Group #16853983 — subgroups

Subgroups — #16853983

Order 3968.

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

normal subgroups only

This group has 356 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	#14520991	C₂	1	4	yes	maximal
2	#14520988	C₂	1	2	yes	maximal
3	#14520970	C₂	1	4	yes	maximal
4	#14520969	C₂	1	2	yes	maximal
5	#14520915	C₂	1	2	yes	maximal
6	C₃₁×#382	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₆₂	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
14	C₂×Dic₃₁⋊C₄	C₂²	1	4	yes	—
15	C₃₁×#95	C₂²	1	4	yes	—
16	C₃₁⋊#95	C₂²	1	4	yes	—
17	C₃₁⋊#95	C₂²	1	4	yes	—
18	C₃₁⋊#95	C₂²	1	2	yes	—
19	C₃₁⋊#95	C₂²	1	4	yes	—
20	C₃₁⋊#95	C₂²	1	2	yes	—
21	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
25	C₂²×Dic₃₁	C₂³	1	4	yes	—
26	C₂²×Dic₃₁	C₂³	1	4	yes	—
27	C₂×Dic₆₂	—	4	4	no	—
28	C₂×Dic₆₂	—	4	2	no	—
29	C₄⋊C₄×C₃₁	—	4	1	no	—
30	C₄⋊C₄×C₃₁	—	4	1	no	—
31	C₄²×C₃₁	Q₈	1	4	yes	complemented
32	C₄⋊Dic₃₁	—	4	2	no	—
33	C₄⋊Dic₃₁	—	4	2	no	—
34	C₄⋊Dic₃₁	—	4	2	no	—
35	Dic₃₁⋊C₄	—	4	4	no	—
36	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	1	no	—
39	C₂×C₄×C₃₁	—	2	1	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	2	4	no	—
42	C₂×C₄×C₃₁	—	2	2	no	—
43	C₂×C₄×C₃₁	—	2	2	no	—
44	C₂×C₄×C₃₁	—	2	4	no	—
45	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
47	C₂×Dic₃₁	—	2	4	no	—
48	C₂×Dic₃₁	—	2	4	no	—
49	C₂×Dic₃₁	—	2	4	no	—
50	C₂×Dic₃₁	—	2	4	no	—
51	C₂×Dic₃₁	—	2	4	no	—
52	C₂×Dic₃₁	—	2	4	no	—
53	Dic₆₂	—	4	8	no	—
54	#2844	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
56	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
58	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
59	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
60	C₄×C₃₁	—	2	4	no	—
61	C₄×C₃₁	—	2	2	no	—
62	C₄×C₃₁	—	4	2	no	—
63	C₄×C₃₁	—	4	2	no	—
64	Dic₃₁	—	4	4	no	—
65	Dic₃₁	—	4	4	no	—
66	#400	—	31	2	no	—
67	#400	—	31	4	no	—
68	#395	—	31	2	no	—
69	#395	—	31	4	no	—
70	#382	—	31	2	no	—
71	#382	D₃₁	1	1	yes	characteristic
72	C₂×C₃₁	#544	1	2	yes	—
73	C₂×C₃₁	#544	1	1	yes	characteristic
74	C₂×C₃₁	H(3,𝔽₄)	1	1	yes	characteristic
75	C₂×C₃₁	#563	1	1	yes	characteristic
76	C₂×C₃₁	#563	1	2	yes	—
77	C₂²×Q₈	—	31	2	no	—
78	C₂×C₄⋊C₄	—	31	2	no	—
79	C₂×C₄⋊C₄	—	31	2	no	—
80	C₂×C₄⋊C₄	—	31	4	no	—
81	C₂×C₄⋊C₄	—	31	2	no	—
82	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
85	#95	—	31	4	no	—
86	#95	—	31	4	no	—
87	#95	—	31	2	no	—
88	#95	—	31	4	no	—
89	#95	—	31	2	no	—
90	#95	C₂×D₃₁	1	4	yes	—
91	C₃₁	#2844	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₈	—	124	2	no	—
93	C₂×Q₈	—	124	4	no	—
94	C₂²×C₄	—	31	4	no	—
95	C₂²×C₄	—	31	4	no	—
96	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
97	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
98	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	124	4	no	—
103	C₄⋊C₄	—	124	2	no	—
104	C₄⋊C₄	—	124	2	no	—
105	C₄⋊C₄	—	124	2	no	—
106	C₄²	Dic₆₂	1	4	yes	complemented
107	C₂³	C₂³×D₃₁	1	1	yes	characteristic
108	Q₈	—	124	8	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	62	4	no	—
118	C₂×C₄	—	62	4	no	—
119	C₂×C₄	—	62	4	no	—
120	C₂×C₄	—	62	4	no	—
121	C₂×C₄	—	62	4	no	—
122	C₂×C₄	—	62	4	no	—
123	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
124	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
125	C₂²	C₂²×Dic₆₂	1	1	yes	characteristic
126	C₂²	#11758561	1	1	yes	characteristic
127	C₂²	#11758564	1	1	yes	characteristic
128	C₂²	#11758566	1	2	yes	—
129	C₂²	#11758567	1	2	yes	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	124	4	no	—
135	C₄	—	124	4	no	—
136	C₂	#14521677	1	2	yes	—
137	C₂	#14521680	1	1	yes	characteristic
138	C₂	#14521791	1	1	yes	characteristic
139	C₂	#14521799	1	1	yes	characteristic
140	C₂	#14521803	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14520991

