Group #16856111 — subgroups

Subgroups — #16856111

Order 3968.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11758447	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521417	C₂	1	2	yes	maximal, complemented
3	#14521413	C₂	1	2	yes	maximal, complemented
4	#14521403	C₂	1	4	yes	maximal, complemented
5	#14521050	C₂	1	4	yes	maximal, complemented
6	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#532	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₃₁	C₂²	1	2	yes	—
9	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃₁×#127	C₂²	1	1	yes	characteristic
11	C₃₁×#124	C₂²	1	2	yes	—
12	C₃₁×#121	—	2	2	no	—
13	C₄×D₄×C₃₁	—	2	2	no	—
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
16	C₂×Dic₃₁⋊C₄	—	2	2	no	—
17	C₂×#92474	C₂²	1	2	yes	—
18	#11758447	—	2	2	no	—
19	#11758447	C₂²	1	2	yes	—
20	#11758447	C₂²	1	2	yes	—
21	#11758423	C₂²	1	8	yes	—
22	#11758420	C₂²	1	8	yes	—
23	#11758391	C₂²	1	2	yes	—
24	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
25	#41×C₃₁	—	2	8	no	—
26	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
27	C₂×D₄×C₃₁	—	2	2	no	—
28	C₂×D₄×C₃₁	—	2	2	no	—
29	C₂×D₄×C₃₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₃₁	—	2	2	no	—
31	C₂²×C₄×C₃₁	D₄	1	2	yes	—
32	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃₁	—	2	2	no	—
34	C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂²⋊C₄×C₃₁	—	2	4	no	—
36	C₄²×C₃₁	D₄	1	2	yes	—
37	C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₃₁	—	2	2	no	—
40	C₄⋊Dic₃₁	C₂³	1	4	yes	—
41	Dic₃₁⋊C₄	—	2	4	no	—
42	Dic₃₁⋊C₄	—	2	4	no	—
43	#92474	—	2	4	no	—
44	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₃₁	—	2	2	no	—
46	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₃₁	—	2	4	no	—
48	D₄×C₃₁	—	2	4	no	—
49	D₄×C₃₁	—	4	4	no	—
50	D₄×C₃₁	—	4	4	no	—
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	—	4	2	no	—
53	C₂×C₄×C₃₁	—	4	4	no	—
54	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₃₁	#41	1	4	yes	—
58	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₃₁	—	2	4	no	—
60	C₂×Dic₃₁	—	4	4	no	—
61	C₃₁⋊C₈	—	2	4	no	—
62	#3142	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	—	4	2	no	—
65	C₂²×C₃₁	—	4	4	no	—
66	C₂²×C₃₁	#124	1	2	yes	—
67	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₃₁	—	2	2	no	—
69	C₄×C₃₁	—	2	2	no	—
70	C₄×C₃₁	—	4	2	no	—
71	C₄×C₃₁	#124	1	2	yes	—
72	C₄×C₃₁	C₂×#41	1	2	yes	—
73	Dic₃₁	—	4	4	no	—
74	#532	D₃₁	1	1	yes	characteristic
75	C₂×#125	—	31	1	no	—
76	#491	—	31	2	no	—
77	#490	—	31	2	no	—
78	#488	—	31	4	no	—
79	#417	—	31	4	no	—
80	C₄×M₁₆	—	31	1	no	—
81	C₂×C₃₁	—	2	2	no	—
82	C₂×C₃₁	—	8	2	no	—
83	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
84	C₂×C₃₁	#575	1	2	yes	—
85	C₂×#41	C₂×D₃₁	1	2	yes	—
86	C₂×M₁₆	—	31	2	no	—
87	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
88	#127	C₂×D₃₁	1	1	yes	characteristic
89	#125	—	31	2	no	—
90	#125	—	31	2	no	—
91	#125	—	62	2	no	—
92	#124	C₂×D₃₁	1	2	yes	—
93	#121	—	2	2	no	—
94	C₄×D₄	—	2	2	no	—
95	C₂×C₄⋊C₄	—	31	2	no	—
96	C₂×C₄⋊C₄	—	62	2	no	—
97	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
98	#103	—	31	8	no	—
99	#102	—	31	8	no	—
100	#97	—	31	2	no	—
101	C₄×C₈	—	31	2	no	—
102	C₃₁	#3142	1	1	yes	characteristic, complemented, Sylow
103	#41	—	2	8	no	—
104	C₂×Q₈	C₂²×D₃₁	1	2	yes	—
105	C₂×D₄	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	C₂²×D₃₁	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	62	2	no	—
110	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
111	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
112	M₁₆	—	62	4	no	—
113	C₂×C₈	—	31	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	31	4	no	—
116	C₄⋊C₄	—	62	4	no	—
117	C₄⋊C₄	—	62	4	no	—
118	C₄⋊C₄	—	62	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₄²	C₂²⋊D₃₁	1	2	yes	—
121	C₄²	C₂²×D₃₁	1	1	yes	characteristic
122	C₄²	C₂²×D₃₁	1	1	yes	characteristic
123	C₂³	—	2	2	no	—
124	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	D₄	—	2	4	no	—
127	D₄	—	4	4	no	—
128	D₄	—	4	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	62	4	no	—
133	C₂×C₄	—	124	4	no	—
134	C₂×C₄	#92497	1	4	yes	—
135	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
139	C₈	—	62	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	#11758513	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	2	no	—
148	C₄	—	124	4	no	—
149	C₄	#11758513	1	2	yes	—
150	C₄	C₂×#92497	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	8	2	no	—
153	C₂	C₂×#11758513	1	1	yes	characteristic
154	C₂	#14521847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758447

