Group #16845879 — subgroups

Subgroups — #16845879

Order 3968.

256 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11758406	C₂	1	2	yes	maximal
2	C₂×#11758405	C₂	1	2	yes	maximal
3	#14520919	C₂	1	1	yes	maximal, characteristic
4	#14520915	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₃₁	—	2	2	no	—
9	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
12	C₃₁×#97	C₄	1	4	yes	complemented
13	C₃₁⋊#95	—	2	2	no	—
14	#11758406	—	2	4	no	—
15	#11758405	—	2	4	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	C₂²×Dic₃₁	—	2	2	no	—
21	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₃₁	D₄	1	4	yes	—
23	C₂×C₈×C₃₁	Q₈	1	4	yes	—
24	C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₄²×C₃₁	Q₈	1	2	yes	—
26	C₄⋊Dic₃₁	—	2	2	no	—
27	C₄⋊Dic₃₁	—	2	2	no	—
28	C₄⋊Dic₃₁	—	2	2	no	—
29	C₄⋊Dic₃₁	—	2	2	no	—
30	C₄⋊Dic₃₁	—	4	4	no	—
31	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	—	2	2	no	—
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
35	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×Dic₃₁	—	4	2	no	—
40	C₂×Dic₃₁	—	4	1	no	—
41	C₂×Dic₃₁	—	4	1	no	—
42	C₂×Dic₃₁	—	4	4	no	—
43	C₂×Dic₃₁	—	4	2	no	—
44	C₂×Dic₃₁	—	4	2	no	—
45	C₈×C₃₁	C₄⋊C₄	1	8	yes	complemented
46	#1847	—	31	1	no	maximal, Sylow
47	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₃₁	#127	1	1	yes	characteristic
49	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
51	C₂²×C₃₁	#137	1	2	yes	—
52	C₄×C₃₁	—	2	4	no	—
53	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
55	Dic₃₁	—	8	2	no	—
56	Dic₃₁	—	8	1	no	—
57	Dic₃₁	—	8	1	no	—
58	C₂×#107	—	31	2	no	—
59	C₂×#106	—	31	2	no	—
60	C₂×#97	D₃₁	1	1	yes	characteristic
61	#384	—	31	1	no	—
62	#382	—	31	1	no	—
63	C₂×C₃₁	#384	1	1	yes	characteristic
64	C₂×C₃₁	#428	1	2	yes	—
65	C₂×C₃₁	#496	1	1	yes	characteristic
66	C₂×C₃₁	#497	1	1	yes	characteristic
67	C₂×C₃₁	#501	1	2	yes	—
68	C₂²×C₈	C₂×D₃₁	1	2	yes	—
69	C₂×C₄⋊C₄	—	31	1	no	—
70	C₂×C₄⋊C₄	—	31	1	no	—
71	C₂×C₄⋊C₄	—	31	2	no	—
72	C₂×C₄⋊C₄	—	62	2	no	—
73	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
74	#107	—	62	4	no	—
75	#106	—	62	4	no	—
76	#97	Dic₃₁	1	4	yes	complemented
77	#95	—	62	2	no	—
78	C₃₁	#1847	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	62	1	no	—
80	C₂²×C₄	—	62	1	no	—
81	C₂²×C₄	—	62	2	no	—
82	C₂²×C₄	D₁₂₄	1	2	yes	—
83	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
84	C₂×C₈	Dic₆₂	1	4	yes	—
85	C₂×C₈	D₁₂₄	1	4	yes	—
86	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
87	C₄⋊C₄	—	62	2	no	—
88	C₄⋊C₄	—	62	2	no	—
89	C₄⋊C₄	—	62	2	no	—
90	C₄⋊C₄	—	62	2	no	—
91	C₄⋊C₄	—	124	4	no	—
92	C₄²	Dic₆₂	1	2	yes	—
93	C₄²	C₂×Dic₃₁	1	2	yes	—
94	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	124	2	no	—
98	C₂×C₄	—	124	1	no	—
99	C₂×C₄	—	124	1	no	—
100	C₂×C₄	—	124	4	no	—
101	C₂×C₄	—	124	2	no	—
102	C₂×C₄	—	124	2	no	—
103	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
104	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
105	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
106	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
107	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
108	C₈	C₄⋊Dic₃₁	1	8	yes	complemented
109	C₂²	#11758446	1	1	yes	characteristic
110	C₂²	#11758451	1	1	yes	characteristic
111	C₂²	#11758485	1	2	yes	—
112	C₂²	#11758486	1	2	yes	—
113	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
114	C₄	—	2	4	no	—
115	C₄	—	248	2	no	—
116	C₄	—	248	1	no	—
117	C₄	—	248	1	no	—
118	C₄	#11758446	1	2	yes	—
119	C₄	C₂×C₄⋊Dic₃₁	1	2	yes	—
120	C₂	#14520919	1	1	yes	characteristic
121	C₂	#14521091	1	2	yes	—
122	C₂	#14521438	1	1	yes	characteristic
123	C₂	#14521443	1	1	yes	characteristic
124	C₂	#14521458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758406

