Group #16856114 — subgroups

Subgroups — #16856114

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	#14521599	C₂	1	1	yes	maximal, characteristic
2	#14521419	C₂	1	2	yes	maximal, complemented
3	#14521415	C₂	1	2	yes	maximal, complemented
4	#14521406	C₂	1	4	yes	maximal, complemented
5	#14521048	C₂	1	4	yes	maximal, complemented
6	C₄×#92474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#92495	C₂²	1	2	yes	—
9	C₃₁×#125	—	2	2	no	—
10	C₃₁×#125	C₂²	1	2	yes	—
11	C₃₁×#125	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	#11758507	—	2	2	no	—
16	C₂×#92474	C₂²	1	2	yes	—
17	#11758452	C₂²	1	2	yes	—
18	#11758451	C₂²	1	1	yes	characteristic
19	C₄×D₁₂₄	—	2	2	no	—
20	#11758446	C₂²	1	1	yes	characteristic
21	#11758398	C₂²	1	8	yes	—
22	#11758397	C₂²	1	8	yes	—
23	#11758391	C₂²	1	2	yes	—
24	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
25	C₂²×C₄×C₃₁	—	2	2	no	—
26	C₂²×C₄×C₃₁	D₄	1	2	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₃₁	—	2	2	no	—
29	#92495	—	2	8	no	—
30	C₂×D₁₂₄	—	2	2	no	—
31	C₂×D₁₂₄	C₂³	1	2	yes	—
32	C₂×C₄×D₃₁	—	2	2	no	—
33	C₂×Dic₆₂	C₂³	1	2	yes	—
34	C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
38	C₄²×C₃₁	D₄	1	2	yes	—
39	C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	(C₂×D₃₁)⋊C₄	—	2	4	no	—
42	C₄⋊Dic₃₁	—	2	2	no	—
43	#92474	—	2	4	no	—
44	C₂×C₃₁⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	—	2	4	no	—
48	C₂×C₄×C₃₁	—	4	2	no	—
49	C₂×C₄×C₃₁	—	4	4	no	—
50	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₁	#41	1	4	yes	—
54	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₃₁	—	4	4	no	—
56	C₂×Dic₃₁	—	2	2	no	—
57	D₁₂₄	—	2	4	no	—
58	D₁₂₄	—	4	4	no	—
59	C₄×D₃₁	—	4	4	no	—
60	Dic₆₂	—	2	4	no	—
61	C₃₁⋊C₈	—	2	4	no	—
62	#3142	—	31	1	no	maximal, Sylow
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	#124	1	2	yes	—
65	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₃₁	—	4	2	no	—
67	C₂×D₃₁	—	4	4	no	—
68	C₄×C₃₁	—	2	2	no	—
69	C₄×C₃₁	—	2	2	no	—
70	C₄×C₃₁	—	4	4	no	—
71	C₄×C₃₁	#124	1	2	yes	—
72	C₄×C₃₁	C₂×#41	1	2	yes	—
73	Dic₃₁	—	4	2	no	—
74	#532	—	31	1	no	—
75	C₂×#125	D₃₁	1	1	yes	characteristic, complemented
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	C₂×#124	1	1	yes	characteristic
83	C₂×C₃₁	#575	1	2	yes	—
84	D₃₁	—	8	2	no	—
85	C₂×#41	—	31	2	no	—
86	C₂×M₁₆	—	31	2	no	—
87	C₄⋊Q₈	—	31	1	no	—
88	#127	—	31	1	no	—
89	#125	—	2	2	no	—
90	#125	C₂×D₃₁	1	2	yes	—
91	#125	C₂×D₃₁	1	2	yes	—
92	#124	—	31	2	no	—
93	#121	—	62	2	no	—
94	C₄×D₄	—	62	2	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
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	—	62	8	no	—
104	C₂×Q₈	—	31	2	no	—
105	C₂×D₄	—	31	2	no	—
106	C₂×D₄	—	62	2	no	—
107	C₂×D₄	—	62	2	no	—
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	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	62	2	no	—
118	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
119	C₂²⋊C₄	—	62	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	—	62	2	no	—
124	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
125	Q₈	—	62	4	no	—
126	D₄	—	62	4	no	—
127	D₄	—	124	4	no	—
128	D₄	—	124	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	62	2	no	—
133	C₂×C₄	—	124	4	no	—
134	C₂×C₄	#92499	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	—	124	2	no	—
142	C₂²	—	124	4	no	—
143	C₂²	#11758524	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	4	no	—
148	C₄	—	124	2	no	—
149	C₄	#11758524	1	2	yes	—
150	C₄	C₂×#92499	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	248	2	no	—
153	C₂	C₂×#11758524	1	1	yes	characteristic
154	C₂	#14521847	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521599

