

#529

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

C₂×#41

yes

characteristic

D₃₁

—

#126

—

#125

—

#124

D₃₁

yes

characteristic, complemented

#123

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₃₁

#529

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₃₁

yes

—

104

C₂²⋊C₄

C₂×D₃₁

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

C₂×D₃₁

yes

characteristic

110

C₂³

—

111

C₂³

C₂²×D₃₁

yes

—

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²×D₃₁

yes

characteristic

125

C₂×C₄

C₂²×D₃₁

yes

characteristic

126

C₂×C₄

C₂²×D₃₁

yes

—

127

C₂×C₄

C₂²×D₃₁

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂³×D₃₁

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#92497

yes

—

144

C₂

—

145

C₂

—

146

C₂

C₂×#92497

yes

characteristic

147

C₂

#41×D₃₁

yes

characteristic

148

C₂

#41×D₃₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#124	C₂
prime radical	#11758515	C₂
hypercenter	#124	D₃₁
hypocenter	C₃₁	#529
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×#124

C₂

C₂

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₃₁×#124	C₂	semidirect
2. #11758523	C₂	semidirect
3. Q₈×Dic₃₁	C₂	semidirect
5. D₄×Dic₃₁	C₂	semidirect
6. #11758460	C₂	semidirect
7. (C₂²⋊D₃₁)⋊C₄	C₂	semidirect
8. #11758456	C₂	semidirect
9. #11758454	C₂	semidirect
10. C₄²×D₃₁	C₂	semidirect
11. #11758447	C₂	semidirect
77. #124	D₃₁	semidirect
86. C₃₁	#529	semidirect

H ≅ G/N

type

1. C₃₁×#124

C₂

semidirect

2. #11758523

C₂

semidirect

3. Q₈×Dic₃₁

C₂

semidirect

5. D₄×Dic₃₁

C₂

semidirect

6. #11758460

C₂

semidirect

7. (C₂²⋊D₃₁)⋊C₄

C₂

semidirect

8. #11758456

C₂

semidirect

9. #11758454

C₂

semidirect

10. C₄²×D₃₁

C₂

semidirect

11. #11758447

C₂

semidirect

77. #124

D₃₁

semidirect

86. C₃₁

#529

semidirect

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

subgroup	index
C₃₁×#124	2
#11758523	2
Q₈×Dic₃₁	2
#11758515	2
D₄×Dic₃₁	2
#11758460	2
(C₂²⋊D₃₁)⋊C₄	2
#11758456	2
#11758454	2
C₄²×D₃₁	2
#11758447	2
#529	31

subgroup

index

C₃₁×#124

Q₈×Dic₃₁

D₄×Dic₃₁

(C₂²⋊D₃₁)⋊C₄

C₄²×D₃₁

Subgroups — #14521578

Conjugacy classes of subgroups

Complements

Maximal subgroups