Group #16845590 — subgroups

Subgroups — #16845590

Order 3968.

232 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 125 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×#11758407	C₂	1	1	yes	maximal, characteristic
2	#14521087	C₂	1	1	yes	maximal, characteristic
3	#14521004	C₂	1	2	yes	maximal
4	#14520809	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	#11758503	C₂²	1	2	yes	—
10	C₂×#92474	C₂²	1	2	yes	—
11	C₄×C₈×C₃₁	C₄	1	4	yes	complemented
12	#11758407	C₄	1	4	yes	complemented
13	#11758406	C₄	1	2	yes	complemented
14	#11758405	C₄	1	2	yes	complemented
15	#11758403	—	2	2	no	—
16	C₈×Dic₃₁	—	2	2	no	—
17	C₂²×C₄×C₃₁	—	2	1	no	—
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₃₁	Q₈	1	2	yes	—
27	C₄²×C₃₁	—	2	1	no	—
28	C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂²⋊Dic₃₁	—	2	2	no	—
30	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
31	C₄×Dic₃₁	—	2	2	no	—
32	#92474	—	2	2	no	—
33	#92474	C₂×C₄	1	4	yes	—
34	C₂×C₃₁⋊C₈	—	2	2	no	—
35	C₂³×C₃₁	#41	1	1	yes	characteristic
36	C₂×C₄×C₃₁	—	2	4	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
44	C₂×Dic₃₁	—	2	4	no	—
45	C₈×C₃₁	C₄²	1	2	yes	—
46	C₈×C₃₁	C₄²	1	2	yes	—
47	C₈×C₃₁	C₄⋊C₄	1	2	yes	—
48	C₈×C₃₁	C₄⋊C₄	1	2	yes	—
49	C₃₁⋊C₈	—	2	4	no	—
50	#1772	—	31	1	no	maximal, Sylow
51	C₂²×C₃₁	—	2	2	no	—
52	C₂²×C₃₁	#117	1	1	yes	characteristic
53	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₃₁	C₄×Q₈	1	1	yes	characteristic
55	C₄×C₃₁	—	2	4	no	—
56	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
57	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
59	C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
60	Dic₃₁	—	4	4	no	—
61	C₂×#108	—	31	1	no	—
62	#427	—	31	1	no	—
63	#405	—	31	2	no	—
64	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic
65	#339	—	31	2	no	—
66	C₂×C₃₁	—	2	1	no	—
67	C₂×C₃₁	—	2	1	no	—
68	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₃₁	#443	1	2	yes	—
70	C₂×M₁₆	—	31	2	no	—
71	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
73	#117	—	31	2	no	—
74	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
75	#108	—	31	4	no	—
76	#107	—	31	2	no	—
77	#106	—	31	2	no	—
78	#97	—	62	2	no	—
79	C₄×C₈	—	62	2	no	—
80	C₄×C₈	Dic₃₁	1	4	yes	complemented
81	C₃₁	#1772	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
84	M₁₆	—	31	4	no	—
85	M₁₆	—	62	2	no	—
86	C₂×C₈	—	62	2	no	—
87	C₂×C₈	Dic₆₂	1	2	yes	—
88	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
89	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
90	C₂×C₈	C₄×D₃₁	1	2	yes	—
91	C₂×C₈	D₁₂₄	1	1	yes	characteristic
92	C₂×C₈	D₁₂₄	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
94	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
95	C₄⋊C₄	—	31	4	no	—
96	C₂²⋊C₄	—	62	2	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	62	2	no	—
99	C₄²	C₂×Dic₃₁	1	2	yes	—
100	C₂³	#92495	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	62	4	no	—
104	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
107	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
108	C₂×C₄	#92495	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
110	C₈	—	62	4	no	—
111	C₈	C₄×Dic₃₁	1	2	yes	—
112	C₈	C₄×Dic₃₁	1	2	yes	—
113	C₈	C₄⋊Dic₃₁	1	2	yes	—
114	C₈	C₄⋊Dic₃₁	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Dic₆₂	1	1	yes	characteristic
117	C₂²	C₄×D₁₂₄	1	1	yes	characteristic
118	C₂²	#11758503	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	124	4	no	—
121	C₄	C₄×Dic₆₂	1	1	yes	characteristic
122	C₄	C₄×D₁₂₄	1	1	yes	characteristic
123	C₄	C₂×C₄×Dic₃₁	1	1	yes	characteristic
124	C₄	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄×C₄⋊Dic₃₁	1	1	yes	characteristic
128	C₂	#14521167	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758407

