Group #16844373 — subgroups

Subgroups — #16844373

Order 3968.

246 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14521102	C₂	1	2	yes	maximal, complemented
2	#14520798	C₂	1	2	yes	maximal, complemented
3	#14520793	C₂	1	2	yes	maximal, complemented
4	C₃₁×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₃₁	C₄	1	2	yes	—
6	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₃₁×#127	C₂²	1	1	yes	characteristic
8	C₃₁×#124	C₄	1	2	yes	—
9	C₃₁×#121	—	2	2	no	—
10	C₄×D₄×C₃₁	—	2	2	no	—
11	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²⋊(C₃₁⋊C₈)	—	2	2	no	—
13	C₄⋊(C₃₁⋊C₈)	C₂²	1	4	yes	—
14	C₄×C₃₁⋊C₈	—	2	2	no	—
15	#41×C₃₁	—	4	4	no	—
16	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₃₁	—	2	2	no	—
18	C₂×D₄×C₃₁	—	2	2	no	—
19	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₁	—	2	2	no	—
21	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	D₄	1	2	yes	—
23	C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₂²⋊C₄×C₃₁	—	2	4	no	—
25	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₃₁	D₄	1	2	yes	—
27	C₄²×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×C₃₁⋊C₈	—	2	4	no	—
29	C₂³×C₃₁	—	2	2	no	—
30	C₂³×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
31	Q₈×C₃₁	—	4	2	no	—
32	D₄×C₃₁	—	4	2	no	—
33	D₄×C₃₁	—	4	4	no	—
34	D₄×C₃₁	—	4	4	no	—
35	C₂×C₄×C₃₁	—	2	2	no	—
36	C₂×C₄×C₃₁	—	2	4	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	4	4	no	—
39	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
43	C₃₁⋊C₈	—	4	4	no	—
44	#1543	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	1	no	—
46	C₂²×C₃₁	—	2	2	no	—
47	C₂²×C₃₁	—	4	2	no	—
48	C₂²×C₃₁	—	4	4	no	—
49	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₄×C₃₁	—	2	1	no	—
51	C₄×C₃₁	—	2	2	no	—
52	C₄×C₃₁	—	4	2	no	—
53	C₄×C₃₁	#100	1	2	yes	—
54	C₄×C₃₁	#135	1	4	yes	—
55	#532	D₃₁	1	1	yes	characteristic
56	#431	—	31	2	no	—
57	#333	—	31	2	no	—
58	#331	—	31	2	no	—
59	C₂×C₃₁	—	4	1	no	—
60	C₂×C₃₁	—	8	2	no	—
61	C₂×C₃₁	C₂×#100	1	1	yes	characteristic
62	C₂×C₃₁	#416	1	2	yes	—
63	C₂×#41	Dic₃₁	1	2	yes	—
64	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
65	#127	C₂×D₃₁	1	1	yes	characteristic
66	#124	Dic₃₁	1	2	yes	—
67	#121	—	2	2	no	—
68	C₄×D₄	—	2	2	no	—
69	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
70	C₄⋊C₈	—	31	4	no	—
71	C₂²⋊C₈	—	62	2	no	—
72	C₄×C₈	—	62	2	no	—
73	C₃₁	#1543	1	1	yes	characteristic, complemented, Sylow
74	#41	—	4	4	no	—
75	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
76	C₂×D₄	—	2	2	no	—
77	C₂×D₄	—	2	2	no	—
78	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
81	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
82	C₂×C₈	—	62	4	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₂²⋊C₄	—	2	4	no	—
85	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
86	C₄²	C₂²⋊D₃₁	1	2	yes	—
87	C₄²	C₂²×D₃₁	1	1	yes	characteristic
88	C₂³	—	2	2	no	—
89	C₂³	C₂²⋊Dic₃₁	1	1	yes	characteristic
90	Q₈	—	4	2	no	—
91	D₄	—	4	2	no	—
92	D₄	—	4	4	no	—
93	D₄	—	4	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	C₂²⋊Dic₃₁	1	2	yes	—
99	C₂×C₄	C₂²⋊Dic₃₁	1	1	yes	characteristic
100	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
102	C₈	—	124	4	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	2	no	—
110	C₄	—	4	2	no	—
111	C₄	#11758421	1	2	yes	—
112	C₄	#11758527	1	4	yes	—
113	C₂	—	4	1	no	—
114	C₂	—	8	2	no	—
115	C₂	C₂×#11758421	1	1	yes	characteristic
116	C₂	#14521045	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521102

C₂

yes

maximal, complemented

#14520798

C₂

yes

maximal, complemented

#14520793

C₂

yes

maximal, complemented

C₃₁×#532

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₄

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₃₁×#127

C₂²

yes

characteristic

C₃₁×#124

C₄

yes

—

C₃₁×#121

—

C₄×D₄×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂²⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄×C₃₁⋊C₈

—

#41×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊C₄

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×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

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₃₁⋊C₈

—

#1543

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#100

yes

—

C₄×C₃₁

#135

yes

—

#532

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#100

yes

characteristic

C₂×C₃₁

#416

yes

—

C₂×#41

Dic₃₁

yes

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

#127

C₂×D₃₁

yes

characteristic

#124

Dic₃₁

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₃₁

#1543

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂²⋊D₃₁

yes

—

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

—

C₂³

C₂²⋊Dic₃₁

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

C₂²⋊Dic₃₁

yes

characteristic

100

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

102

C₈

—

124

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₂²⋊Dic₃₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#11758421

yes

—

112

C₄

#11758527

yes

—

113

C₂

—

114

C₂

—

115

C₂

C₂×#11758421

yes

characteristic

116

C₂

#14521045

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting, prime radical	C₃₁×#532	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	#532	D₃₁
hypocenter	C₃₁	#1543
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting, prime radical

C₃₁×#532

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

cosocle

C₄²×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #14521102	C₂	semidirect
2. #14520798	C₂	semidirect
3. #14520793	C₂	semidirect
73. C₃₁	#1543	semidirect

H ≅ G/N

type

1. #14521102

C₂

semidirect

2. #14520798

C₂

semidirect

3. #14520793

C₂

semidirect

73. C₃₁

#1543

semidirect

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

subgroup	index
#14521102	2
#14520798	2
#14520793	2
C₃₁×#532	2
#1543	31

subgroup

index

C₃₁×#532

Subgroups — #16844373

Conjugacy classes of subgroups

Complements

Maximal subgroups