Group #16857510 — subgroups

Subgroups — #16857510

Order 3968.

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

normal subgroups only

This group has 476 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	#14521850	C₂	1	4	yes	maximal, complemented
2	#14521316	C₂	1	4	yes	maximal, complemented
3	#14521216	C₂	1	4	yes	maximal, complemented
4	#14521167	C₂	1	2	yes	maximal, complemented
5	C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758561	C₂²	1	4	yes	complemented
7	C₂×D₈×C₃₁	C₂²	1	4	yes	complemented
8	C₃₁×#127	C₂²	1	2	yes	complemented
9	#11758509	C₂²	1	8	yes	complemented
10	#11758489	—	2	4	no	—
11	#11758488	—	2	8	no	—
12	D₈×D₃₁	—	2	4	no	—
13	#11758481	C₂²	1	2	yes	—
14	#11758478	C₂²	1	4	yes	complemented
15	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
16	#11758421	C₂²	1	4	yes	complemented
17	#11758407	C₂²	1	2	yes	complemented
18	#11758393	C₂²	1	4	yes	complemented
19	C₂×D₄×C₃₁	—	4	2	no	—
20	C₂×D₄×C₃₁	C₂³	1	4	yes	—
21	C₂×C₂²⋊D₃₁	—	4	4	no	—
22	#92497	—	2	8	no	—
23	D₄×D₃₁	—	2	8	no	—
24	#92495	C₂³	1	4	yes	—
25	D₈×C₃₁	—	2	4	no	—
26	D₈×C₃₁	—	2	4	no	—
27	C₂×C₈×C₃₁	C₂³	1	2	yes	—
28	C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	#92480	—	2	8	no	—
30	#92479	—	2	8	no	—
31	#92474	C₂³	1	4	yes	—
32	Dic₁₂₄	D₄	1	2	yes	complemented
33	D₂₄₈	D₄	1	2	yes	complemented
34	#92470	D₄	1	4	yes	complemented
35	#92469	—	2	4	no	—
36	C₈×D₃₁	—	2	4	no	—
37	C₂³×C₃₁	—	2	4	no	—
38	C₂²×D₃₁	—	4	4	no	—
39	D₄×C₃₁	—	2	8	no	—
40	D₄×C₃₁	—	8	4	no	—
41	C₂×C₄×C₃₁	—	2	1	no	—
42	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
43	C₂²⋊D₃₁	—	2	4	no	—
44	C₂²⋊D₃₁	—	4	8	no	—
45	C₂×Dic₃₁	—	4	4	no	—
46	D₁₂₄	C₂×D₄	1	4	yes	—
47	C₄×D₃₁	—	2	4	no	—
48	Dic₆₂	C₂×D₄	1	4	yes	—
49	C₈×C₃₁	C₂×D₄	1	2	yes	—
50	C₈×C₃₁	C₂×D₄	1	2	yes	—
51	C₃₁⋊C₈	—	2	4	no	—
52	#3292	—	31	1	no	maximal, Sylow
53	C₂²×C₃₁	—	4	4	no	—
54	C₂²×C₃₁	—	8	4	no	—
55	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
56	C₂×D₃₁	—	2	4	no	—
57	C₂×D₃₁	—	16	4	no	—
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	C₂²×D₄	1	2	yes	—
60	Dic₃₁	—	2	4	no	—
61	#576	—	31	4	no	—
62	#493	D₃₁	1	1	yes	characteristic, complemented
63	#472	—	31	4	no	—
64	#453	—	31	4	no	—
65	#443	—	31	2	no	—
66	C₂×C₃₁	—	2	1	no	—
67	C₂×C₃₁	—	8	4	no	—
68	C₂×C₃₁	D₄²	1	1	yes	characteristic
69	D₃₁	—	4	4	no	—
70	H(5,𝔽₂)	—	31	4	no	—
71	AGL(1,ℤ/8)	—	31	8	no	—
72	AGL(1,ℤ/8)	—	62	8	no	—
73	#135	—	31	2	no	—
74	#135	—	62	4	no	—
75	C₂×D₈	—	62	4	no	—
76	C₂×D₈	C₂×D₃₁	1	4	yes	complemented
77	#131	—	31	4	no	—
78	#127	C₂×D₃₁	1	2	yes	complemented
79	#108	—	31	2	no	—
80	#104	—	31	4	no	—
81	#100	—	31	4	no	—
82	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
83	C₃₁	#3292	1	1	yes	characteristic, complemented, Sylow
84	#41	—	31	4	no	—
85	#41	—	62	8	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	62	8	no	—
88	C₂×D₄	—	124	4	no	—
89	C₂×D₄	C₂²×D₃₁	1	4	yes	—
90	Q₁₆	—	31	2	no	—
91	SD₁₆	—	31	4	no	—
92	SD₁₆	—	62	8	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	31	2	no	—
96	D₈	—	62	8	no	—
97	M₁₆	—	31	4	no	—
98	M₁₆	—	62	4	no	—
99	C₂×C₈	—	62	4	no	—
100	C₂×C₈	C₂²×D₃₁	1	2	yes	—
101	C₄²	C₂²×D₃₁	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	124	4	no	—
104	Q₈	—	31	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	31	4	no	—
108	D₄	—	62	4	no	—
109	D₄	—	124	8	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	62	4	no	—
112	C₂×C₄	—	124	4	no	—
113	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
114	C₈	—	62	4	no	—
115	C₈	D₄×D₃₁	1	2	yes	—
116	C₈	D₄×D₃₁	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	62	4	no	—
120	C₂²	—	496	4	no	—
121	C₂²	#11758561	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	62	4	no	—
124	C₄	C₂×D₄×D₃₁	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	124	4	no	—
128	C₂	#14521684	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521850

