Group #14521782 — subgroups

Subgroups — #14521782

Order 1984.

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

normal subgroups only

This group has 236 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	C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758517	C₂	1	1	yes	maximal, characteristic
3	#11758513	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758506	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
6	#11758470	C₂	1	1	yes	maximal, characteristic, complemented
7	#11758464	C₂	1	1	yes	maximal, characteristic, complemented
8	#11758462	C₂	1	2	yes	maximal, complemented
9	#11758460	C₂	1	2	yes	maximal, complemented
10	#11758452	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
16	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃₁	C₂²	1	2	yes	—
18	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
19	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₄²×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
24	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
25	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
28	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
29	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
30	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₃₁⋊C₄	C₂²	1	2	yes	—
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	2	yes	—
37	D₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	4	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	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²⋊D₃₁	—	2	4	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	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
50	D₁₂₄	—	2	2	no	—
51	C₄×D₃₁	—	4	2	no	—
52	Dic₆₂	—	2	2	no	—
53	Dic₆₂	—	2	2	no	—
54	C₂²×C₃₁	—	2	2	no	—
55	C₂²×C₃₁	—	2	2	no	—
56	C₂²×C₃₁	—	2	2	no	—
57	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
58	C₂×D₃₁	—	2	2	no	—
59	C₂×D₃₁	—	2	2	no	—
60	C₂×D₃₁	—	2	2	no	—
61	C₄×C₃₁	—	2	1	no	—
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	—	2	1	no	—
64	C₄×C₃₁	#41	1	2	yes	—
65	Dic₃₁	—	2	2	no	—
66	Dic₃₁	—	2	1	no	—
67	Dic₃₁	—	2	1	no	—
68	Dic₃₁	—	2	1	no	—
69	Dic₃₁	#41	1	2	yes	—
70	#559	—	31	1	no	maximal, Sylow
71	C₂×C₃₁	—	4	2	no	—
72	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
73	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
74	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
75	D₃₁	—	4	2	no	—
76	#127	—	31	1	no	—
77	#124	—	31	1	no	—
78	#124	—	31	2	no	—
79	#124	—	31	1	no	—
80	#123	—	31	2	no	—
81	#123	—	31	2	no	—
82	C₄×Q₈	—	31	1	no	—
83	C₄×Q₈	—	31	1	no	—
84	C₄×D₄	—	31	2	no	—
85	C₄×D₄	—	31	1	no	—
86	C₄×D₄	D₃₁	1	1	yes	characteristic, complemented
87	C₃₁	#559	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₈	—	31	1	no	—
89	C₂×Q₈	—	31	1	no	—
90	C₂×D₄	—	31	2	no	—
91	C₂×D₄	—	31	2	no	—
92	C₂×D₄	—	31	1	no	—
93	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
94	C₂²×C₄	—	31	2	no	—
95	C₂²×C₄	C₂×D₃₁	1	2	yes	—
96	C₄⋊C₄	—	31	1	no	—
97	C₄⋊C₄	—	31	2	no	—
98	C₄⋊C₄	—	31	1	no	—
99	C₄⋊C₄	—	31	1	no	—
100	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
101	C₂²⋊C₄	—	31	2	no	—
102	C₂²⋊C₄	—	31	2	no	—
103	C₂²⋊C₄	—	31	2	no	—
104	C₂²⋊C₄	—	31	2	no	—
105	C₂²⋊C₄	—	31	2	no	—
106	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
107	C₄²	—	31	2	no	—
108	C₄²	—	31	1	no	—
109	C₄²	—	31	1	no	—
110	C₄²	C₂×D₃₁	1	1	yes	characteristic
111	C₂³	—	31	2	no	—
112	C₂³	C₂²×D₃₁	1	2	yes	—
113	Q₈	—	62	2	no	—
114	Q₈	—	62	2	no	—
115	D₄	—	2	2	no	—
116	D₄	—	62	4	no	—
117	D₄	—	62	2	no	—
118	D₄	—	124	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	31	1	no	—
121	C₂×C₄	—	31	2	no	—
122	C₂×C₄	—	31	1	no	—
123	C₂×C₄	—	31	1	no	—
124	C₂×C₄	—	31	1	no	—
125	C₂×C₄	—	124	2	no	—
126	C₂×C₄	C₂²×D₃₁	1	2	yes	—
127	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
128	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
129	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	62	2	no	—
134	C₂²	—	62	2	no	—
135	C₂²	—	62	2	no	—
136	C₂²	C₂³×D₃₁	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	31	2	no	—
141	C₄	—	62	2	no	—
142	C₄	—	62	1	no	—
143	C₄	—	62	1	no	—
144	C₄	—	62	1	no	—
145	C₄	#92495	1	2	yes	—
146	C₂	—	4	2	no	—
147	C₂	—	124	2	no	—
148	C₂	C₂×#92495	1	1	yes	characteristic
149	C₂	#11758561	1	1	yes	characteristic
150	C₂	#41×D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄×C₃₁

