Group #16846532 — subgroups

Subgroups — #16846532

Order 3968.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11758423	C₂	1	4	yes	maximal
2	C₂×#11758391	C₂	1	1	yes	maximal, characteristic
3	#14520915	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
6	C₃₁×C₄⋊Q₈	—	2	2	no	—
7	C₃₁×C₄⋊Q₈	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
12	#11758423	—	2	8	no	—
13	C₃₁⋊#95	—	2	2	no	—
14	#11758391	—	2	2	no	—
15	C₂×Q₈×C₃₁	—	2	2	no	—
16	C₂×Q₈×C₃₁	—	2	4	no	—
17	C₂×Q₈×C₃₁	—	2	4	no	—
18	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₃₁	—	2	2	no	—
20	C₂²×C₄×C₃₁	D₄	1	2	yes	—
21	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃₁	—	2	2	no	—
23	C₄⋊C₄×C₃₁	—	2	8	no	—
24	C₄²×C₃₁	C₂×C₄	1	2	yes	—
25	C₄²×C₃₁	D₄	1	2	yes	—
26	C₄⋊Dic₃₁	—	2	4	no	—
27	C₂×C₃₁⋊C₈	—	2	2	no	—
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
30	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₃₁	—	2	8	no	—
32	Q₈×C₃₁	—	4	4	no	—
33	C₂×C₄×C₃₁	—	2	4	no	—
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	—	4	2	no	—
36	C₂×C₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	#41	1	2	yes	—
42	C₂×Dic₃₁	—	4	2	no	—
43	C₂×Dic₃₁	—	4	4	no	—
44	C₃₁⋊C₈	—	2	4	no	—
45	#1960	—	31	1	no	maximal, Sylow
46	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₃₁	#124	1	1	yes	characteristic
48	C₂²×C₃₁	#127	1	1	yes	characteristic
49	C₂²×C₃₁	#137	1	4	yes	—
50	C₄×C₃₁	—	2	4	no	—
51	C₄×C₃₁	—	4	4	no	—
52	C₄×C₃₁	C₄×D₄	1	2	yes	—
53	C₄×C₃₁	#121	1	2	yes	—
54	Dic₃₁	—	8	2	no	—
55	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
56	C₂×#103	—	31	4	no	—
57	C₂×#97	—	31	1	no	—
58	#382	—	31	1	no	—
59	C₂×C₃₁	#390	1	1	yes	characteristic
60	C₂×C₃₁	#419	1	2	yes	—
61	C₂×C₃₁	#491	1	2	yes	—
62	C₂×C₃₁	#497	1	2	yes	—
63	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
64	C₂²×C₈	—	31	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	Dic₃₁	1	4	yes	complemented
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	31	2	no	—
69	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
70	#103	—	62	8	no	—
71	#97	—	62	2	no	—
72	#95	—	62	2	no	—
73	C₃₁	#1960	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₃₁	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	62	2	no	—
80	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
82	C₂×C₈	—	31	4	no	—
83	C₂×C₈	—	62	2	no	—
84	C₂×C₈	—	62	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	62	4	no	—
87	C₄²	C₂×Dic₃₁	1	2	yes	—
88	C₄²	C₂²⋊D₃₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	124	2	no	—
97	C₂×C₄	—	124	4	no	—
98	C₂×C₄	C₂²⋊Dic₃₁	1	4	yes	—
99	C₂×C₄	D₄×D₃₁	1	2	yes	—
100	C₂×C₄	#92497	1	2	yes	—
101	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
103	C₈	—	62	4	no	—
104	C₂²	#11758513	1	1	yes	characteristic
105	C₂²	#11758517	1	1	yes	characteristic
106	C₂²	#11758519	1	4	yes	—
107	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	248	2	no	—
111	C₄	D₄×Dic₃₁	1	2	yes	—
112	C₄	#11758515	1	2	yes	—
113	C₂	#14520952	1	1	yes	characteristic
114	C₂	#14521058	1	2	yes	—
115	C₂	#14521417	1	2	yes	—
116	C₂	#14521441	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758423

C₂

yes

maximal

C₂×#11758391

C₂

yes

maximal, characteristic

#14520915

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₃₁×C₄⋊Q₈

—

C₃₁×C₄⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758423

—

C₃₁⋊#95

—

#11758391

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

Q₈×C₃₁

—

Q₈×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

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₃₁⋊C₈

—

#1960

—

maximal, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

#121

yes

—

Dic₃₁

—

C₂×C₄⋊Q₈

D₃₁

yes

characteristic

C₂×#103

—

C₂×#97

—

#382

—

C₂×C₃₁

#390

yes

characteristic

C₂×C₃₁

#419

yes

—

C₂×C₃₁

#491

yes

—

C₂×C₃₁

#497

yes

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₃₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

C₃₁

#1960

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₃₁

yes

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

D₄×D₃₁

yes

—

100

C₂×C₄

#92497

yes

—

101

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

103

C₈

—

104

C₂²

#11758513

yes

characteristic

105

C₂²

#11758517

yes

characteristic

106

C₂²

#11758519

yes

—

107

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

248

—

111

C₄

D₄×Dic₃₁

yes

—

112

C₄

#11758515

yes

—

113

C₂

#14520952

yes

characteristic

114

C₂

#14521058

yes

—

115

C₂

#14521417

yes

—

116

C₂

#14521441

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#1960
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₃₁

hypocenter

C₃₁

#1960

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
7. C₃₁×C₄⋊Q₈	C₄	semidirect
66. C₄⋊Q₈	Dic₃₁	semidirect
73. C₃₁	#1960	semidirect

H ≅ G/N

type

7. C₃₁×C₄⋊Q₈

C₄

semidirect

66. C₄⋊Q₈

Dic₃₁

semidirect

73. C₃₁

#1960

semidirect

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

subgroup	index
C₂×#11758423	2
C₂×#11758391	2
#14520915	2
C₂×C₃₁×C₄⋊Q₈	2
#1960	31

subgroup

index

C₂×C₃₁×C₄⋊Q₈

Subgroups — #16846532

Conjugacy classes of subgroups

Complements

Maximal subgroups