Group #16845542 — subgroups

Subgroups — #16845542

Order 3968.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758404	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic
4	#14520812	C₂	1	2	yes	maximal
5	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₁⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₃₁	—	2	1	no	—
13	C₂×C₄×Dic₃₁	—	2	1	no	—
14	C₄×Dic₆₂	—	2	2	no	—
15	C₄×Dic₆₂	C₄	1	4	yes	complemented
16	C₃₁×#97	—	2	2	no	—
17	C₃₁⋊#95	—	2	1	no	—
18	#11758404	C₄	1	8	yes	complemented
19	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₁	—	2	1	no	—
23	C₂²×Dic₃₁	—	2	2	no	—
24	C₂²×Dic₃₁	—	2	1	no	—
25	C₂×Dic₆₂	—	2	1	no	—
26	C₂×Dic₆₂	—	2	2	no	—
27	C₂×Dic₆₂	C₂×C₄	1	4	yes	—
28	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₃₁	—	2	2	no	—
30	C₂×C₈×C₃₁	—	2	2	no	—
31	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
32	C₄²×C₃₁	C₂×C₄	1	2	yes	—
33	C₄²×C₃₁	D₄	1	2	yes	—
34	C₄⋊Dic₃₁	—	2	1	no	—
35	C₄⋊Dic₃₁	—	2	2	no	—
36	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
38	Dic₃₁⋊C₄	—	2	4	no	—
39	C₄×Dic₃₁	—	2	4	no	—
40	C₄×Dic₃₁	—	4	2	no	—
41	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	—	2	1	no	—
43	C₂×C₄×C₃₁	—	2	1	no	—
44	C₂×C₄×C₃₁	—	2	2	no	—
45	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	#41	1	2	yes	—
51	C₂×Dic₃₁	—	2	2	no	—
52	C₂×Dic₃₁	—	2	4	no	—
53	C₂×Dic₃₁	—	2	2	no	—
54	C₂×Dic₃₁	—	2	4	no	—
55	C₂×Dic₃₁	—	4	2	no	—
56	C₂×Dic₃₁	—	4	1	no	—
57	C₂×Dic₃₁	—	4	1	no	—
58	C₂×Dic₃₁	—	4	2	no	—
59	Dic₆₂	—	2	2	no	—
60	Dic₆₂	—	2	4	no	—
61	Dic₆₂	C₄²	1	4	yes	complemented
62	C₈×C₃₁	—	2	4	no	—
63	#1763	—	31	1	no	maximal, Sylow
64	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₃₁	C₄×D₄	1	2	yes	—
66	C₂²×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂²×C₃₁	#137	1	1	yes	characteristic
68	C₂²×C₃₁	#137	1	2	yes	—
69	C₄×C₃₁	—	2	2	no	—
70	C₄×C₃₁	—	2	2	no	—
71	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₃₁	#117	1	1	yes	characteristic
73	C₄×C₃₁	C₄×D₄	1	2	yes	—
74	Dic₃₁	—	2	4	no	—
75	Dic₃₁	—	4	2	no	—
76	Dic₃₁	—	4	4	no	—
77	C₂×C₄×Q₈	—	31	1	no	—
78	C₂×#103	—	31	2	no	—
79	C₂×#97	D₃₁	1	1	yes	characteristic
80	C₄×C₄⋊C₄	—	31	1	no	—
81	#340	—	31	2	no	—
82	C₂×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃₁	#417	1	1	yes	characteristic
84	C₂×C₃₁	#419	1	1	yes	characteristic
85	C₂×C₃₁	#440	1	2	yes	—
86	C₂×C₃₁	#441	1	2	yes	—
87	C₂²×Q₈	—	31	1	no	—
88	C₂²×C₈	C₂×D₃₁	1	2	yes	—
89	C₄×Q₈	—	31	4	no	—
90	C₄×Q₈	—	62	2	no	—
91	C₂×C₄⋊C₄	—	31	1	no	—
92	C₂×C₄⋊C₄	—	31	2	no	—
93	C₂×C₄⋊C₄	—	62	1	no	—
94	C₂×C₄²	—	62	1	no	—
95	C₂×C₄²	—	62	1	no	—
96	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
97	#103	—	31	8	no	—
98	#97	—	2	2	no	—
99	#95	—	62	1	no	—
100	C₃₁	#1763	1	1	yes	characteristic, complemented, Sylow
101	C₂×Q₈	—	31	4	no	—
102	C₂×Q₈	—	31	2	no	—
103	C₂×Q₈	—	62	2	no	—
104	C₂×Q₈	—	62	1	no	—
105	C₂²×C₄	—	62	1	no	—
106	C₂²×C₄	—	62	2	no	—
107	C₂²×C₄	—	62	1	no	—
108	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
109	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	C₄×D₃₁	1	4	yes	—
114	C₄⋊C₄	—	31	4	no	—
115	C₄⋊C₄	—	31	2	no	—
116	C₄⋊C₄	—	62	4	no	—
117	C₄⋊C₄	—	62	2	no	—
118	C₄⋊C₄	—	62	1	no	—
119	C₄²	—	62	4	no	—
120	C₄²	—	124	2	no	—
121	C₄²	C₄×D₃₁	1	2	yes	—
122	C₄²	C₂²⋊D₃₁	1	2	yes	—
123	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
124	Q₈	—	31	4	no	—
125	Q₈	—	62	2	no	—
126	Q₈	—	62	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	62	4	no	—
131	C₂×C₄	—	62	2	no	—
132	C₂×C₄	—	62	4	no	—
133	C₂×C₄	—	62	2	no	—
134	C₂×C₄	—	124	2	no	—
135	C₂×C₄	—	124	1	no	—
136	C₂×C₄	—	124	2	no	—
137	C₂×C₄	—	124	1	no	—
138	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
142	C₂×C₄	#92495	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₂²	C₄×D₁₂₄	1	2	yes	—
146	C₂²	#11758485	1	1	yes	characteristic
147	C₂²	#11758486	1	2	yes	—
148	C₂²	#11758486	1	1	yes	characteristic
149	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	62	4	no	—
153	C₄	—	124	4	no	—
154	C₄	—	124	2	no	—
155	C₄	C₄²×D₃₁	1	1	yes	characteristic
156	C₄	#11758449	1	1	yes	characteristic
157	C₄	C₄×C₂²⋊D₃₁	1	2	yes	—
158	C₂	C₄×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
159	C₂	#14521051	1	1	yes	characteristic
160	C₂	#14521059	1	1	yes	characteristic
161	C₂	#14521151	1	2	yes	—
162	C₂	#14521157	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₆₂