C₂

yes

maximal, characteristic, complemented

#11758517

C₂

yes

maximal, characteristic

#11758513

C₂

yes

maximal, characteristic, complemented

#11758506

C₂

yes

maximal, complemented

C₄×C₂²⋊D₃₁

C₂

yes

maximal, complemented

#11758470

C₂

yes

maximal, characteristic, complemented

#11758464

C₂

yes

maximal, characteristic, complemented

#11758462

C₂

yes

maximal, complemented

#11758460

C₂

yes

maximal, complemented

#11758452

C₂

yes

maximal, characteristic, complemented

C₄×Dic₆₂

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊Dic₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

—

C₂³×C₃₁

C₂³

yes

—

C₂²×D₃₁

C₂³

yes

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

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

—

C₂×Dic₃₁

C₂³

yes

characteristic

D₁₂₄

—

C₄×D₃₁

—

Dic₆₂

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#41

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

#41

yes

—

#559

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

characteristic

D₃₁

—

—

—

—

—

—

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₄×D₄

D₃₁

yes

characteristic, complemented

C₃₁

#559

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

C₂×D₃₁

yes

characteristic

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

C₂×D₃₁

yes

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

C₂×D₃₁

yes

characteristic

111

C₂³

—

112

C₂³

C₂²×D₃₁

yes

—

113

Q₈

—

114

Q₈

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

124

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

124

—

126

C₂×C₄

C₂²×D₃₁

yes

—

127

C₂×C₄

C₂²×D₃₁

yes

characteristic

128

C₂×C₄

C₂²×D₃₁

yes

characteristic

129

C₂×C₄

C₂²×D₃₁

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂³×D₃₁

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

#92495

yes

—

146

C₂

—

147

C₂

—

124

—

148

C₂

C₂×#92495

yes

characteristic

149

C₂

#11758561

yes

characteristic

150

C₂

#41×D₃₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₄×D₄×C₃₁	C₂
prime radical	#11758517	C₂
hypercenter	C₄×D₄	D₃₁
hypocenter	C₃₁	#559
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₄×D₄×C₃₁

C₂

prime radical

#11758517

C₂

hypercenter

C₄×D₄

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₄×D₄×C₃₁	C₂	semidirect
3. #11758513	C₂	semidirect
4. #11758506	C₂	semidirect
5. C₄×C₂²⋊D₃₁	C₂	semidirect
6. #11758470	C₂	semidirect
7. #11758464	C₂	semidirect
8. #11758462	C₂	semidirect
9. #11758460	C₂	semidirect
10. #11758452	C₂	semidirect
11. C₄×Dic₆₂	C₂	semidirect
86. C₄×D₄	D₃₁	semidirect
87. C₃₁	#559	semidirect

H ≅ G/N

type

1. C₄×D₄×C₃₁

C₂

semidirect

3. #11758513

C₂

semidirect

4. #11758506

C₂

semidirect

5. C₄×C₂²⋊D₃₁

C₂

semidirect

6. #11758470

C₂

semidirect

7. #11758464

C₂

semidirect

8. #11758462

C₂

semidirect

9. #11758460

C₂

semidirect

10. #11758452

C₂

semidirect

11. C₄×Dic₆₂

C₂

semidirect

86. C₄×D₄

D₃₁

semidirect

87. C₃₁

#559

semidirect

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

subgroup	index
C₄×D₄×C₃₁	2
#11758517	2
#11758513	2
#11758506	2
C₄×C₂²⋊D₃₁	2
#11758470	2
#11758464	2
#11758462	2
#11758460	2
#11758452	2
C₄×Dic₆₂	2
#559	31

subgroup

index

C₄×D₄×C₃₁

C₄×C₂²⋊D₃₁

C₄×Dic₆₂

Subgroups — #14521782

Conjugacy classes of subgroups

Complements

Maximal subgroups