Group #16856221 — subgroups

Subgroups — #16856221

Order 3968.

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

normal subgroups only

This group has 356 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	#14521452	C₂	1	2	yes	maximal, complemented
3	#14521448	C₂	1	4	yes	maximal, complemented
4	#14521445	C₂	1	2	yes	maximal, complemented
5	#14521085	C₂	1	4	yes	maximal
6	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₃₁×#125	C₂²	1	2	yes	—
10	C₃₁×C₂²⋊Q₈	—	2	2	no	—
11	C₄×Q₈×C₃₁	—	2	2	no	—
12	C₃₁×#117	C₂²	1	2	yes	—
13	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	#11758503	C₂²	1	2	yes	—
15	#11758501	—	2	2	no	—
16	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
17	#11758447	C₂²	1	2	yes	—
18	#11758446	C₂²	1	2	yes	—
19	C₄×Dic₆₂	—	2	2	no	—
20	#11758396	C₂²	1	8	yes	—
21	#11758395	C₂²	1	8	yes	—
22	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
23	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
24	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₃₁	—	2	2	no	—
26	C₂²×C₄×C₃₁	D₄	1	2	yes	—
27	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₆₂	—	2	2	no	—
29	C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₃₁	—	2	2	no	—
33	C₄²×C₃₁	—	2	2	no	—
34	C₄²×C₃₁	D₄	1	2	yes	—
35	C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₃₁	—	2	2	no	—
38	C₄⋊Dic₃₁	—	2	2	no	—
39	C₄⋊Dic₃₁	C₂³	1	4	yes	—
40	Dic₃₁⋊C₄	—	2	4	no	—
41	C₄×Dic₃₁	—	2	2	no	—
42	C₂×C₃₁⋊C₈	Q₈	1	8	yes	complemented
43	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
44	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
45	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₃₁	—	4	4	no	—
47	C₂×C₄×C₃₁	—	2	4	no	—
48	C₂×C₄×C₃₁	—	2	4	no	—
49	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₃₁	—	2	4	no	—
55	Dic₆₂	—	4	4	no	—
56	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
57	C₃₁⋊C₈	C₂×Q₈	1	4	yes	—
58	#3160	—	31	1	no	maximal, Sylow
59	C₂²×C₃₁	—	2	2	no	—
60	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
61	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	—	4	4	no	—
64	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
65	C₄×C₃₁	#135	1	4	yes	—
66	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
67	Dic₃₁	—	4	4	no	—
68	#533	—	31	1	no	—
69	#533	D₃₁	1	1	yes	characteristic
70	#500	—	31	2	no	—
71	#499	—	31	4	no	—
72	#498	—	31	2	no	—
73	#427	—	31	4	no	—
74	C₂×C₄×C₈	—	31	1	no	—
75	C₂×C₃₁	—	2	2	no	—
76	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₃₁	C₂×#135	1	2	yes	—
78	C₂²×C₈	—	31	2	no	—
79	C₄⋊Q₈	—	31	2	no	—
80	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
81	#125	—	31	2	no	—
82	#125	C₂×D₃₁	1	2	yes	—
83	C₂²⋊Q₈	—	2	2	no	—
84	C₂²⋊Q₈	—	62	2	no	—
85	C₄×Q₈	—	2	2	no	—
86	C₄×Q₈	—	62	2	no	—
87	#117	—	31	2	no	—
88	#117	C₂×D₃₁	1	2	yes	—
89	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
90	#107	—	31	8	no	—
91	#106	—	31	8	no	—
92	C₄×C₈	—	31	1	no	—
93	C₄×C₈	—	31	2	no	—
94	C₄×C₈	—	31	1	no	—
95	C₃₁	#3160	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	62	2	no	—
98	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
99	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
100	C₂×C₈	—	31	8	no	—
101	C₂×C₈	—	31	2	no	—
102	C₂×C₈	—	31	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	31	4	no	—
106	C₄⋊C₄	—	62	4	no	—
107	C₄⋊C₄	—	62	2	no	—
108	C₄⋊C₄	C₂²×D₃₁	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	62	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	62	2	no	—
113	C₄²	C₂²⋊D₃₁	1	2	yes	—
114	C₄²	C₂²×D₃₁	1	1	yes	characteristic
115	C₄²	C₂²×D₃₁	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	124	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	62	4	no	—
122	C₂×C₄	Q₈×D₃₁	1	4	yes	—
123	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
125	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
126	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
127	C₈	—	31	4	no	—
128	C₈	—	31	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#11758521	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	124	4	no	—
135	C₄	#11758521	1	2	yes	—
136	C₄	#11758527	1	4	yes	—
137	C₄	C₂×Q₈×D₃₁	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#11758521	1	1	yes	characteristic
140	C₂	C₂×#11758527	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

#14521452

C₂

yes

maximal, complemented

#14521448

C₂

yes

maximal, complemented

#14521445

C₂

yes

maximal, complemented

#14521085

C₂

yes

maximal

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

C₃₁×#533

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₃₁×#125

C₂²

yes

—

C₃₁×C₂²⋊Q₈

—

C₄×Q₈×C₃₁

—

C₃₁×#117

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758501

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758447

C₂²

yes

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

—

#11758396

C₂²

yes

—

#11758395

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

C₂³

yes

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

Q₈

yes

complemented

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂×C₃₁⋊C₈

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×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

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

Dic₆₂

—

C₃₁⋊C₈

C₂×Q₈

yes

—

C₃₁⋊C₈

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄⋊Q₈

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

#135

yes

—

C₄×C₃₁

C₂²×Q₈

yes

—

Dic₃₁

—

#533

—

#533

D₃₁

yes

—

—

—

—

C₂×C₄×C₈

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

C₂×#135

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

—

#125

—

#125

C₂×D₃₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₁

#3160

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂²×D₃₁

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

C₂²⋊D₃₁

yes

—

114

C₄²

C₂²×D₃₁

yes

characteristic

115

C₄²

C₂²×D₃₁

yes

characteristic

116

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

117

Q₈

—

118

Q₈

—

124

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

Q₈×D₃₁

yes

—

123

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

124

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

125

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

126

C₂×C₄

C₂³×D₃₁

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

#11758521

yes

—

131

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

124

—

135

C₄

#11758521

yes

—

136

C₄

#11758527

yes

—

137

C₄

C₂×Q₈×D₃₁

yes

—

138

C₂

—

139

C₂

C₂×#11758521

yes

characteristic

140

C₂

C₂×#11758527

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#533	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	#533	D₃₁
hypocenter	C₃₁	#3160
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#533

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14521452	C₂	semidirect
3. #14521448	C₂	semidirect
4. #14521445	C₂	semidirect
42. C₂×C₃₁⋊C₈	Q₈	semidirect
95. C₃₁	#3160	semidirect

H ≅ G/N

type

2. #14521452

C₂

semidirect

3. #14521448

C₂

semidirect

4. #14521445

C₂

semidirect

42. C₂×C₃₁⋊C₈

Q₈

semidirect

95. C₃₁

#3160

semidirect

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

subgroup	index
#14521606	2
#14521452	2
#14521448	2
#14521445	2
#14521085	2
C₂×C₄×C₃₁⋊C₈	2
C₃₁×#533	2
#3160	31

subgroup

index

C₂×C₄×C₃₁⋊C₈

C₃₁×#533

#3160

Subgroups — #16856221

Conjugacy classes of subgroups

Complements

Maximal subgroups