Group #16856263 — subgroups

Subgroups — #16856263

Order 3968.

348 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂×#11758447	C₂	1	1	yes	maximal, characteristic
3	#14521458	C₂	1	4	yes	maximal, complemented
4	#14521451	C₂	1	4	yes	maximal, complemented
5	#14521091	C₂	1	4	yes	maximal
6	C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	#11758503	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	#11758501	—	2	2	no	—
12	C₂×Dic₃₁⋊C₄	—	2	2	no	—
13	#11758447	—	2	2	no	—
14	#11758447	C₂²	1	2	yes	—
15	#11758447	C₂²	1	2	yes	—
16	#11758447	C₂²	1	2	yes	—
17	#11758446	C₂²	1	2	yes	—
18	C₄×Dic₆₂	—	2	2	no	—
19	C₃₁×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	#11758406	C₂²	1	8	yes	—
22	#11758405	C₂²	1	8	yes	—
23	C₂²×C₄×C₃₁	D₄	1	2	yes	—
24	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₁	—	2	2	no	—
26	C₂×Dic₆₂	—	2	2	no	—
27	M₁₆×C₃₁	Q₈	1	8	yes	complemented
28	C₂×C₈×C₃₁	C₂³	1	4	yes	—
29	C₄²×C₃₁	D₄	1	2	yes	—
30	C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²⋊Dic₃₁	—	2	2	no	—
33	C₄⋊Dic₃₁	—	2	2	no	—
34	C₄⋊Dic₃₁	—	2	2	no	—
35	C₄⋊Dic₃₁	C₂³	1	4	yes	—
36	C₄⋊Dic₃₁	C₂³	1	4	yes	—
37	Dic₃₁⋊C₄	—	2	4	no	—
38	Dic₃₁⋊C₄	—	2	4	no	—
39	Dic₃₁⋊C₄	—	2	4	no	—
40	C₄×Dic₃₁	—	2	2	no	—
41	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	—	4	2	no	—
43	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₃₁	—	2	4	no	—
49	C₂×Dic₃₁	—	2	4	no	—
50	C₂×Dic₃₁	—	4	4	no	—
51	Dic₆₂	—	4	4	no	—
52	C₈×C₃₁	C₂×Q₈	1	8	yes	—
53	#3166	—	31	1	no	maximal, Sylow
54	C₂²×C₃₁	—	2	2	no	—
55	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
60	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
61	Dic₃₁	—	4	4	no	—
62	Dic₃₁	—	4	4	no	—
63	#533	—	31	1	no	—
64	C₂×#125	—	31	1	no	—
65	#501	—	31	4	no	—
66	#499	—	31	4	no	—
67	#428	—	31	4	no	—
68	C₄×M₁₆	D₃₁	1	1	yes	characteristic
69	C₂×C₃₁	—	2	2	no	—
70	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₃₁	#575	1	2	yes	—
72	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
73	C₄⋊Q₈	—	31	2	no	—
74	#125	—	31	2	no	—
75	#125	—	31	2	no	—
76	#125	—	31	2	no	—
77	#125	—	62	2	no	—
78	C₂²⋊Q₈	—	62	2	no	—
79	C₄×Q₈	—	62	2	no	—
80	#117	—	31	2	no	—
81	C₂×C₄⋊C₄	—	31	2	no	—
82	C₂×C₄⋊C₄	—	62	2	no	—
83	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
84	#107	—	31	8	no	—
85	#106	—	31	8	no	—
86	#97	C₂×D₃₁	1	2	yes	—
87	C₄×C₈	C₂×D₃₁	1	2	yes	—
88	C₃₁	#3166	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	62	2	no	—
90	C₂²×C₄	—	62	2	no	—
91	C₂²×C₄	D₁₂₄	1	2	yes	—
92	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
93	M₁₆	Dic₆₂	1	8	yes	complemented
94	C₂×C₈	C₂²×D₃₁	1	4	yes	—
95	C₄⋊C₄	—	31	4	no	—
96	C₄⋊C₄	—	31	4	no	—
97	C₄⋊C₄	—	62	4	no	—
98	C₄⋊C₄	—	62	4	no	—
99	C₄⋊C₄	—	62	4	no	—
100	C₄⋊C₄	—	62	2	no	—
101	C₄⋊C₄	—	62	2	no	—
102	C₂²⋊C₄	—	62	2	no	—
103	C₄²	—	62	2	no	—
104	C₄²	D₁₂₄	1	2	yes	—
105	C₄²	C₂²×D₃₁	1	1	yes	characteristic
106	C₄²	C₂²×D₃₁	1	1	yes	characteristic
107	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
108	Q₈	—	124	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	62	4	no	—
111	C₂×C₄	—	62	4	no	—
112	C₂×C₄	—	124	4	no	—
113	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
114	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
115	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
116	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
118	C₈	C₂×Dic₆₂	1	8	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	#11758446	1	2	yes	—
121	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	124	4	no	—
125	C₄	—	124	4	no	—
126	C₄	#11758446	1	2	yes	—
127	C₄	C₂²×Dic₆₂	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#11758446	1	1	yes	characteristic
130	C₂	#14521849	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

