Group #16857515 — subgroups

Subgroups — #16857515

Order 3968.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#14521862	C₂	1	4	yes	maximal, complemented
2	#14521321	C₂	1	4	yes	maximal, complemented
3	#14521231	C₂	1	4	yes	maximal, complemented
4	#14521167	C₂	1	2	yes	maximal, complemented
5	C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758564	C₂²	1	4	yes	—
7	C₂×Q₁₆×C₃₁	C₂²	1	4	yes	—
8	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	#11758519	C₂²	1	8	yes	—
10	#11758496	—	2	4	no	—
11	#11758495	—	2	8	no	—
12	Q₁₆×D₃₁	—	2	4	no	—
13	#11758481	C₂²	1	2	yes	—
14	#11758478	C₂²	1	4	yes	—
15	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	#11758424	C₂²	1	4	yes	—
17	#11758407	C₂²	1	2	yes	—
18	#11758393	C₂²	1	4	yes	—
19	C₂×Q₈×C₃₁	C₂³	1	4	yes	—
20	#92499	—	2	8	no	—
21	Q₈×D₃₁	—	2	8	no	—
22	#92495	—	4	4	no	—
23	#92495	C₂³	1	4	yes	—
24	Q₁₆×C₃₁	—	2	4	no	—
25	Q₁₆×C₃₁	—	2	4	no	—
26	C₂×C₈×C₃₁	C₂³	1	2	yes	—
27	C₄⋊C₄×C₃₁	—	4	2	no	—
28	C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	#92482	—	2	8	no	—
30	#92481	—	2	8	no	—
31	#92474	C₂³	1	4	yes	—
32	Dic₁₂₄	D₄	1	2	yes	—
33	D₂₄₈	D₄	1	2	yes	—
34	#92470	D₄	1	4	yes	—
35	#92469	—	2	4	no	—
36	C₈×D₃₁	—	2	4	no	—
37	Q₈×C₃₁	—	2	8	no	—
38	C₂×C₄×C₃₁	—	2	1	no	—
39	C₂×C₄×C₃₁	—	2	4	no	—
40	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₃₁	—	2	4	no	—
42	D₁₂₄	—	4	4	no	—
43	D₁₂₄	C₂×D₄	1	4	yes	—
44	C₄×D₃₁	—	2	4	no	—
45	C₄×D₃₁	—	4	8	no	—
46	Dic₆₂	—	4	4	no	—
47	Dic₆₂	C₂×D₄	1	4	yes	—
48	C₈×C₃₁	C₂×D₄	1	2	yes	—
49	C₈×C₃₁	C₂×D₄	1	2	yes	—
50	C₃₁⋊C₈	—	2	4	no	—
51	#3293	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂×D₃₁	—	2	4	no	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	—	4	4	no	—
56	C₄×C₃₁	C₂²×D₄	1	2	yes	—
57	Dic₃₁	—	2	4	no	—
58	#578	—	31	4	no	—
59	#494	D₃₁	1	1	yes	characteristic, complemented
60	#473	—	31	4	no	—
61	#456	—	31	4	no	—
62	#443	—	31	2	no	—
63	C₂×C₃₁	—	2	1	no	—
64	C₂×C₃₁	D₄²	1	1	yes	characteristic
65	D₃₁	—	4	4	no	—
66	2₋¹⁺⁴	—	31	4	no	—
67	#137	—	31	8	no	—
68	#137	—	62	8	no	—
69	#135	—	31	2	no	—
70	#135	—	62	4	no	—
71	C₂×Q₁₆	—	62	4	no	—
72	C₂×Q₁₆	C₂×D₃₁	1	4	yes	—
73	#131	—	31	4	no	—
74	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
75	#108	—	31	2	no	—
76	#104	—	31	4	no	—
77	#101	—	31	4	no	—
78	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
79	C₃₁	#3293	1	1	yes	characteristic, complemented, Sylow
80	#41	—	31	4	no	—
81	#41	—	62	8	no	—
82	#41	—	124	4	no	—
83	C₂×Q₈	—	62	8	no	—
84	C₂×Q₈	C₂²×D₃₁	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	31	2	no	—
88	Q₁₆	—	62	8	no	—
89	SD₁₆	—	31	4	no	—
90	SD₁₆	—	62	8	no	—
91	D₈	—	31	2	no	—
92	M₁₆	—	31	4	no	—
93	M₁₆	—	62	4	no	—
94	C₂×C₈	—	62	4	no	—
95	C₂×C₈	C₂²×D₃₁	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₃₁	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	31	4	no	—
100	Q₈	—	124	4	no	—
101	D₄	—	31	4	no	—
102	D₄	—	62	4	no	—
103	D₄	—	124	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	62	4	no	—
107	C₂×C₄	—	124	8	no	—
108	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
109	C₈	—	62	4	no	—
110	C₈	D₄×D₃₁	1	2	yes	—
111	C₈	D₄×D₃₁	1	2	yes	—
112	C₂²	—	62	4	no	—
113	C₂²	#11758561	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	62	4	no	—
117	C₄	C₂×D₄×D₃₁	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	124	4	no	—
120	C₂	#14521684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521862

