Group #16846683 — subgroups

Subgroups — #16846683

Order 3968.

276 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 105 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×#11758446	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758391	C₂	1	1	yes	maximal, characteristic
3	#14520811	C₂	1	4	yes	maximal
4	C₃₁×#382	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₆₂	—	2	1	no	—
6	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₃₁	—	2	1	no	—
9	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
10	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
11	#11758446	—	2	2	no	—
12	#11758446	C₄	1	4	yes	complemented
13	C₃₁×#95	—	2	1	no	—
14	C₃₁×#95	—	2	1	no	—
15	#11758391	—	2	2	no	—
16	C₂²×C₄×C₃₁	—	2	2	no	—
17	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₁	—	2	2	no	—
21	C₂×Dic₆₂	—	2	4	no	—
22	C₂×Dic₆₂	—	4	4	no	—
23	C₄⋊C₄×C₃₁	—	2	4	no	—
24	C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₄²×C₃₁	D₄	1	2	yes	—
26	C₄⋊Dic₃₁	—	2	4	no	—
27	C₄⋊Dic₃₁	—	2	2	no	—
28	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂×C₃₁⋊C₈	Q₈	1	4	yes	complemented
32	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	—	2	2	no	—
35	C₂×C₄×C₃₁	—	4	2	no	—
36	C₂×C₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	—	4	2	no	—
38	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
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	2	yes	—
43	C₂×C₄×C₃₁	#41	1	2	yes	—
44	C₂×Dic₃₁	—	2	4	no	—
45	C₂×Dic₃₁	—	4	4	no	—
46	Dic₆₂	—	4	8	no	—
47	C₃₁⋊C₈	—	2	4	no	—
48	#1993	—	31	1	no	maximal, Sylow
49	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₃₁	#124	1	1	yes	characteristic
51	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
52	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
53	C₂²×C₃₁	#137	1	2	yes	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	—	2	2	no	—
56	C₄×C₃₁	—	8	2	no	—
57	C₄×C₃₁	C₄×Q₈	1	2	yes	—
58	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
59	Dic₃₁	—	4	4	no	—
60	C₂×C₄⋊Q₈	—	31	1	no	—
61	C₂×#97	—	31	1	no	—
62	#382	D₃₁	1	1	yes	characteristic
63	#340	—	31	4	no	—
64	C₂×C₃₁	#391	1	1	yes	characteristic
65	C₂×C₃₁	#418	1	1	yes	characteristic
66	C₂×C₃₁	#419	1	1	yes	characteristic
67	C₂×C₃₁	#489	1	2	yes	—
68	C₂×C₃₁	#501	1	2	yes	—
69	C₂²×Q₈	—	62	1	no	—
70	C₂²×C₈	—	31	2	no	—
71	C₄⋊Q₈	—	31	4	no	—
72	C₄⋊Q₈	—	62	2	no	—
73	C₂×C₄⋊C₄	—	31	2	no	—
74	C₂×C₄⋊C₄	—	62	1	no	—
75	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
76	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
77	#97	—	62	2	no	—
78	#95	—	2	1	no	—
79	#95	—	2	1	no	—
80	C₃₁	#1993	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₈	—	62	4	no	—
82	C₂×Q₈	—	124	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	62	2	no	—
85	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
88	C₂×C₈	—	31	4	no	—
89	C₂×C₈	—	62	2	no	—
90	C₂×C₈	—	62	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	31	4	no	—
93	C₄⋊C₄	—	62	4	no	—
94	C₄⋊C₄	—	62	2	no	—
95	C₄²	C₄×D₃₁	1	2	yes	—
96	C₄²	D₁₂₄	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
98	Q₈	—	124	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	62	4	no	—
105	C₂×C₄	—	124	4	no	—
106	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
110	C₂×C₄	#92497	1	2	yes	—
111	C₂×C₄	Q₈×D₃₁	1	2	yes	—
112	C₈	—	62	4	no	—
113	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
114	C₂²	#11758513	1	1	yes	characteristic
115	C₂²	#11758521	1	1	yes	characteristic
116	C₂²	#11758526	1	2	yes	—
117	C₂²	#11758528	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	2	no	—
121	C₄	—	124	4	no	—
122	C₄	#11758464	1	2	yes	—
123	C₄	#11758474	1	2	yes	—
124	C₂	#14520954	1	1	yes	characteristic
125	C₂	#14521053	1	1	yes	characteristic
126	C₂	#14521057	1	1	yes	characteristic
127	C₂	#14521408	1	2	yes	—
128	C₂	#14521459	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

C₂×#11758391

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal

C₃₁×#382

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758446

—

#11758446

C₄

yes

complemented

C₃₁×#95

—

C₃₁×#95

—

#11758391

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

Q₈

yes

complemented

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

C₃₁⋊C₈

—

#1993

—

maximal, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

#124

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

AGL(1,ℤ/8)

yes

—

C₂²×C₃₁

#137

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

D₃₁

yes

characteristic

#340

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#501

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₃₁

#1993

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃₁

yes

—

C₄²

D₁₂₄

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

Q₈

—

124

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

124

—

106

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

108

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

109

C₂×C₄

C₂×D₁₂₄

yes

characteristic

110

C₂×C₄

#92497

yes

—

111

C₂×C₄

Q₈×D₃₁

yes

—

112

C₈

—

113

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

114

C₂²

#11758513

yes

characteristic

115

C₂²

#11758521

yes

characteristic

116

C₂²

#11758526

yes

—

117

C₂²

#11758528

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

124

—

122

C₄

#11758464

yes

—

123

C₄

#11758474

yes

—

124

C₂

#14520954

yes

characteristic

125

C₂

#14521053

yes

characteristic

126

C₂

#14521057

yes

characteristic

127

C₂

#14521408

yes

—

128

C₂

#14521459

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#382	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#382	D₃₁
hypocenter	C₃₁	#1993
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#382

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
12. #11758446	C₄	semidirect
31. C₂×C₃₁⋊C₈	Q₈	semidirect
80. C₃₁	#1993	semidirect

H ≅ G/N

type

12. #11758446

C₄

semidirect

31. C₂×C₃₁⋊C₈

Q₈

semidirect

80. C₃₁

#1993

semidirect

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

subgroup	index
C₂×#11758446	2
C₂×#11758391	2
#14520811	2
C₃₁×#382	2
#1993	31

subgroup

index

C₃₁×#382

Subgroups — #16846683

Conjugacy classes of subgroups

Complements

Maximal subgroups