Group #16845840 — subgroups

Subgroups — #16845840

Order 3968.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
2	#14520914	C₂	1	2	yes	maximal
3	#14520812	C₂	1	1	yes	maximal, characteristic
4	#14520811	C₂	1	2	yes	maximal
5	C₃₁×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
9	C₂×C₄×Dic₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₃₁⋊#95	—	2	2	no	—
12	C₃₁⋊#95	—	2	2	no	—
13	C₈×Dic₃₁	C₄	1	4	yes	complemented
14	C₂²×C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
16	C₂²×Dic₃₁	—	2	2	no	—
17	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
18	C₂²×Dic₃₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
21	C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₄⋊Dic₃₁	—	2	2	no	—
24	C₄⋊Dic₃₁	—	2	2	no	—
25	C₄×Dic₃₁	C₂×C₄	1	2	yes	—
26	C₄×Dic₃₁	Q₈	1	2	yes	—
27	C₂×C₃₁⋊C₈	—	2	2	no	—
28	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
29	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₃₁	—	4	2	no	—
31	C₂×C₄×C₃₁	—	4	2	no	—
32	C₂×C₄×C₃₁	—	4	2	no	—
33	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₃₁	#41	1	2	yes	—
36	C₂×C₄×C₃₁	#41	1	2	yes	—
37	C₂×Dic₃₁	—	2	1	no	—
38	C₂×Dic₃₁	—	2	2	no	—
39	C₂×Dic₃₁	—	2	1	no	—
40	C₂×Dic₃₁	—	4	2	no	—
41	C₂×Dic₃₁	—	4	2	no	—
42	C₂×Dic₃₁	—	4	2	no	—
43	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
44	C₂×Dic₃₁	C₄⋊C₄	1	2	yes	—
45	C₈×C₃₁	—	2	2	no	—
46	C₃₁⋊C₈	—	2	2	no	—
47	#1835	—	31	1	no	maximal, Sylow
48	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₃₁	#124	1	2	yes	—
50	C₂²×C₃₁	#135	1	1	yes	characteristic
51	C₂²×C₃₁	#135	1	2	yes	—
52	C₂²×C₃₁	#135	1	1	yes	characteristic
53	C₄×C₃₁	—	8	2	no	—
54	C₄×C₃₁	#117	1	1	yes	characteristic
55	C₄×C₃₁	#117	1	1	yes	characteristic
56	C₄×C₃₁	#125	1	2	yes	—
57	Dic₃₁	—	2	2	no	—
58	Dic₃₁	—	2	2	no	—
59	Dic₃₁	—	8	2	no	—
60	C₂×C₄×C₈	—	31	1	no	—
61	#382	—	31	2	no	—
62	#340	—	31	1	no	—
63	#340	—	31	2	no	—
64	#340	D₃₁	1	1	yes	characteristic
65	C₂×C₃₁	#382	1	1	yes	characteristic
66	C₂×C₃₁	#427	1	1	yes	characteristic
67	C₂×C₃₁	#427	1	1	yes	characteristic
68	C₂×C₃₁	#488	1	2	yes	—
69	C₂×C₃₁	#488	1	2	yes	—
70	C₂²×C₈	—	31	1	no	—
71	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
72	C₂×C₄⋊C₄	—	31	2	no	—
73	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
74	C₂×C₄²	—	31	1	no	—
75	C₄×C₈	—	31	4	no	—
76	#95	—	62	2	no	—
77	#95	—	62	2	no	—
78	C₃₁	#1835	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	31	1	no	—
81	C₂²×C₄	—	31	1	no	—
82	C₂²×C₄	—	62	2	no	—
83	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	31	2	no	—
86	C₂×C₈	—	62	2	no	—
87	C₂×C₈	C₄×D₃₁	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	62	2	no	—
91	C₄⋊C₄	—	62	2	no	—
92	C₄²	—	31	2	no	—
93	C₄²	—	31	2	no	—
94	C₂³	D₄×D₃₁	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	31	2	no	—
99	C₂×C₄	—	31	2	no	—
100	C₂×C₄	—	62	1	no	—
101	C₂×C₄	—	62	2	no	—
102	C₂×C₄	—	62	1	no	—
103	C₂×C₄	—	124	2	no	—
104	C₂×C₄	—	124	2	no	—
105	C₂×C₄	—	124	2	no	—
106	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
107	C₂×C₄	#92495	1	2	yes	—
108	C₂×C₄	#92497	1	2	yes	—
109	C₂×C₄	Q₈×D₃₁	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	62	2	no	—
112	C₂²	#11758462	1	2	yes	—
113	C₂²	C₄⋊C₄×D₃₁	1	1	yes	characteristic
114	C₂²	#11758489	1	1	yes	characteristic
115	C₂²	#11758493	1	2	yes	—
116	C₂²	#11758496	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	62	2	no	—
119	C₄	—	62	2	no	—
120	C₄	—	248	2	no	—
121	C₄	#11758449	1	1	yes	characteristic
122	C₄	#11758454	1	1	yes	characteristic
123	C₄	#11758466	1	2	yes	—
124	C₂	#14520913	1	1	yes	characteristic
125	C₂	#14521084	1	1	yes	characteristic
126	C₂	#14521086	1	1	yes	characteristic
127	C₂	#14521402	1	2	yes	—
128	C₂	#14521405	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₃₁

C₂

yes

maximal, characteristic

#14520914

C₂

yes

maximal

#14520812

C₂

yes

maximal, characteristic

#14520811

C₂

yes

maximal

C₃₁×#340

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₃₁⋊#95

—

C₃₁⋊#95

—

C₈×Dic₃₁

C₄

yes

complemented

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂²×Dic₃₁

D₄

yes

characteristic

C₂²×Dic₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1835

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#124

yes

—

C₂²×C₃₁

#135

yes

characteristic

C₂²×C₃₁

#135

yes

—

C₂²×C₃₁

#135

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄×C₈

—

—

—

—

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#488

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₄×C₈

—

#95

—

#95

—

C₃₁

#1835

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

D₄×D₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

124

—

104

C₂×C₄

—

124

—

105

C₂×C₄

—

124

—

106

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

107

C₂×C₄

#92495

yes

—

108

C₂×C₄

#92497

yes

—

109

C₂×C₄

Q₈×D₃₁

yes

characteristic

110

C₈

—

111

C₈

—

112

C₂²

#11758462

yes

—

113

C₂²

C₄⋊C₄×D₃₁

yes

characteristic

114

C₂²

#11758489

yes

characteristic

115

C₂²

#11758493

yes

—

116

C₂²

#11758496

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

248

—

121

C₄

#11758449

yes

characteristic

122

C₄

#11758454

yes

characteristic

123

C₄

#11758466

yes

—

124

C₂

#14520913

yes

characteristic

125

C₂

#14521084

yes

characteristic

126

C₂

#14521086

yes

characteristic

127

C₂

#14521402

yes

—

128

C₂

#14521405

yes

—

	subgroup	G/H
center	C₂³	D₄×D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#340	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#340	D₃₁
hypocenter	C₃₁	#1835
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#340

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 ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
13. C₈×Dic₃₁	C₄	semidirect
78. C₃₁	#1835	semidirect

H ≅ G/N

type

13. C₈×Dic₃₁

C₄

semidirect

78. C₃₁

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₃₁	2
#14520914	2
#14520812	2
#14520811	2
C₃₁×#340	2
#1835	31

subgroup

index

C₂×C₈×Dic₃₁

C₃₁×#340

Subgroups — #16845840

Conjugacy classes of subgroups

Complements

Maximal subgroups