Group #16846627 — subgroups

Subgroups — #16846627

Order 3968.

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

normal subgroups only

This group has 276 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	C₂×#11758407	C₂	1	1	yes	maximal, characteristic
2	#14521058	C₂	1	2	yes	maximal
3	C₂×#11758424	C₂	1	2	yes	maximal
4	#14521004	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
7	C₂×Q₁₆×C₃₁	—	2	1	no	—
8	C₂×Q₁₆×C₃₁	—	2	1	no	—
9	C₂×Q₁₆×C₃₁	—	2	2	no	—
10	C₂×Q₁₆×C₃₁	C₄	1	4	yes	complemented
11	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	#11758503	C₂²	1	1	yes	characteristic
13	C₂×#92474	C₂²	1	2	yes	—
14	C₂×#92474	C₂²	1	1	yes	characteristic
15	#11758424	—	2	4	no	—
16	#11758423	—	2	4	no	—
17	#11758407	—	2	2	no	—
18	#11758403	—	2	1	no	—
19	C₈×Dic₃₁	—	2	1	no	—
20	C₂×Q₈×C₃₁	—	2	2	no	—
21	C₂×Q₈×C₃₁	—	2	2	no	—
22	C₂×Q₈×C₃₁	—	2	2	no	—
23	C₂×Q₈×C₃₁	—	2	2	no	—
24	C₂×Q₈×C₃₁	—	2	2	no	—
25	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₃₁	—	2	2	no	—
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	Q₁₆×C₃₁	—	2	4	no	—
29	Q₁₆×C₃₁	—	2	4	no	—
30	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₂×C₈×C₃₁	D₄	1	2	yes	—
34	C₂²⋊Dic₃₁	—	2	1	no	—
35	C₄⋊Dic₃₁	D₄	1	2	yes	—
36	C₄×Dic₃₁	—	2	1	no	—
37	#92474	—	2	1	no	—
38	#92474	—	2	2	no	—
39	#92474	—	2	2	no	—
40	#92474	D₄	1	2	yes	—
41	C₂×C₃₁⋊C₈	—	2	1	no	—
42	C₂×C₃₁⋊C₈	—	2	2	no	—
43	C₂³×C₃₁	#41	1	1	yes	characteristic
44	Q₈×C₃₁	—	2	4	no	—
45	Q₈×C₃₁	—	2	4	no	—
46	Q₈×C₃₁	—	4	2	no	—
47	Q₈×C₃₁	—	4	2	no	—
48	C₂×C₄×C₃₁	—	2	4	no	—
49	C₂×C₄×C₃₁	—	4	2	no	—
50	C₂×C₄×C₃₁	—	4	2	no	—
51	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
57	C₂×Dic₃₁	—	2	2	no	—
58	C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
59	C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₃₁⋊C₈	—	2	2	no	—
61	C₃₁⋊C₈	—	4	2	no	—
62	#1979	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
65	C₂²×C₃₁	#121	1	1	yes	characteristic
66	C₂²×C₃₁	#124	1	1	yes	characteristic
67	C₄×C₃₁	—	4	4	no	—
68	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
70	C₄×C₃₁	#121	1	1	yes	characteristic
71	C₄×C₃₁	#127	1	1	yes	characteristic
72	Dic₃₁	—	4	2	no	—
73	C₂²×Q₁₆	D₃₁	1	1	yes	characteristic
74	C₂×#108	—	31	1	no	—
75	#419	—	31	2	no	—
76	C₂×#101	—	31	2	no	—
77	#405	—	31	1	no	—
78	C₂×C₃₁	—	2	1	no	—
79	C₂×C₃₁	—	2	1	no	—
80	C₂×C₃₁	#390	1	1	yes	characteristic
81	C₂×C₃₁	#473	1	2	yes	—
82	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
83	C₂×Q₁₆	—	2	1	no	—
84	C₂×Q₁₆	—	2	1	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×Q₁₆	Dic₃₁	1	4	yes	complemented
87	C₂×M₁₆	—	31	1	no	—
88	C₂×M₁₆	—	31	2	no	—
89	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
90	#117	—	31	1	no	—
91	#108	—	62	2	no	—
92	#103	—	62	4	no	—
93	#101	—	62	4	no	—
94	#97	—	62	1	no	—
95	C₄×C₈	—	62	1	no	—
96	C₃₁	#1979	1	1	yes	characteristic, complemented, Sylow
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	C₂×Dic₃₁	1	4	yes	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
105	Q₁₆	—	2	4	no	—
106	Q₁₆	—	2	4	no	—
107	M₁₆	—	31	2	no	—
108	M₁₆	—	62	1	no	—
109	M₁₆	—	62	2	no	—
110	M₁₆	—	62	2	no	—
111	C₂×C₈	—	62	1	no	—
112	C₂×C₈	—	62	2	no	—
113	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
114	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
117	C₄⋊C₄	—	31	2	no	—
118	C₂²⋊C₄	—	62	1	no	—
119	C₄²	—	62	1	no	—
120	C₂³	#92497	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	62	2	no	—
129	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
130	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
131	C₂×C₄	#92497	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
135	C₈	—	62	2	no	—
136	C₈	—	124	2	no	—
137	C₈	C₂²⋊Dic₃₁	1	2	yes	—
138	C₈	C₂²⋊Dic₃₁	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	D₄×Dic₃₁	1	1	yes	characteristic
141	C₂²	#11758513	1	1	yes	characteristic
142	C₂²	#11758515	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	124	2	no	—
145	C₄	D₄×Dic₃₁	1	1	yes	characteristic
146	C₄	#11758515	1	1	yes	characteristic
147	C₄	#11758517	1	1	yes	characteristic
148	C₄	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#14520952	1	1	yes	characteristic
152	C₂	#14521321	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758407

