Group #16846688 — subgroups

Subgroups — #16846688

Order 3968.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#14521606	C₂	1	1	yes	maximal, characteristic
2	#14521073	C₂	1	1	yes	maximal, characteristic
3	#14520809	C₂	1	4	yes	maximal
4	C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄²×C₃₁	—	2	2	no	—
6	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	#11758503	C₂²	1	2	yes	—
9	#11758501	—	2	2	no	—
10	C₂×#92474	C₂²	1	2	yes	—
11	#11758447	C₄	1	2	yes	complemented
12	#11758446	C₄	1	2	yes	complemented
13	C₄×Dic₆₂	—	2	2	no	—
14	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
15	C₂²×C₄×C₃₁	—	2	2	no	—
16	C₂²×C₄×C₃₁	D₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×Dic₆₂	—	2	2	no	—
19	C₄²×C₃₁	—	2	8	no	—
20	C₄²×C₃₁	—	2	2	no	—
21	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₃₁	D₄	1	2	yes	—
24	C₄²×C₃₁	Q₈	1	4	yes	complemented
25	C₂²⋊Dic₃₁	—	2	2	no	—
26	C₄⋊Dic₃₁	—	2	2	no	—
27	C₄⋊Dic₃₁	C₂×C₄	1	4	yes	—
28	Dic₃₁⋊C₄	—	2	4	no	—
29	C₄×Dic₃₁	—	2	2	no	—
30	#92474	—	2	4	no	—
31	C₂×C₃₁⋊C₈	—	2	2	no	—
32	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	—	2	8	no	—
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	—	2	4	no	—
36	C₂×C₄×C₃₁	C₂²⋊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	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₃₁	#41	1	2	yes	—
42	C₂×Dic₃₁	—	2	4	no	—
43	Dic₆₂	—	4	4	no	—
44	C₃₁⋊C₈	—	4	2	no	—
45	#1994	—	31	1	no	maximal, Sylow
46	C₂²×C₃₁	—	2	2	no	—
47	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₃₁	—	2	8	no	—
50	C₄×C₃₁	—	2	2	no	—
51	C₄×C₃₁	#104	1	4	yes	—
52	C₄×C₃₁	C₄×Q₈	1	2	yes	—
53	C₄×C₃₁	#124	1	1	yes	characteristic
54	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
55	Dic₃₁	—	4	4	no	—
56	#533	—	31	1	no	—
57	#423	—	31	1	no	—
58	C₄³	D₃₁	1	1	yes	characteristic
59	#339	—	31	4	no	—
60	C₂×C₃₁	—	2	2	no	—
61	C₂×C₃₁	#391	1	1	yes	characteristic
62	C₂×C₃₁	C₂×#104	1	2	yes	—
63	C₂×M₁₆	—	31	2	no	—
64	C₄⋊Q₈	—	31	2	no	—
65	#125	—	31	2	no	—
66	C₂²⋊Q₈	—	62	2	no	—
67	C₄×Q₈	—	62	2	no	—
68	#117	—	31	2	no	—
69	C₂×C₄²	—	2	2	no	—
70	C₂×C₄²	C₂×D₃₁	1	2	yes	—
71	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
72	C₄⋊C₈	—	62	2	no	—
73	C₃₁	#1994	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	62	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
77	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
78	M₁₆	—	62	4	no	—
79	C₂×C₈	—	62	2	no	—
80	C₄⋊C₄	—	31	4	no	—
81	C₄⋊C₄	—	62	4	no	—
82	C₄⋊C₄	—	62	2	no	—
83	C₂²⋊C₄	—	62	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	62	2	no	—
87	C₄²	Dic₆₂	1	4	yes	complemented
88	C₄²	C₄×D₃₁	1	1	yes	characteristic
89	C₄²	C₄×D₃₁	1	1	yes	characteristic
90	C₄²	D₁₂₄	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
92	Q₈	—	124	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	62	4	no	—
97	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
100	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
102	C₂×C₄	#92495	1	2	yes	—
103	C₈	—	124	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	#11758501	1	2	yes	—
106	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	124	4	no	—
110	C₄	#11758393	1	4	yes	—
111	C₄	C₄×Dic₆₂	1	2	yes	—
112	C₄	#11758446	1	1	yes	characteristic
113	C₄	#11758452	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#14520955	1	1	yes	characteristic
116	C₂	C₂×#11758393	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal, characteristic

#14521073

C₂

yes

maximal, characteristic

#14520809

C₂

yes

maximal

C₄³×C₃₁

C₂

yes

maximal, characteristic

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758501

—

C₂×#92474

C₂²

yes

—

#11758447

C₄

yes

complemented

#11758446

C₄

yes

complemented

C₄×Dic₆₂

—

C₄⋊(C₃₁⋊C₈)

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

Q₈

yes

complemented

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

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

—

Dic₆₂

—

C₃₁⋊C₈

—

#1994

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#104

yes

—

C₄×C₃₁

C₄×Q₈

yes

—

C₄×C₃₁

#124

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

#533

—

#423

—

C₄³

D₃₁

yes

characteristic

#339

—

C₂×C₃₁

—

C₂×C₃₁

#391

yes

characteristic

C₂×C₃₁

C₂×#104

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₃₁

#1994

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₆₂

yes

complemented

C₄²

C₄×D₃₁

yes

characteristic

C₄²

C₄×D₃₁

yes

characteristic

C₄²

D₁₂₄

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

C₂×C₄

C₂×Dic₆₂

yes

—

100

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

101

C₂×C₄

C₂×D₁₂₄

yes

characteristic

102

C₂×C₄

#92495

yes

—

103

C₈

—

124

—

104

C₂²

—

105

C₂²

#11758501

yes

—

106

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

124

—

110

C₄

#11758393

yes

—

111

C₄

C₄×Dic₆₂

yes

—

112

C₄

#11758446

yes

characteristic

113

C₄

#11758452

yes

characteristic

114

C₂

—

115

C₂

#14520955

yes

characteristic

116

C₂

C₂×#11758393

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₄³×C₃₁	C₂
socle	C₂²×C₃₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₄³	D₃₁
hypocenter	C₃₁	#1994
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₄³×C₃₁

C₂

socle

C₂²×C₃₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₄³

D₃₁

hypocenter

C₃₁

#1994

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
11. #11758447	C₄	semidirect
12. #11758446	C₄	semidirect
24. C₄²×C₃₁	Q₈	semidirect
73. C₃₁	#1994	semidirect
87. C₄²	Dic₆₂	semidirect

H ≅ G/N

type

11. #11758447

C₄

semidirect

12. #11758446

C₄

semidirect

24. C₄²×C₃₁

Q₈

semidirect

73. C₃₁

#1994

semidirect

87. C₄²

Dic₆₂

semidirect

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

subgroup	index
#14521606	2
#14521073	2
#14520809	2
C₄³×C₃₁	2
#1994	31

subgroup

index

C₄³×C₃₁

Subgroups — #16846688

Conjugacy classes of subgroups

Complements

Maximal subgroups