Group #16854816 — subgroups

Subgroups — #16854816

Order 3968.

314 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 157 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	#14521606	C₂	1	1	yes	maximal, characteristic
2	#14521178	C₂	1	4	yes	maximal, complemented
3	#14521177	C₂	1	4	yes	maximal, complemented
4	#14521107	C₂	1	4	yes	maximal
5	#14521106	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#432	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	#11758503	C₂²	1	2	yes	—
9	#11758501	C₄	1	2	yes	—
10	#11758501	C₄	1	2	yes	—
11	#11758447	C₄	1	2	yes	—
12	#11758446	C₄	1	1	yes	characteristic
13	#11758446	C₄	1	1	yes	characteristic
14	C₄×Dic₆₂	C₂²	1	2	yes	—
15	C₄×Dic₆₂	C₂²	1	2	yes	—
16	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
18	C₃₁×#97	C₂²	1	2	yes	—
19	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
20	#11758403	C₂²	1	4	yes	—
21	Dic₃₁⋊C₈	C₂²	1	8	yes	—
22	C₈×Dic₃₁	C₂²	1	4	yes	—
23	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
24	#11758391	C₂²	1	2	yes	—
25	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
29	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₃₁	C₂³	1	4	yes	—
31	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₃₁	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
40	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
41	Dic₃₁⋊C₄	C₂×C₄	1	4	yes	—
42	C₄×Dic₃₁	C₂³	1	4	yes	—
43	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
44	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	—	2	2	no	—
46	C₂×C₄×C₃₁	—	4	1	no	—
47	C₂×C₄×C₃₁	—	4	1	no	—
48	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
55	C₂×Dic₃₁	C₂²×C₄	1	4	yes	—
56	Dic₆₂	—	4	2	no	—
57	Dic₆₂	—	4	2	no	—
58	C₈×C₃₁	—	2	4	no	—
59	C₃₁⋊C₈	—	2	4	no	—
60	#2987	—	31	1	no	maximal, Sylow
61	C₂²×C₃₁	—	2	1	no	—
62	C₂²×C₃₁	—	2	1	no	—
63	C₂²×C₃₁	—	2	1	no	—
64	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
65	C₄×C₃₁	—	2	1	no	—
66	C₄×C₃₁	—	2	1	no	—
67	C₄×C₃₁	—	2	1	no	—
68	C₄×C₃₁	—	2	1	no	—
69	C₄×C₃₁	—	2	1	no	—
70	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
71	Dic₃₁	—	2	4	no	—
72	Dic₃₁	—	2	4	no	—
73	#533	—	31	1	no	—
74	#446	—	31	4	no	—
75	#446	—	31	4	no	—
76	#433	—	31	4	no	—
77	#432	—	31	1	no	—
78	#432	D₃₁	1	1	yes	characteristic
79	C₂×C₃₁	—	4	1	no	—
80	C₂×C₃₁	#519	1	1	yes	characteristic
81	C₂×C₃₁	#568	1	1	yes	characteristic
82	C₂×C₃₁	#568	1	1	yes	characteristic
83	C₄⋊Q₈	—	31	1	no	—
84	C₄⋊Q₈	—	31	1	no	—
85	#125	—	31	2	no	—
86	C₂²⋊Q₈	—	31	2	no	—
87	C₂²⋊Q₈	—	31	2	no	—
88	C₄×Q₈	—	31	2	no	—
89	C₄×Q₈	—	31	2	no	—
90	#117	—	31	2	no	—
91	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
92	C₄⋊C₈	—	31	8	no	—
93	C₄⋊C₈	—	31	2	no	—
94	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
95	C₂²⋊C₈	—	31	2	no	—
96	C₂²⋊C₈	C₂×D₃₁	1	2	yes	—
97	#97	—	31	4	no	—
98	#97	—	31	2	no	—
99	#97	C₂×D₃₁	1	2	yes	—
100	C₄×C₈	—	31	4	no	—
101	C₃₁	#2987	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₈	—	31	2	no	—
103	C₂×Q₈	—	31	2	no	—
104	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
105	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
107	C₂×C₈	—	31	4	no	—
108	C₂×C₈	C₂²×D₃₁	1	4	yes	—
109	C₄⋊C₄	—	31	4	no	—
110	C₄⋊C₄	—	31	4	no	—
111	C₄⋊C₄	—	31	2	no	—
112	C₄⋊C₄	—	31	2	no	—
113	C₄⋊C₄	—	31	2	no	—
114	C₄⋊C₄	—	31	2	no	—
115	C₂²⋊C₄	—	31	4	no	—
116	C₄²	—	31	4	no	—
117	C₄²	C₄×D₃₁	1	1	yes	characteristic
118	C₄²	C₄×D₃₁	1	1	yes	characteristic
119	C₄²	C₂²×D₃₁	1	1	yes	characteristic
120	C₄²	C₂²×D₃₁	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
122	Q₈	—	124	2	no	—
123	Q₈	—	124	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	31	4	no	—
128	C₂×C₄	—	31	4	no	—
129	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
135	C₈	—	2	4	no	—
136	C₈	—	62	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	62	4	no	—
147	C₄	—	62	4	no	—
148	C₄	#11758567	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#14521526	1	1	yes	characteristic
151	C₂	#14521815	1	1	yes	characteristic
152	C₂	#14521815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

