Group #16845706 — subgroups

Subgroups — #16845706

Order 3968.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#11758464	C₂	1	1	yes	maximal, characteristic
2	#14521072	C₂	1	2	yes	maximal
3	C₂×#11758412	C₂	1	2	yes	maximal
4	#14520818	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₁⋊C₄	C₄	1	2	yes	—
10	C₂×C₄×Dic₃₁	C₄	1	2	yes	—
11	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
12	C₂×#92474	C₂²	1	2	yes	—
13	#11758464	—	2	2	no	—
14	#11758464	—	2	2	no	—
15	#11758412	—	2	4	no	—
16	Dic₃₁⋊C₈	—	2	2	no	—
17	Dic₃₁⋊C₈	—	2	2	no	—
18	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
22	C₂²×Dic₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₆₂	—	2	2	no	—
24	C₂×Dic₆₂	—	2	2	no	—
25	C₂×Dic₆₂	D₄	1	2	yes	—
26	C₂×Dic₆₂	Q₈	1	2	yes	—
27	M₁₆×C₃₁	—	2	2	no	—
28	M₁₆×C₃₁	—	2	2	no	—
29	C₂×C₈×C₃₁	—	2	2	no	—
30	C₄⋊C₄×C₃₁	—	2	2	no	—
31	Dic₃₁⋊C₄	—	2	4	no	—
32	C₄×Dic₃₁	—	2	2	no	—
33	C₄×Dic₃₁	—	2	2	no	—
34	C₄×Dic₃₁	D₄	1	2	yes	—
35	C₄×Dic₃₁	D₄	1	2	yes	—
36	#92474	—	2	2	no	—
37	#92474	—	2	2	no	—
38	C₂×C₃₁⋊C₈	—	2	2	no	—
39	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	4	2	no	—
42	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₃₁	#41	1	2	yes	—
46	C₂×Dic₃₁	—	2	2	no	—
47	C₂×Dic₃₁	—	2	2	no	—
48	C₂×Dic₃₁	—	2	2	no	—
49	C₂×Dic₃₁	—	4	1	no	—
50	C₂×Dic₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₃₁	C₂²⋊C₄	1	2	yes	—
52	C₂×Dic₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₃₁	C₄⋊C₄	1	4	yes	—
54	Dic₆₂	—	2	4	no	—
55	Dic₆₂	—	2	4	no	—
56	C₈×C₃₁	—	4	2	no	—
57	C₃₁⋊C₈	—	4	2	no	—
58	#1801	—	31	1	no	maximal, Sylow
59	C₂²×C₃₁	—	2	1	no	—
60	C₂²×C₃₁	—	2	1	no	—
61	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₃₁	#117	1	1	yes	characteristic
64	C₄×C₃₁	—	4	2	no	—
65	C₄×C₃₁	#121	1	2	yes	—
66	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
67	Dic₃₁	—	2	4	no	—
68	Dic₃₁	—	2	1	no	—
69	Dic₃₁	—	2	2	no	—
70	Dic₃₁	#101	1	2	yes	complemented
71	C₂×C₄×Q₈	—	31	1	no	—
72	#423	—	31	2	no	—
73	C₂×#101	—	31	2	no	—
74	#343	—	31	1	no	—
75	#343	D₃₁	1	1	yes	characteristic
76	C₂×C₃₁	—	2	1	no	—
77	C₂×C₃₁	—	2	1	no	—
78	C₂×C₃₁	#380	1	1	yes	characteristic
79	C₂×C₃₁	C₂×#101	1	1	yes	characteristic
80	C₂×C₃₁	#413	1	1	yes	characteristic
81	C₂²×Q₈	—	31	1	no	—
82	C₂×M₁₆	—	31	2	no	—
83	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
84	C₄×Q₈	—	62	2	no	—
85	C₄×Q₈	—	62	2	no	—
86	C₂×C₄⋊C₄	—	31	2	no	—
87	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
88	C₂×C₄²	—	31	1	no	—
89	C₂×C₄²	—	31	2	no	—
90	C₄⋊C₈	—	62	2	no	—
91	C₄⋊C₈	—	62	2	no	—
92	#101	—	62	4	no	—
93	C₃₁	#1801	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	31	2	no	—
95	C₂×Q₈	—	31	2	no	—
96	C₂×Q₈	—	62	2	no	—
97	C₂×Q₈	—	62	2	no	—
98	C₂²×C₄	—	31	1	no	—
99	C₂²×C₄	—	31	2	no	—
100	C₂²×C₄	—	31	1	no	—
101	C₂²×C₄	C₄×D₃₁	1	2	yes	—
102	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	62	2	no	—
106	M₁₆	—	62	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	62	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	62	4	no	—
111	C₄²	—	31	2	no	—
112	C₄²	—	31	2	no	—
113	C₄²	—	62	2	no	—
114	C₄²	—	62	2	no	—
115	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
116	Q₈	—	62	4	no	—
117	Q₈	—	62	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	31	1	no	—
121	C₂×C₄	—	31	1	no	—
122	C₂×C₄	—	31	2	no	—
123	C₂×C₄	—	31	4	no	—
124	C₂×C₄	—	62	2	no	—
125	C₂×C₄	—	62	2	no	—
126	C₂×C₄	—	62	2	no	—
127	C₂×C₄	—	124	1	no	—
128	C₂×C₄	#92495	1	2	yes	—
129	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
130	C₂×C₄	D₄×D₃₁	1	2	yes	—
131	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
132	C₈	—	4	2	no	—
133	C₈	—	124	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#11758449	1	1	yes	characteristic
137	C₂²	C₂²⋊C₄×D₃₁	1	1	yes	characteristic
138	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	31	2	no	—
141	C₄	—	62	4	no	—
142	C₄	—	62	1	no	—
143	C₄	—	62	2	no	—
144	C₄	#11758461	1	2	yes	—
145	C₄	#11758473	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#14520908	1	1	yes	characteristic
149	C₂	#101×D₃₁	1	1	yes	characteristic
150	C₂	#14521030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758464

