Group #16846684 — subgroups

Subgroups — #16846684

Order 3968.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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	#14520915	C₂	1	1	yes	maximal, characteristic
3	#14520812	C₂	1	4	yes	maximal
4	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₆₂	—	2	1	no	—
6	C₂²×C₈×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	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	#11758446	—	2	2	no	—
12	#11758446	C₄	1	4	yes	complemented
13	C₃₁×#97	—	2	2	no	—
14	C₃₁⋊#95	—	2	1	no	—
15	C₃₁⋊#95	—	2	1	no	—
16	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₁	—	2	2	no	—
20	C₂²×Dic₃₁	—	2	2	no	—
21	C₂×Dic₆₂	—	2	4	no	—
22	C₂×Dic₆₂	—	4	4	no	—
23	C₂×C₈×C₃₁	—	2	2	no	—
24	C₂×C₈×C₃₁	—	2	2	no	—
25	C₂×C₈×C₃₁	Q₈	1	4	yes	complemented
26	C₄²×C₃₁	C₂×C₄	1	2	yes	—
27	C₄²×C₃₁	D₄	1	2	yes	—
28	C₄⋊Dic₃₁	—	2	4	no	—
29	C₄⋊Dic₃₁	—	2	2	no	—
30	C₄⋊Dic₃₁	—	2	4	no	—
31	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
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	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
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	2	yes	—
40	C₂×C₄×C₃₁	#41	1	2	yes	—
41	C₂×Dic₃₁	—	2	4	no	—
42	C₂×Dic₃₁	—	4	4	no	—
43	C₂×Dic₃₁	—	4	2	no	—
44	C₂×Dic₃₁	—	4	2	no	—
45	C₂×Dic₃₁	—	4	2	no	—
46	Dic₆₂	—	4	8	no	—
47	C₈×C₃₁	—	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	C₄×Q₈	1	2	yes	—
57	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
58	Dic₃₁	—	4	4	no	—
59	Dic₃₁	—	8	2	no	—
60	C₂×C₄⋊Q₈	—	31	1	no	—
61	C₂×#97	D₃₁	1	1	yes	characteristic
62	#382	—	31	1	no	—
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	C₂×D₃₁	1	2	yes	—
71	C₄⋊Q₈	—	31	4	no	—
72	C₄⋊Q₈	—	62	2	no	—
73	C₂×C₄⋊C₄	—	31	2	no	—
74	C₂×C₄⋊C₄	—	31	2	no	—
75	C₂×C₄⋊C₄	—	62	1	no	—
76	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
77	#97	—	2	2	no	—
78	#95	—	62	1	no	—
79	#95	—	62	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	—	62	2	no	—
84	C₂²×C₄	—	62	2	no	—
85	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
86	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₆₂	1	4	yes	complemented
91	C₄⋊C₄	—	31	4	no	—
92	C₄⋊C₄	—	62	4	no	—
93	C₄⋊C₄	—	62	2	no	—
94	C₄⋊C₄	—	62	4	no	—
95	C₄²	C₄×D₃₁	1	2	yes	—
96	C₄²	C₂²⋊D₃₁	1	2	yes	—
97	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
98	Q₈	—	124	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	62	4	no	—
102	C₂×C₄	—	124	4	no	—
103	C₂×C₄	—	124	2	no	—
104	C₂×C₄	—	124	2	no	—
105	C₂×C₄	—	124	2	no	—
106	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
110	C₂×C₄	#92495	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₂²	#11758446	1	1	yes	characteristic
114	C₂²	#11758452	1	1	yes	characteristic
115	C₂²	#11758485	1	2	yes	—
116	C₂²	#11758486	1	2	yes	—
117	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	124	4	no	—
121	C₄	—	248	2	no	—
122	C₄	C₄×Dic₆₂	1	2	yes	—
123	C₄	#11758501	1	2	yes	—
124	C₂	#14520955	1	1	yes	characteristic
125	C₂	#14521055	1	1	yes	characteristic
126	C₂	#14521059	1	1	yes	characteristic
127	C₂	#14521409	1	2	yes	—
128	C₂	#14521458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

#14520915

C₂

yes

maximal, characteristic

#14520812

C₂

yes

maximal

C₂×C₃₁×#97

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

—

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

#11758446

—

#11758446

C₄

yes

complemented

C₃₁×#97

—

C₃₁⋊#95

—

C₃₁⋊#95

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

Q₈

yes

complemented

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

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

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

C₈×C₃₁

—

#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×Q₈

yes

—

C₄×C₃₁

C₂²⋊Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×#97

D₃₁

yes

characteristic

#382

—

#340

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#501

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₃₁

#1993

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆₂

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃₁

yes

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

Q₈

—

124

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

124

—

103

C₂×C₄

—

124

—

104

C₂×C₄

—

124

—

105

C₂×C₄

—

124

—

106

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

108

C₂×C₄

C₂×Dic₆₂

yes

—

109

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

110

C₂×C₄

#92495

yes

—

111

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

112

C₈

—

113

C₂²

#11758446

yes

characteristic

114

C₂²

#11758452

yes

characteristic

115

C₂²

#11758485

yes

—

116

C₂²

#11758486

yes

—

117

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

124

—

121

C₄

—

248

—

122

C₄

C₄×Dic₆₂

yes

—

123

C₄

#11758501

yes

—

124

C₂

#14520955

yes

characteristic

125

C₂

#14521055

yes

characteristic

126

C₂

#14521059

yes

characteristic

127

C₂

#14521409

yes

—

128

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	#1993
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

#1993

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. #11758446	C₄	semidirect
25. C₂×C₈×C₃₁	Q₈	semidirect
80. C₃₁	#1993	semidirect
90. C₂×C₈	Dic₆₂	semidirect

H ≅ G/N

type

12. #11758446

C₄

semidirect

25. C₂×C₈×C₃₁

Q₈

semidirect

80. C₃₁

#1993

semidirect

90. C₂×C₈

Dic₆₂

semidirect

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

subgroup	index
C₂×#11758446	2
#14520915	2
#14520812	2
C₂×C₃₁×#97	2
#1993	31

subgroup

index

C₂×C₃₁×#97

Subgroups — #16846684

Conjugacy classes of subgroups

Complements

Maximal subgroups