C₂

yes

maximal

C₂×#11758405

C₂

yes

maximal

#14520919

C₂

yes

maximal, characteristic

#14520915

C₂

yes

maximal, characteristic

C₂×C₃₁×#97

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₃₁×#97

C₄

yes

complemented

C₃₁⋊#95

—

#11758406

—

#11758405

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

Q₈

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

C₄⋊C₄

yes

complemented

#1847

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#127

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₄×C₃₁

C₄⋊Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×#107

—

C₂×#106

—

C₂×#97

D₃₁

yes

characteristic

#384

—

#382

—

C₂×C₃₁

#384

yes

characteristic

C₂×C₃₁

#428

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

—

—

Dic₃₁

yes

complemented

#95

—

C₃₁

#1847

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄²

Dic₆₂

yes

—

C₄²

C₂×Dic₃₁

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

100

C₂×C₄

—

124

—

101

C₂×C₄

—

124

—

102

C₂×C₄

—

124

—

103

C₂×C₄

C₄⋊Dic₃₁

yes

—

104

C₂×C₄

C₂×Dic₆₂

yes

characteristic

105

C₂×C₄

C₂×Dic₆₂

yes

—

106

C₂×C₄

C₂×D₁₂₄

yes

—

107

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

108

C₈

C₄⋊Dic₃₁

yes

complemented

109

C₂²

#11758446

yes

characteristic

110

C₂²

#11758451

yes

characteristic

111

C₂²

#11758485

yes

—

112

C₂²

#11758486

yes

—

113

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

114

C₄

—

115

C₄

—

248

—

116

C₄

—

248

—

117

C₄

—

248

—

118

C₄

#11758446

yes

—

119

C₄

C₂×C₄⋊Dic₃₁

yes

—

120

C₂

#14520919

yes

characteristic

121

C₂

#14521091

yes

—

122

C₂

#14521438

yes

characteristic

123

C₂

#14521443

yes

characteristic

124

C₂

#14521458

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#97	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#97	D₃₁
hypocenter	C₃₁	#1847
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#97

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×#97

D₃₁

hypocenter

C₃₁

#1847

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. C₃₁×#97	C₄	semidirect
45. C₈×C₃₁	C₄⋊C₄	semidirect
76. #97	Dic₃₁	semidirect
78. C₃₁	#1847	semidirect
108. C₈	C₄⋊Dic₃₁	semidirect

H ≅ G/N

type

12. C₃₁×#97

C₄

semidirect

45. C₈×C₃₁

C₄⋊C₄

semidirect

76. #97

Dic₃₁

semidirect

78. C₃₁

#1847

semidirect

108. C₈

C₄⋊Dic₃₁

semidirect

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

subgroup	index
C₂×#11758406	2
C₂×#11758405	2
#14520919	2
#14520915	2
C₂×C₃₁×#97	2
#1847	31

subgroup

index

C₂×C₃₁×#97

Subgroups — #16845879

Conjugacy classes of subgroups

Complements

Maximal subgroups