C₂

yes

maximal, characteristic

#14521072

C₂

yes

maximal

C₂×#11758412

C₂

yes

maximal

#14520818

C₂

yes

maximal, characteristic

C₃₁×#343

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×Dic₃₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₃₁

C₄

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

#11758464

—

#11758464

—

#11758412

—

Dic₃₁⋊C₈

—

Dic₃₁⋊C₈

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂²×Dic₃₁

C₂×C₄

yes

characteristic

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

D₄

yes

—

C₂×Dic₆₂

Q₈

yes

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

D₄

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₁

C₂²⋊C₄

yes

—

C₂×Dic₃₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₁

C₄⋊C₄

yes

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1801

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#117

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

#121

yes

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

#101

yes

complemented

C₂×C₄×Q₈

—

—

—

—

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₃₁

#1801

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₃₁

yes

—

102

C₂²×C₄

C₂²×D₃₁

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₂³

C₂×C₄×D₃₁

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

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

#92495

yes

—

129

C₂×C₄

D₄×D₃₁

yes

characteristic

130

C₂×C₄

D₄×D₃₁

yes

—

131

C₂×C₄

Q₈×D₃₁

yes

characteristic

132

C₈

—

133

C₈

—

124

—

134

C₂²

—

135

C₂²

—

136

C₂²

#11758449

yes

characteristic

137

C₂²

C₂²⋊C₄×D₃₁

yes

characteristic

138

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#11758461

yes

—

145

C₄

#11758473

yes

—

146

C₂

—

147

C₂

—

148

C₂

#14520908

yes

characteristic

149

C₂

#101×D₃₁

yes

characteristic

150

C₂

#14521030

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₃₁
commutator, prime radical	C₂³×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#343	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#343	D₃₁
hypocenter	C₃₁	#1801
LCS	G ▷ C₂³×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₃₁

commutator, prime radical

C₂³×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#343

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
70. Dic₃₁	#101	semidirect
93. C₃₁	#1801	semidirect

H ≅ G/N

type

70. Dic₃₁

#101

semidirect

93. C₃₁

#1801

semidirect

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

subgroup	index
C₂×#11758464	2
#14521072	2
C₂×#11758412	2
#14520818	2
C₃₁×#343	2
#1801	31

subgroup

index

C₃₁×#343

Subgroups — #16845706

Conjugacy classes of subgroups

Complements

Maximal subgroups