C₂

yes

maximal, complemented

#14521321

C₂

yes

maximal, complemented

#14521231

C₂

yes

maximal, complemented

#14521167

C₂

yes

maximal, complemented

C₃₁×#494

C₂

yes

maximal, characteristic, complemented

#11758564

C₂²

yes

—

C₂×Q₁₆×C₃₁

C₂²

yes

—

C₃₁×C₄⋊Q₈

C₂²

yes

—

#11758519

C₂²

yes

—

#11758496

—

#11758495

—

Q₁₆×D₃₁

—

#11758481

C₂²

yes

—

#11758478

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758424

C₂²

yes

—

#11758407

C₂²

yes

—

#11758393

C₂²

yes

—

C₂×Q₈×C₃₁

C₂³

yes

—

#92499

—

Q₈×D₃₁

—

#92495

—

#92495

C₂³

yes

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂³

yes

—

—

C₂³

yes

—

Dic₁₂₄

D₄

yes

—

D₂₄₈

D₄

yes

—

#92470

D₄

yes

—

#92469

—

C₈×D₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂²⋊D₃₁

—

D₁₂₄

—

D₁₂₄

C₂×D₄

yes

—

C₄×D₃₁

—

C₄×D₃₁

—

Dic₆₂

—

Dic₆₂

C₂×D₄

yes

—

C₈×C₃₁

C₂×D₄

yes

—

C₈×C₃₁

C₂×D₄

yes

—

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²×D₄

yes

—

Dic₃₁

—

#578

—

#494

D₃₁

yes

characteristic, complemented

#473

—

#456

—

#443

—

C₂×C₃₁

—

C₂×C₃₁

D₄²

yes

characteristic

D₃₁

—

2₋¹⁺⁴

—

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₃₁

yes

—

#131

—

C₄⋊Q₈

C₂×D₃₁

yes

—

#108

—

#104

—

#101

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#3293

yes

characteristic, complemented, Sylow

#41

—

#41

—

#41

—

124

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₃₁

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₃₁

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

124

—

101

D₄

—

102

D₄

—

103

D₄

—

124

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

124

—

108

C₂×C₄

C₂³×D₃₁

yes

characteristic

109

C₈

—

110

C₈

D₄×D₃₁

yes

—

111

C₈

D₄×D₃₁

yes

—

112

C₂²

—

113

C₂²

#11758561

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×D₄×D₃₁

yes

—

118

C₂

—

119

C₂

—

124

—

120

C₂

#14521684

yes

characteristic

	subgroup	G/H
center	C₂	#14521684
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#494	C₂
socle	C₂×C₃₁	D₄²
hypercenter	#494	D₃₁
hypocenter	C₃₁	#3293
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#14521684

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#494

C₂

socle

C₂×C₃₁

D₄²

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #14521862	C₂	semidirect
2. #14521321	C₂	semidirect
3. #14521231	C₂	semidirect
4. #14521167	C₂	semidirect
5. C₃₁×#494	C₂	semidirect
59. #494	D₃₁	semidirect
79. C₃₁	#3293	semidirect

H ≅ G/N

type

1. #14521862

C₂

semidirect

2. #14521321

C₂

semidirect

3. #14521231

C₂

semidirect

4. #14521167

C₂

semidirect

5. C₃₁×#494

C₂

semidirect

59. #494

D₃₁

semidirect

79. C₃₁

#3293

semidirect

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

subgroup	index
#14521862	2
#14521321	2
#14521231	2
#14521167	2
C₃₁×#494	2
#3293	31

subgroup

index

C₃₁×#494

Subgroups — #16857515

Conjugacy classes of subgroups

Complements

Maximal subgroups