C₂×#11758447

C₂

yes

maximal, characteristic

#14521458

C₂

yes

maximal, complemented

#14521451

C₂

yes

maximal, complemented

#14521091

C₂

yes

maximal

C₄×M₁₆×C₃₁

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

#11758501

—

C₂×Dic₃₁⋊C₄

—

#11758447

—

#11758447

C₂²

yes

—

#11758447

C₂²

yes

—

#11758447

C₂²

yes

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

—

C₃₁×#97

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

#11758406

C₂²

yes

—

#11758405

C₂²

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

M₁₆×C₃₁

Q₈

yes

complemented

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

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

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

C₈×C₃₁

C₂×Q₈

yes

—

#3166

—

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

C₂²×Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

—

—

—

—

—

C₄×M₁₆

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

#575

yes

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

—

—

C₂×D₃₁

yes

—

C₄×C₈

C₂×D₃₁

yes

—

C₃₁

#3166

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

Dic₆₂

yes

complemented

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

D₁₂₄

yes

—

105

C₄²

C₂²×D₃₁

yes

characteristic

106

C₄²

C₂²×D₃₁

yes

characteristic

107

C₂³

C₂×D₁₂₄

yes

characteristic

108

Q₈

—

124

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

124

—

113

C₂×C₄

C₂×Dic₆₂

yes

—

114

C₂×C₄

C₂×D₁₂₄

yes

—

115

C₂×C₄

C₂×D₁₂₄

yes

—

116

C₂×C₄

C₂×D₁₂₄

yes

characteristic

117

C₂×C₄

C₂³×D₃₁

yes

characteristic

118

C₈

C₂×Dic₆₂

yes

—

119

C₂²

—

120

C₂²

#11758446

yes

—

121

C₂²

C₂²×D₁₂₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

124

—

125

C₄

—

124

—

126

C₄

#11758446

yes

—

127

C₄

C₂²×Dic₆₂

yes

—

128

C₂

—

129

C₂

C₂×#11758446

yes

characteristic

130

C₂

#14521849

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₄×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₄×M₁₆	D₃₁
hypocenter	C₃₁	#3166
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₁₂₄

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₄×M₁₆×C₃₁

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂³×C₃₁

C₂×D₄

hypercenter

C₄×M₁₆

D₃₁

hypocenter

C₃₁

#3166

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #14521458	C₂	semidirect
4. #14521451	C₂	semidirect
27. M₁₆×C₃₁	Q₈	semidirect
88. C₃₁	#3166	semidirect
93. M₁₆	Dic₆₂	semidirect

H ≅ G/N

type

3. #14521458

C₂

semidirect

4. #14521451

C₂

semidirect

27. M₁₆×C₃₁

Q₈

semidirect

88. C₃₁

#3166

semidirect

93. M₁₆

Dic₆₂

semidirect

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

subgroup	index
#14521606	2
C₂×#11758447	2
#14521458	2
#14521451	2
#14521091	2
C₄×M₁₆×C₃₁	2
#3166	31

subgroup

index

C₄×M₁₆×C₃₁

Subgroups — #16856263

Conjugacy classes of subgroups

Complements

Maximal subgroups