C₂

yes

maximal, characteristic

#14521419

C₂

yes

maximal, complemented

#14521415

C₂

yes

maximal, complemented

#14521406

C₂

yes

maximal, complemented

#14521048

C₂

yes

maximal, complemented

C₄×#92474

C₂

yes

maximal, characteristic, complemented

C₂×C₃₁×#125

C₂

yes

maximal, characteristic, complemented

C₂×#92495

C₂²

yes

—

C₃₁×#125

—

C₃₁×#125

C₂²

yes

—

C₃₁×#125

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758507

—

C₂×#92474

C₂²

yes

—

#11758452

C₂²

yes

—

#11758451

C₂²

yes

characteristic

C₄×D₁₂₄

—

#11758446

C₂²

yes

characteristic

#11758398

C₂²

yes

—

#11758397

C₂²

yes

—

#11758391

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₂²⋊D₃₁

—

#92495

—

C₂×D₁₂₄

—

C₂×D₁₂₄

C₂³

yes

—

C₂×C₄×D₃₁

—

C₂×Dic₆₂

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

(C₂×D₃₁)⋊C₄

—

C₄⋊Dic₃₁

—

#92474

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂²×D₃₁

—

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

C₂²⋊D₃₁

—

C₂×Dic₃₁

—

D₁₂₄

—

D₁₂₄

—

C₄×D₃₁

—

Dic₆₂

—

C₃₁⋊C₈

—

#3142

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

C₂×#41

yes

—

Dic₃₁

—

#532

—

C₂×#125

D₃₁

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#124

yes

characteristic

C₂×C₃₁

#575

yes

—

D₃₁

—

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#125

—

#125

C₂×D₃₁

yes

—

#125

C₂×D₃₁

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

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

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

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

C₂²×D₃₁

yes

—

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

—

124

—

128

D₄

—

124

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

124

—

134

C₂×C₄

#92499

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

—

124

—

142

C₂²

—

124

—

143

C₂²

#11758524

yes

—

144

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

124

—

149

C₄

#11758524

yes

—

150

C₄

C₂×#92499

yes

—

151

C₂

—

152

C₂

—

248

—

153

C₂

C₂×#11758524

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	C₂×C₃₁×#125	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	#14521599	C₂
hypercenter	C₂×#125	D₃₁
hypocenter	C₃₁	#3142
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₃₁×#125

C₂

socle

C₂²×C₃₁

C₂²×D₄

C₂

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #14521419	C₂	semidirect
3. #14521415	C₂	semidirect
4. #14521406	C₂	semidirect
5. #14521048	C₂	semidirect
6. C₄×#92474	C₂	semidirect
7. C₂×C₃₁×#125	C₂	semidirect
75. C₂×#125	D₃₁	semidirect
102. C₃₁	#3142	semidirect

H ≅ G/N

type

2. #14521419

C₂

semidirect

3. #14521415

C₂

semidirect

4. #14521406

C₂

semidirect

5. #14521048

C₂

semidirect

6. C₄×#92474

C₂

semidirect

7. C₂×C₃₁×#125

C₂

semidirect

75. C₂×#125

D₃₁

semidirect

102. C₃₁

#3142

semidirect

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

subgroup	index
#14521599	2
#14521419	2
#14521415	2
#14521406	2
#14521048	2
C₄×#92474	2
C₂×C₃₁×#125	2
#3142	31

subgroup

index

C₂×C₃₁×#125

Subgroups — #16856114

Conjugacy classes of subgroups

Complements

Maximal subgroups