C₂

yes

maximal, characteristic

C₂×#11758404

C₂

yes

maximal

C₄×C₄⋊Dic₃₁

C₂

yes

maximal, characteristic

#14520812

C₂

yes

maximal

C₂×C₃₁×#97

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂×Dic₃₁⋊C₄

—

C₂×C₄×Dic₃₁

—

C₂×C₄×Dic₃₁

—

C₄×Dic₆₂

—

C₄×Dic₆₂

C₄

yes

complemented

C₃₁×#97

—

C₃₁⋊#95

—

#11758404

C₄

yes

complemented

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂×C₄

yes

—

C₂×Dic₆₂

C₂×C₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

C₄²

yes

complemented

C₈×C₃₁

—

#1763

—

maximal, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₄×D₄

yes

—

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄²

yes

characteristic

C₄×C₃₁

#117

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄×Q₈

—

C₂×#103

—

C₂×#97

D₃₁

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₂×C₃₁

C₄×C₂²⋊C₄

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#441

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₃₁

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

100

C₃₁

#1763

yes

characteristic, complemented, Sylow

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

D₁₂₄

yes

characteristic

109

C₂²×C₄

D₁₂₄

yes

characteristic

110

C₂²×C₄

C₂²×D₃₁

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₄×D₃₁

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

124

—

121

C₄²

C₄×D₃₁

yes

—

122

C₄²

C₂²⋊D₃₁

yes

—

123

C₂³

C₂×D₁₂₄

yes

characteristic

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

124

—

135

C₂×C₄

—

124

—

136

C₂×C₄

—

124

—

137

C₂×C₄

—

124

—

138

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₃₁

yes

—

142

C₂×C₄

#92495

yes

—

143

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

144

C₈

—

145

C₂²

C₄×D₁₂₄

yes

—

146

C₂²

#11758485

yes

characteristic

147

C₂²

#11758486

yes

—

148

C₂²

#11758486

yes

characteristic

149

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

124

—

154

C₄

—

124

—

155

C₄

C₄²×D₃₁

yes

characteristic

156

C₄

#11758449

yes

characteristic

157

C₄

C₄×C₂²⋊D₃₁

yes

—

158

C₂

C₄×(C₂×D₃₁)⋊C₄

yes

characteristic

159

C₂

#14521051

yes

characteristic

160

C₂

#14521059

yes

characteristic

161

C₂

#14521151

yes

—

162

C₂

#14521157

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#97	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#97	D₃₁
hypocenter	C₃₁	#1763
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator

C₄×C₃₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#97

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×#97

D₃₁

hypocenter

C₃₁

#1763

LCS

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
15. C₄×Dic₆₂	C₄	semidirect
18. #11758404	C₄	semidirect
61. Dic₆₂	C₄²	semidirect
100. C₃₁	#1763	semidirect

H ≅ G/N

type

15. C₄×Dic₆₂

C₄

semidirect

18. #11758404

C₄

semidirect

61. Dic₆₂

C₄²

semidirect

100. C₃₁

#1763

semidirect

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

subgroup	index
C₂×C₄×Dic₆₂	2
C₂×#11758404	2
C₄×C₄⋊Dic₃₁	2
#14520812	2
C₂×C₃₁×#97	2
#1763	31

subgroup

index

C₂×C₄×Dic₆₂

C₂×#11758404

C₄×C₄⋊Dic₃₁

#14520812

C₂×C₃₁×#97

#1763

Subgroups — #16845542

Conjugacy classes of subgroups

Complements

Maximal subgroups