Group #16845836 — subgroups

Subgroups — #16845836

Order 3968.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	#14521087	C₂	1	1	yes	maximal, characteristic
2	#14521004	C₂	1	1	yes	maximal, characteristic
3	#14520821	C₂	1	2	yes	maximal
4	#14520806	C₂	1	2	yes	maximal
5	C₂×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	#11758503	C₂²	1	2	yes	—
9	#11758503	C₂²	1	1	yes	characteristic
10	C₂×#92474	C₂²	1	1	yes	characteristic
11	C₃₁×#108	—	2	1	no	—
12	C₃₁×#108	—	2	1	no	—
13	#11758406	—	2	1	no	—
14	#11758405	—	2	1	no	—
15	#11758403	C₄	1	2	yes	complemented
16	C₈×Dic₃₁	C₄	1	2	yes	complemented
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	M₁₆×C₃₁	—	2	2	no	—
19	M₁₆×C₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	—	2	2	no	—
21	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₃₁	Q₈	1	1	yes	characteristic
26	C₂×C₈×C₃₁	Q₈	1	1	yes	characteristic
27	C₂²⋊Dic₃₁	—	2	1	no	—
28	C₂²⋊Dic₃₁	—	2	2	no	—
29	C₄⋊Dic₃₁	—	2	1	no	—
30	C₄⋊Dic₃₁	—	2	2	no	—
31	C₄×Dic₃₁	—	2	2	no	—
32	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
33	#92474	—	2	1	no	—
34	#92474	—	2	1	no	—
35	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
36	C₂³×C₃₁	#41	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
43	C₂×Dic₃₁	—	2	2	no	—
44	C₂×Dic₃₁	—	4	2	no	—
45	C₈×C₃₁	—	4	2	no	—
46	C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
50	C₃₁⋊C₈	—	2	2	no	—
51	#1833	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	—	4	1	no	—
53	C₂²×C₃₁	#117	1	1	yes	characteristic
54	C₂²×C₃₁	#124	1	1	yes	characteristic
55	C₂²×C₃₁	#125	1	1	yes	characteristic
56	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₃₁	#117	1	1	yes	characteristic
58	C₄×C₃₁	#124	1	1	yes	characteristic
59	C₄×C₃₁	#125	1	1	yes	characteristic
60	Dic₃₁	—	4	2	no	—
61	Dic₃₁	—	8	2	no	—
62	C₂×#108	D₃₁	1	1	yes	characteristic
63	#427	—	31	1	no	—
64	#405	—	31	1	no	—
65	#344	—	31	2	no	—
66	H(3,ℤ/4)	—	31	2	no	—
67	C₂×C₃₁	—	2	1	no	—
68	C₂×C₃₁	—	2	1	no	—
69	C₂×C₃₁	—	2	1	no	—
70	C₂×C₃₁	#382	1	1	yes	characteristic
71	C₂×M₁₆	—	31	1	no	—
72	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
73	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
74	#117	—	31	1	no	—
75	#117	—	31	2	no	—
76	#108	—	2	1	no	—
77	#108	—	2	1	no	—
78	#107	—	62	1	no	—
79	#106	—	62	1	no	—
80	#97	—	31	2	no	—
81	C₄×C₈	—	31	2	no	—
82	C₃₁	#1833	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	62	1	no	—
87	M₁₆	—	62	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	31	2	no	—
90	C₂×C₈	Dic₆₂	1	1	yes	characteristic
91	C₂×C₈	Dic₆₂	1	1	yes	characteristic
92	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
93	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
96	C₄⋊C₄	—	62	1	no	—
97	C₄⋊C₄	—	62	2	no	—
98	C₂²⋊C₄	—	62	1	no	—
99	C₂²⋊C₄	—	62	2	no	—
100	C₄²	—	31	2	no	—
101	C₄²	—	62	2	no	—
102	C₂³	#92499	1	1	yes	characteristic
103	C₂×C₄	—	62	2	no	—
104	C₂×C₄	—	124	2	no	—
105	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
107	C₂×C₄	#92497	1	1	yes	characteristic
108	C₂×C₄	#92497	1	1	yes	characteristic
109	C₂×C₄	#92499	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	62	2	no	—
113	C₈	Dic₃₁⋊C₄	1	1	yes	characteristic
114	C₈	Dic₃₁⋊C₄	1	1	yes	characteristic
115	C₈	Dic₃₁⋊C₄	1	1	yes	characteristic
116	C₈	Dic₃₁⋊C₄	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#11758467	1	1	yes	characteristic
119	C₂²	#11758469	1	1	yes	characteristic
120	C₂²	#11758524	1	1	yes	characteristic
121	C₄	—	124	2	no	—
122	C₄	—	248	2	no	—
123	C₄	#11758467	1	1	yes	characteristic
124	C₄	#11758469	1	1	yes	characteristic
125	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
126	C₄	#11758513	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#14520914	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521087