C₂

yes

maximal, characteristic

#14521087

C₂

yes

maximal, characteristic

#14521004

C₂

yes

maximal

#14520809

C₂

yes

maximal

C₂×C₄×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

C₂×#92474

C₂²

yes

—

C₄×C₈×C₃₁

C₄

yes

complemented

#11758407

C₄

yes

complemented

#11758406

C₄

yes

complemented

#11758405

C₄

yes

complemented

#11758403

—

C₈×Dic₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

Q₈

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

—

#92474

—

#92474

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

#41

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₈×C₃₁

C₄²

yes

—

C₈×C₃₁

C₄²

yes

—

C₈×C₃₁

C₄⋊C₄

yes

—

C₈×C₃₁

C₄⋊C₄

yes

—

C₃₁⋊C₈

—

#1772

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

#117

yes

characteristic

C₂²×C₃₁

C₄×D₄

yes

characteristic

C₂²×C₃₁

C₄×Q₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄²

yes

characteristic

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

characteristic

C₄×C₃₁

C₄×Q₈

yes

characteristic

Dic₃₁

—

C₂×#108

—

#427

—

#405

—

C₂×C₄×C₈

D₃₁

yes

characteristic

#339

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₁

#443

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

#117

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

Dic₃₁

yes

complemented

C₃₁

#1772

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

C₄×D₃₁

yes

characteristic

C₂×C₈

C₄×D₃₁

yes

characteristic

C₂×C₈

C₄×D₃₁

yes

—

C₂×C₈

D₁₂₄

yes

characteristic

C₂×C₈

D₁₂₄

yes

characteristic

C₂×C₈

C₂×Dic₃₁

yes

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₃₁

yes

—

100

C₂³

#92495

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×Dic₆₂

yes

characteristic

105

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

106

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

107

C₂×C₄

C₂×D₁₂₄

yes

characteristic

108

C₂×C₄

#92495

yes

characteristic

109

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

110

C₈

—

111

C₈

C₄×Dic₃₁

yes

—

112

C₈

C₄×Dic₃₁

yes

—

113

C₈

C₄⋊Dic₃₁

yes

—

114

C₈

C₄⋊Dic₃₁

yes

—

115

C₂²

—

116

C₂²

C₄×Dic₆₂

yes

characteristic

117

C₂²

C₄×D₁₂₄

yes

characteristic

118

C₂²

#11758503

yes

characteristic

119

C₄

—

120

C₄

—

124

—

121

C₄

C₄×Dic₆₂

yes

characteristic

122

C₄

C₄×D₁₂₄

yes

characteristic

123

C₄

C₂×C₄×Dic₃₁

yes

characteristic

124

C₄

C₂×C₄⋊Dic₃₁

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄×C₄⋊Dic₃₁

yes

characteristic

128

C₂

#14521167

yes

—

	subgroup	G/H
center	C₂×C₈	D₁₂₄
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
socle	C₂²×C₃₁	C₄×D₄
prime radical	C₂³×C₃₁	#41
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	#1772
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

D₁₂₄

commutator

C₄×C₃₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₄×C₈×C₃₁

C₂

socle

C₂²×C₃₁

C₄×D₄

prime radical

C₂³×C₃₁

#41

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄×C₈

D₃₁

hypocenter

C₃₁

#1772

LCS

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
11. C₄×C₈×C₃₁	C₄	semidirect
12. #11758407	C₄	semidirect
13. #11758406	C₄	semidirect
14. #11758405	C₄	semidirect
80. C₄×C₈	Dic₃₁	semidirect
81. C₃₁	#1772	semidirect

H ≅ G/N

type

11. C₄×C₈×C₃₁

C₄

semidirect

12. #11758407

C₄

semidirect

13. #11758406

C₄

semidirect

14. #11758405

C₄

semidirect

80. C₄×C₈

Dic₃₁

semidirect

81. C₃₁

#1772

semidirect

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

subgroup	index
C₂×#11758407	2
#14521087	2
#14521004	2
#14520809	2
C₂×C₄×C₈×C₃₁	2
#1772	31

subgroup

index

C₂×C₄×C₈×C₃₁

Subgroups — #16845590

Conjugacy classes of subgroups

Complements

Maximal subgroups