Group #16846859 — subgroups

Subgroups — #16846859

Order 3968.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#11758398	C₂	1	2	yes	maximal
2	#14520955	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	2	yes	maximal
4	#14520804	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
7	C₃₁×#125	—	2	4	no	—
8	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
9	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
13	C₃₁⋊#95	—	2	2	no	—
14	#11758398	—	2	4	no	—
15	C₂²×C₄×C₃₁	—	2	2	no	—
16	C₂²×C₄×C₃₁	D₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₁	—	2	1	no	—
19	C₂²×Dic₃₁	—	2	1	no	—
20	C₂×Dic₆₂	—	2	2	no	—
21	C₂×Dic₆₂	—	2	2	no	—
22	C₂×Dic₆₂	—	2	1	no	—
23	C₂×Dic₆₂	Q₈	1	2	yes	—
24	C₄⋊C₄×C₃₁	—	2	8	no	—
25	C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₄⋊C₄×C₃₁	D₄	1	4	yes	—
27	C₄²×C₃₁	—	2	2	no	—
28	C₄⋊Dic₃₁	—	2	2	no	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂×C₃₁⋊C₈	—	2	2	no	—
32	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	—	2	2	no	—
36	C₂×C₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	—	4	2	no	—
38	C₂×C₄×C₃₁	—	4	2	no	—
39	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	2	yes	—
42	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
43	C₂×Dic₃₁	—	2	2	no	—
44	C₂×Dic₃₁	—	4	1	no	—
45	C₂×Dic₃₁	—	4	1	no	—
46	C₂×Dic₃₁	—	4	1	no	—
47	C₂×Dic₃₁	—	4	2	no	—
48	Dic₆₂	—	2	4	no	—
49	Dic₆₂	—	4	2	no	—
50	C₃₁⋊C₈	—	4	2	no	—
51	#2026	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	#120	1	1	yes	characteristic
53	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₃₁	#124	1	1	yes	characteristic
55	C₂²×C₃₁	#135	1	2	yes	—
56	C₂²×C₃₁	#137	1	2	yes	—
57	C₄×C₃₁	—	4	2	no	—
58	C₄×C₃₁	—	4	4	no	—
59	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₃₁	#124	1	2	yes	—
61	Dic₃₁	—	4	2	no	—
62	Dic₃₁	—	8	1	no	—
63	C₂×#125	D₃₁	1	1	yes	characteristic
64	C₂×#103	—	31	2	no	—
65	#391	—	31	1	no	—
66	#340	—	31	2	no	—
67	#336	—	31	1	no	—
68	C₂×C₃₁	#393	1	1	yes	characteristic
69	C₂×C₃₁	#452	1	2	yes	—
70	C₂×C₃₁	#479	1	2	yes	—
71	C₂×C₃₁	#488	1	1	yes	characteristic
72	C₂×C₃₁	#491	1	1	yes	characteristic
73	C₂²×Q₈	—	31	1	no	—
74	C₂²×C₈	—	31	2	no	—
75	#125	—	2	4	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	31	1	no	—
78	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
79	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
80	#103	—	62	4	no	—
81	#95	—	62	2	no	—
82	C₃₁	#2026	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	31	2	no	—
84	C₂×Q₈	—	62	1	no	—
85	C₂×Q₈	—	62	2	no	—
86	C₂×Q₈	—	62	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	62	1	no	—
89	C₂²×C₄	—	62	1	no	—
90	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
92	C₂×C₈	—	62	2	no	—
93	C₂×C₈	—	62	2	no	—
94	C₂×C₈	—	62	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	62	2	no	—
98	C₄⋊C₄	C₂²⋊D₃₁	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
101	Q₈	—	62	4	no	—
102	Q₈	—	124	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	62	2	no	—
110	C₂×C₄	—	124	1	no	—
111	C₂×C₄	—	124	1	no	—
112	C₂×C₄	—	124	1	no	—
113	C₂×C₄	—	124	2	no	—
114	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
115	C₂×C₄	#92499	1	2	yes	—
116	C₂×C₄	#92499	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
118	C₈	—	124	2	no	—
119	C₂²	#11758523	1	1	yes	characteristic
120	C₂²	#11758524	1	1	yes	characteristic
121	C₂²	#11758527	1	2	yes	—
122	C₂²	#11758528	1	2	yes	—
123	C₂²	#11758530	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	124	2	no	—
127	C₄	—	248	1	no	—
128	C₄	#11758523	1	2	yes	—
129	C₄	#11758524	1	2	yes	—
130	C₂	#14520960	1	1	yes	characteristic
131	C₂	#14521211	1	2	yes	—
132	C₂	#14521353	1	2	yes	—
133	C₂	#14521406	1	1	yes	characteristic
134	C₂	#14521419	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758398

C₂

yes

maximal

#14520955

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal

#14520804

C₂

yes

maximal, characteristic

C₂×C₃₁×#125

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

characteristic

C₃₁×#125

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁⋊#95

—

#11758398

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

Q₈

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

—

C₄²×C₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₂²×C₃₁

C₂²⋊Q₈

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

#124

yes

—

Dic₃₁

—

Dic₃₁

—

C₂×#125

D₃₁

yes

—

—

—

—

C₂×C₃₁

#393

yes

characteristic

C₂×C₃₁

#452

yes

—

C₂×C₃₁

#479

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#103

—

#95

—

C₃₁

#2026

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₃₁

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

101

Q₈

—

102

Q₈

—

124

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

124

—

111

C₂×C₄

—

124

—

112

C₂×C₄

—

124

—

113

C₂×C₄

—

124

—

114

C₂×C₄

Q₈×D₃₁

yes

characteristic

115

C₂×C₄

#92499

yes

—

116

C₂×C₄

#92499

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

118

C₈

—

124

—

119

C₂²

#11758523

yes

characteristic

120

C₂²

#11758524

yes

characteristic

121

C₂²

#11758527

yes

—

122

C₂²

#11758528

yes

—

123

C₂²

#11758530

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

124

—

127

C₄

—

248

—

128

C₄

#11758523

yes

—

129

C₄

#11758524

yes

—

130

C₂

#14520960

yes

characteristic

131

C₂

#14521211

yes

—

132

C₂

#14521353

yes

—

133

C₂

#14521406

yes

characteristic

134

C₂

#14521419

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#125	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×#125	D₃₁
hypocenter	C₃₁	#2026
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator, cosocle

C₂²×C₄×C₃₁

C₂³

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#125

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

D₃₁

C₃₁

G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
82. C₃₁	#2026	semidirect

H ≅ G/N

type

82. C₃₁

#2026

semidirect

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

subgroup	index
C₂×#11758398	2
#14520955	2
#14520811	2
#14520804	2
C₂×C₃₁×#125	2
#2026	31

subgroup

index

C₂×C₃₁×#125

Subgroups — #16846859

Conjugacy classes of subgroups

Complements

Maximal subgroups