C₂

yes

maximal, characteristic

#14521004

C₂

yes

maximal, characteristic

#14520821

C₂

yes

maximal

#14520806

C₂

yes

maximal

C₂×C₃₁×#108

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758503

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

characteristic

C₃₁×#108

—

C₃₁×#108

—

#11758406

—

#11758405

—

#11758403

C₄

yes

complemented

C₈×Dic₃₁

C₄

yes

complemented

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

Q₈

yes

characteristic

C₂×C₈×C₃₁

Q₈

yes

characteristic

C₂²⋊Dic₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

C₂×C₄

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

#41

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₄⋊C₄

yes

characteristic

C₈×C₃₁

C₄⋊C₄

yes

characteristic

C₈×C₃₁

C₄⋊C₄

yes

characteristic

C₈×C₃₁

C₄⋊C₄

yes

characteristic

C₃₁⋊C₈

—

#1833

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

C₂×C₄⋊C₄

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

C₂×#108

D₃₁

yes

—

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#382

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₃₁

#1833

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆₂

yes

characteristic

C₂×C₈

Dic₆₂

yes

characteristic

C₂×C₈

C₄×D₃₁

yes

characteristic

C₂×C₈

C₄×D₃₁

yes

characteristic

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₂³

#92499

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

124

—

105

C₂×C₄

C₂×Dic₆₂

yes

characteristic

106

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

107

C₂×C₄

#92497

yes

characteristic

108

C₂×C₄

#92497

yes

characteristic

109

C₂×C₄

#92499

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

Dic₃₁⋊C₄

yes

characteristic

114

C₈

Dic₃₁⋊C₄

yes

characteristic

115

C₈

Dic₃₁⋊C₄

yes

characteristic

116

C₈

Dic₃₁⋊C₄

yes

characteristic

117

C₂²

—

118

C₂²

#11758467

yes

characteristic

119

C₂²

#11758469

yes

characteristic

120

C₂²

#11758524

yes

characteristic

121

C₄

—

124

—

122

C₄

—

248

—

123

C₄

#11758467

yes

characteristic

124

C₄

#11758469

yes

characteristic

125

C₄

C₂×Dic₃₁⋊C₄

yes

characteristic

126

C₄

#11758513

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#14520914

yes

characteristic

	subgroup	G/H
center	C₄	#11758469
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#108	C₂
socle	C₂×C₃₁	#382
prime radical	C₂³×C₃₁	#41
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#108	D₃₁
hypocenter	C₃₁	#1833
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#11758469

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#108

C₂

C₂×C₃₁

C₂³×C₃₁

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
15. #11758403	C₄	semidirect
16. C₈×Dic₃₁	C₄	semidirect
82. C₃₁	#1833	semidirect

H ≅ G/N

type

15. #11758403

C₄

semidirect

16. C₈×Dic₃₁

C₄

semidirect

82. C₃₁

#1833

semidirect

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

subgroup	index
#14521087	2
#14521004	2
#14520821	2
#14520806	2
C₂×C₃₁×#108	2
#1833	31

subgroup

index

C₂×C₃₁×#108

Subgroups — #16845836

Conjugacy classes of subgroups

Complements

Maximal subgroups