C₂

yes

maximal, characteristic, complemented

#14521417

C₂

yes

maximal, complemented

#14521413

C₂

yes

maximal, complemented

#14521403

C₂

yes

maximal, complemented

#14521050

C₂

yes

maximal, complemented

C₄×#92474

C₂

yes

maximal, characteristic, complemented

C₃₁×#532

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₃₁×#127

C₂²

yes

characteristic

C₃₁×#124

C₂²

yes

—

C₃₁×#121

—

C₄×D₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

—

C₂×#92474

C₂²

yes

—

#11758447

—

#11758447

C₂²

yes

—

#11758447

C₂²

yes

—

#11758423

C₂²

yes

—

#11758420

C₂²

yes

—

#11758391

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

#41×C₃₁

—

C₂×Q₈×C₃₁

C₂³

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

#92474

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#3142

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

#532

D₃₁

yes

—

—

—

—

—

C₄×M₁₆

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#124

yes

characteristic

C₂×C₃₁

#575

yes

—

C₂×#41

C₂×D₃₁

yes

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

#127

C₂×D₃₁

yes

—

—

—

C₂×D₃₁

yes

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#103

—

#102

—

100

#97

—

101

C₄×C₈

—

102

C₃₁

#3142

yes

characteristic, complemented, Sylow

103

#41

—

104

C₂×Q₈

C₂²×D₃₁

yes

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₃₁

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₃₁

yes

—

111

C₂²×C₄

C₂²×D₃₁

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₃₁

yes

—

121

C₄²

C₂²×D₃₁

yes

characteristic

122

C₄²

C₂²×D₃₁

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

124

—

134

C₂×C₄

#92497

yes

—

135

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

136

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

137

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

138

C₂×C₄

C₂³×D₃₁

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#11758513

yes

—

144

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

124

—

149

C₄

#11758513

yes

—

150

C₄

C₂×#92497

yes

—

151

C₂

—

152

C₂

—

153

C₂

C₂×#11758513

yes

characteristic

154

C₂

#14521847

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting, prime radical	C₃₁×#532	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	#532	D₃₁
hypocenter	C₃₁	#3142
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting, prime radical

C₃₁×#532

C₂

socle

C₂²×C₃₁

C₂²×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#11758447	C₂	semidirect
2. #14521417	C₂	semidirect
3. #14521413	C₂	semidirect
4. #14521403	C₂	semidirect
5. #14521050	C₂	semidirect
6. C₄×#92474	C₂	semidirect
102. C₃₁	#3142	semidirect

H ≅ G/N

type

1. C₂×#11758447

C₂

semidirect

2. #14521417

C₂

semidirect

3. #14521413

C₂

semidirect

4. #14521403

C₂

semidirect

5. #14521050

C₂

semidirect

6. C₄×#92474

C₂

semidirect

102. C₃₁

#3142

semidirect

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

subgroup	index
C₂×#11758447	2
#14521417	2
#14521413	2
#14521403	2
#14521050	2
C₄×#92474	2
C₃₁×#532	2
#3142	31

subgroup

index

C₃₁×#532

Subgroups — #16856111

Conjugacy classes of subgroups

Complements

Maximal subgroups