C₂

yes

maximal

#14520988

C₂

yes

maximal

#14520970

C₂

yes

maximal

#14520969

C₂

yes

maximal

#14520915

C₂

yes

maximal

C₃₁×#382

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

C₂²

yes

—

C₃₁×#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

Q₈

yes

complemented

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

Dic₃₁⋊C₄

—

C₂³×C₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

#2844

—

maximal, Sylow

C₂²×C₃₁

C₂²×Q₈

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

characteristic

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

D₃₁

yes

characteristic

C₂×C₃₁

#544

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

—

C₂×D₃₁

yes

—

C₃₁

#2844

yes

characteristic, complemented, Sylow

C₂×Q₈

—

124

—

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

124

—

103

C₄⋊C₄

—

124

—

104

C₄⋊C₄

—

124

—

105

C₄⋊C₄

—

124

—

106

C₄²

Dic₆₂

yes

complemented

107

C₂³

C₂³×D₃₁

yes

characteristic

108

Q₈

—

124

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×Dic₆₂

yes

—

124

C₂×C₄

C₂×Dic₆₂

yes

—

125

C₂²

C₂²×Dic₆₂

yes

characteristic

126

C₂²

#11758561

yes

characteristic

127

C₂²

#11758564

yes

characteristic

128

C₂²

#11758566

yes

—

129

C₂²

#11758567

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

124

—

135

C₄

—

124

—

136

C₂

#14521677

yes

—

137

C₂

#14521680

yes

characteristic

138

C₂

#14521791

yes

characteristic

139

C₂

#14521799

yes

characteristic

140

C₂

#14521803

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator, socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
Fitting	C₃₁×#382	C₂
hypercenter	#382	D₃₁
hypocenter	C₃₁	#2844
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator, socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

Fitting

C₃₁×#382

C₂

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
31. C₄²×C₃₁	Q₈	semidirect
91. C₃₁	#2844	semidirect
106. C₄²	Dic₆₂	semidirect

H ≅ G/N

type

31. C₄²×C₃₁

Q₈

semidirect

91. C₃₁

#2844

semidirect

106. C₄²

Dic₆₂

semidirect

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

subgroup	index
#14520991	2
#14520988	2
#14520970	2
#14520969	2
#14520915	2
C₃₁×#382	2
#2844	31

subgroup

index

C₃₁×#382

Subgroups — #16853983

Conjugacy classes of subgroups

Complements

Maximal subgroups