Group #16845428 — subgroups

Subgroups — #16845428

Order 3968.

424 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 205 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×#11758503	C₂	1	2	yes	maximal
2	#14520812	C₂	1	4	yes	maximal, complemented
3	C₂³×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
5	C₂²×C₈×C₃₁	—	2	4	no	—
6	C₂²×C₈×C₃₁	C₄	1	4	yes	complemented
7	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₃₁	—	2	2	no	—
9	#11758503	—	2	4	no	—
10	#11758503	C₄	1	8	yes	complemented
11	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
12	C₂×C₄×Dic₃₁	—	2	2	no	—
13	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	D₄	1	4	yes	—
18	C₂²×C₄×C₃₁	Q₈	1	2	yes	—
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₁	—	2	4	no	—
21	C₂×C₈×C₃₁	—	2	8	no	—
22	C₂×C₈×C₃₁	—	2	4	no	—
23	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
24	C₂²⋊Dic₃₁	—	2	8	no	—
25	C₄⋊Dic₃₁	—	2	4	no	—
26	C₄⋊Dic₃₁	C₂×C₄	1	8	yes	—
27	C₄×Dic₃₁	—	2	8	no	—
28	C₂³×C₃₁	—	2	4	no	—
29	C₂³×C₃₁	C₂²⋊C₄	1	4	yes	—
30	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
31	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	C₄²	1	4	yes	—
33	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
36	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	2	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×Q₈	1	1	yes	characteristic
41	C₂×Dic₃₁	—	2	8	no	—
42	C₂×Dic₃₁	—	4	8	no	—
43	C₈×C₃₁	—	2	4	no	—
44	#1735	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	8	no	—
46	C₂²×C₃₁	—	2	4	no	—
47	C₂²×C₃₁	#95	1	4	yes	—
48	C₂²×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
49	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₃₁	#135	1	4	yes	—
51	C₄×C₃₁	#95	1	4	yes	—
52	C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
53	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
55	Dic₃₁	—	4	8	no	—
56	C₂³×C₈	D₃₁	1	1	yes	characteristic
57	C₂×#117	—	31	2	no	—
58	#340	—	31	4	no	—
59	C₂×C₃₁	—	2	4	no	—
60	C₂×C₃₁	C₂×#95	1	1	yes	characteristic
61	C₂×C₃₁	#416	1	4	yes	—
62	C₂×C₃₁	#427	1	2	yes	—
63	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
64	C₂²×C₈	—	2	4	no	—
65	C₂²×C₈	Dic₃₁	1	4	yes	complemented
66	C₂²×C₈	C₂×D₃₁	1	2	yes	—
67	#117	—	31	8	no	—
68	#117	—	62	4	no	—
69	C₂×C₄⋊C₄	—	31	4	no	—
70	C₂×C₂²⋊C₄	—	62	2	no	—
71	C₂×C₄²	—	62	2	no	—
72	C₃₁	#1735	1	1	yes	characteristic, complemented, Sylow
73	C₂⁴	D₁₂₄	1	1	yes	characteristic
74	C₂²×C₄	—	62	4	no	—
75	C₂²×C₄	Dic₆₂	1	2	yes	—
76	C₂²×C₄	C₄×D₃₁	1	4	yes	—
77	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
79	C₂²×C₄	C₂²⋊D₃₁	1	4	yes	—
80	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
81	C₂×C₈	—	2	8	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
84	C₄⋊C₄	—	31	8	no	—
85	C₄⋊C₄	—	62	4	no	—
86	C₂²⋊C₄	—	62	8	no	—
87	C₄²	—	62	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₄⋊Dic₃₁	1	2	yes	—
90	C₂³	(C₂×D₃₁)⋊C₄	1	4	yes	—
91	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
92	C₂×C₄	—	62	8	no	—
93	C₂×C₄	—	124	8	no	—
94	C₂×C₄	C₄×Dic₃₁	1	4	yes	—
95	C₂×C₄	Dic₃₁⋊C₄	1	8	yes	—
96	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
97	C₂×C₄	(C₂×D₃₁)⋊C₄	1	4	yes	—
98	C₂×C₄	C₂²⋊Dic₃₁	1	4	yes	—
99	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
101	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	4	no	—
106	C₂²	C₃₁⋊#95	1	4	yes	—
107	C₂²	#11758481	1	4	yes	—
108	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
109	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	2	yes	—
110	C₄	—	124	8	no	—
111	C₄	C₃₁⋊#95	1	4	yes	—
112	C₄	C₂×C₄×Dic₃₁	1	1	yes	characteristic
113	C₄	C₂×Dic₃₁⋊C₄	1	2	yes	—
114	C₄	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
115	C₂	—	2	4	no	—
116	C₂	C₂×C₃₁⋊#95	1	1	yes	characteristic
117	C₂	#14521046	1	4	yes	—
118	C₂	#14521087	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758503