C₂

yes

maximal, characteristic

#14521058

C₂

yes

maximal

C₂×#11758424

C₂

yes

maximal

#14521004

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

C₄

yes

complemented

C₂²×C₈×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂×#92474

C₂²

yes

—

—

—

—

C₈×Dic₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

D₄

yes

—

C₄×Dic₃₁

—

—

—

—

D₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

#41

yes

characteristic

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₈×C₃₁

C₂²⋊C₄

yes

—

C₈×C₃₁

C₂²⋊C₄

yes

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1979

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×D₄

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

C₂²×Q₁₆

D₃₁

yes

—

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#390

yes

characteristic

C₂×C₃₁

#473

yes

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₃₁

yes

complemented

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₃₁

yes

—

—

—

—

—

C₄×C₈

—

C₃₁

#1979

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂×Dic₃₁

yes

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₃₁

yes

characteristic

105

Q₁₆

—

106

Q₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂×Dic₃₁

yes

—

114

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

115

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

116

C₂×C₈

C₂²⋊D₃₁

yes

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

#92497

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×D₃₁

yes

characteristic

130

C₂×C₄

D₄×D₃₁

yes

characteristic

131

C₂×C₄

#92497

yes

characteristic

132

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

135

C₈

—

136

C₈

—

124

—

137

C₈

C₂²⋊Dic₃₁

yes

—

138

C₈

C₂²⋊Dic₃₁

yes

—

139

C₂²

—

140

C₂²

D₄×Dic₃₁

yes

characteristic

141

C₂²

#11758513

yes

characteristic

142

C₂²

#11758515

yes

characteristic

143

C₄

—

144

C₄

—

124

—

145

C₄

D₄×Dic₃₁

yes

characteristic

146

C₄

#11758515

yes

characteristic

147

C₄

#11758517

yes

characteristic

148

C₄

C₂×C₂²⋊Dic₃₁

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#14520952

yes

characteristic

152

C₂

#14521321

yes

—

	subgroup	G/H
center	C₂²	#11758515
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂²×Q₁₆×C₃₁	C₂
socle	C₂²×C₃₁	#121
prime radical	C₂³×C₃₁	#41
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂²×Q₁₆	D₃₁
hypocenter	C₃₁	#1979
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#11758515

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂²×Q₁₆×C₃₁

C₂

C₂²×C₃₁

C₂³×C₃₁

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂²×Q₁₆

D₃₁

hypocenter

C₃₁

#1979

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
10. C₂×Q₁₆×C₃₁	C₄	semidirect
86. C₂×Q₁₆	Dic₃₁	semidirect
96. C₃₁	#1979	semidirect

H ≅ G/N

type

10. C₂×Q₁₆×C₃₁

C₄

semidirect

86. C₂×Q₁₆

Dic₃₁

semidirect

96. C₃₁

#1979

semidirect

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

subgroup	index
C₂×#11758407	2
#14521058	2
C₂×#11758424	2
#14521004	2
C₂²×Q₁₆×C₃₁	2
#1979	31

subgroup

index

C₂²×Q₁₆×C₃₁

Subgroups — #16846627

Conjugacy classes of subgroups

Complements

Maximal subgroups