#14521178

C₂

yes

maximal, complemented

#14521177

C₂

yes

maximal, complemented

#14521107

C₂

yes

maximal

#14521106

C₂

yes

maximal, characteristic

C₃₁×#432

C₂

yes

maximal, characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758501

C₄

yes

—

#11758501

C₄

yes

—

#11758447

C₄

yes

—

#11758446

C₄

yes

characteristic

#11758446

C₄

yes

characteristic

C₄×Dic₆₂

C₂²

yes

—

C₄×Dic₆₂

C₂²

yes

—

C₃₁×C₄⋊C₈

C₂²

yes

—

C₃₁×C₂²⋊C₈

C₂²

yes

—

C₃₁×#97

C₂²

yes

—

C₂²⋊(C₃₁⋊C₈)

C₂²

yes

—

#11758403

C₂²

yes

—

Dic₃₁⋊C₈

C₂²

yes

—

C₈×Dic₃₁

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

#11758391

C₂²

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

C₂×C₄

yes

—

C₂×Dic₆₂

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

C₂×C₄

yes

—

Dic₃₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₃₁

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

C₂²×C₄

yes

—

C₂×Dic₃₁

C₂²×C₄

yes

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#2987

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

2₋¹⁺⁴

yes

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₃₁

yes

—

#97

—

#97

—

#97

C₂×D₃₁

yes

—

100

C₄×C₈

—

101

C₃₁

#2987

yes

characteristic, complemented, Sylow

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

C₄×D₃₁

yes

characteristic

105

C₂²×C₄

C₄×D₃₁

yes

characteristic

106

C₂²×C₄

C₂²×D₃₁

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₃₁

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₄×D₃₁

yes

characteristic

118

C₄²

C₄×D₃₁

yes

characteristic

119

C₄²

C₂²×D₃₁

yes

characteristic

120

C₄²

C₂²×D₃₁

yes

characteristic

121

C₂³

C₂×C₄×D₃₁

yes

characteristic

122

Q₈

—

124

—

123

Q₈

—

124

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

130

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

131

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

132

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

133

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

134

C₂×C₄

C₂³×D₃₁

yes

characteristic

135

C₈

—

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂²×C₄×D₃₁

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#11758567

yes

—

149

C₂

—

150

C₂

#14521526

yes

characteristic

151

C₂

#14521815

yes

characteristic

152

C₂

#14521815

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#432	C₂
prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#432	D₃₁
hypocenter	C₃₁	#2987
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₁

commutator, socle

C₂²×C₃₁

C₂³×C₄

Fitting

C₃₁×#432

C₂

prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂×C₄×C₃₁

C₂⁴

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #14521178	C₂	semidirect
3. #14521177	C₂	semidirect
101. C₃₁	#2987	semidirect

H ≅ G/N

type

2. #14521178

C₂

semidirect

3. #14521177

C₂

semidirect

101. C₃₁

#2987

semidirect

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

subgroup	index
#14521606	2
#14521178	2
#14521177	2
#14521107	2
#14521106	2
C₃₁×#432	2
#2987	31

subgroup

index

C₃₁×#432

Subgroups — #16854816

Conjugacy classes of subgroups

Complements

Maximal subgroups