C₂

yes

maximal

#14520812

C₂

yes

maximal, complemented

C₂³×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

C₄

yes

complemented

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₂²⋊Dic₃₁

—

#11758503

—

#11758503

C₄

yes

complemented

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

—

C₂⁴×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

C₄×Dic₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²⋊C₄

yes

—

C₂³×C₃₁

C₄⋊C₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

—

#1735

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#95

yes

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#135

yes

—

C₄×C₃₁

#95

yes

—

C₄×C₃₁

C₂×C₄²

yes

characteristic

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

C₂×C₄⋊C₄

yes

—

Dic₃₁

—

C₂³×C₈

D₃₁

yes

characteristic

C₂×#117

—

#340

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#95

yes

characteristic

C₂×C₃₁

#416

yes

—

C₂×C₃₁

#427

yes

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

Dic₃₁

yes

complemented

C₂²×C₈

C₂×D₃₁

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₃₁

#1735

yes

characteristic, complemented, Sylow

C₂⁴

D₁₂₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Dic₆₂

yes

—

C₂²×C₄

C₄×D₃₁

yes

—

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₄⋊Dic₃₁

yes

—

C₂³

(C₂×D₃₁)⋊C₄

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

C₄×Dic₃₁

yes

—

C₂×C₄

Dic₃₁⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

C₂×Dic₆₂

yes

characteristic

100

C₂×C₄

C₂×C₄×D₃₁

yes

—

101

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₃₁⋊#95

yes

—

107

C₂²

#11758481

yes

—

108

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

109

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

—

110

C₄

—

124

—

111

C₄

C₃₁⋊#95

yes

—

112

C₄

C₂×C₄×Dic₃₁

yes

characteristic

113

C₄

C₂×Dic₃₁⋊C₄

yes

—

114

C₄

C₂×C₂²⋊Dic₃₁

yes

characteristic

115

C₂

—

116

C₂

C₂×C₃₁⋊#95

yes

characteristic

117

C₂

#14521046

yes

—

118

C₂

#14521087

yes

—

	subgroup	G/H
center	C₂²×C₄	D₁₂₄
commutator	C₄×C₃₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂³×C₈×C₃₁	C₂
socle	C₂³×C₃₁	C₂×D₄
prime radical	C₂⁴×C₃₁	D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂³×C₈	D₃₁
hypocenter	C₃₁	#1735
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

D₁₂₄

commutator

C₄×C₃₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂³×C₈×C₃₁

C₂

socle

C₂³×C₃₁

C₂×D₄

prime radical

C₂⁴×C₃₁

D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂³×C₈

D₃₁

hypocenter

C₃₁

#1735

LCS

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. #14520812	C₂	semidirect
6. C₂²×C₈×C₃₁	C₄	semidirect
10. #11758503	C₄	semidirect
65. C₂²×C₈	Dic₃₁	semidirect
72. C₃₁	#1735	semidirect

H ≅ G/N

type

2. #14520812

C₂

semidirect

6. C₂²×C₈×C₃₁

C₄

semidirect

10. #11758503

C₄

semidirect

65. C₂²×C₈

Dic₃₁

semidirect

72. C₃₁

#1735

semidirect

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

subgroup	index
C₂×#11758503	2
#14520812	2
C₂³×C₈×C₃₁	2
#1735	31

subgroup

index

C₂×#11758503

#14520812

C₂³×C₈×C₃₁

#1735

Subgroups — #16845428

Conjugacy classes of subgroups

Complements

Maximal subgroups