C₂

yes

maximal, complemented

#14521316

C₂

yes

maximal, complemented

#14521216

C₂

yes

maximal, complemented

#14521167

C₂

yes

maximal, complemented

C₃₁×#493

C₂

yes

maximal, characteristic, complemented

#11758561

C₂²

yes

complemented

C₂×D₈×C₃₁

C₂²

yes

complemented

C₃₁×#127

C₂²

yes

complemented

#11758509

C₂²

yes

complemented

#11758489

—

#11758488

—

D₈×D₃₁

—

#11758481

C₂²

yes

—

#11758478

C₂²

yes

complemented

C₄×C₈×C₃₁

C₂²

yes

characteristic, complemented

#11758421

C₂²

yes

complemented

#11758407

C₂²

yes

complemented

#11758393

C₂²

yes

complemented

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

C₂³

yes

—

C₂×C₂²⋊D₃₁

—

#92497

—

D₄×D₃₁

—

#92495

C₂³

yes

—

D₈×C₃₁

—

D₈×C₃₁

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

C₂³

yes

—

—

C₂³

yes

—

Dic₁₂₄

D₄

yes

complemented

D₂₄₈

D₄

yes

complemented

#92470

D₄

yes

complemented

#92469

—

C₈×D₃₁

—

C₂³×C₃₁

—

C₂²×D₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂²⋊D₃₁

—

C₂²⋊D₃₁

—

C₂×Dic₃₁

—

D₁₂₄

C₂×D₄

yes

—

C₄×D₃₁

—

Dic₆₂

C₂×D₄

yes

—

C₈×C₃₁

C₂×D₄

yes

—

C₈×C₃₁

C₂×D₄

yes

—

C₃₁⋊C₈

—

#3292

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

H(5,𝔽₂)

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂²×D₄

yes

—

Dic₃₁

—

#576

—

#493

D₃₁

yes

characteristic, complemented

#472

—

#453

—

#443

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

D₄²

yes

characteristic

D₃₁

—

—

—

—

—

—

C₂×D₈

—

C₂×D₈

C₂×D₃₁

yes

complemented

#131

—

#127

C₂×D₃₁

yes

—

—

—

C₄×C₈

C₂×D₃₁

yes

characteristic, complemented

C₃₁

#3292

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

124

—

C₂×D₄

C₂²×D₃₁

yes

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

C₂²×D₃₁

yes

—

101

C₄²

C₂²×D₃₁

yes

characteristic

102

C₂³

—

103

C₂³

—

124

—

104

Q₈

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

124

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

124

—

113

C₂×C₄

C₂³×D₃₁

yes

characteristic

114

C₈

—

115

C₈

D₄×D₃₁

yes

—

116

C₈

D₄×D₃₁

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

496

—

121

C₂²

#11758561

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₂×D₄×D₃₁

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

124

—

128

C₂

#14521684

yes

characteristic

	subgroup	G/H
center	C₂	#14521684
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#493	C₂
socle	C₂×C₃₁	D₄²
hypercenter	#493	D₃₁
hypocenter	C₃₁	#3292
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#14521684

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#493

C₂

socle

C₂×C₃₁

D₄²

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #14521850	C₂	semidirect
2. #14521316	C₂	semidirect
3. #14521216	C₂	semidirect
4. #14521167	C₂	semidirect
5. C₃₁×#493	C₂	semidirect
6. #11758561	C₂²	semidirect
7. C₂×D₈×C₃₁	C₂²	semidirect
8. C₃₁×#127	C₂²	semidirect
9. #11758509	C₂²	semidirect
14. #11758478	C₂²	semidirect
15. C₄×C₈×C₃₁	C₂²	semidirect
16. #11758421	C₂²	semidirect
17. #11758407	C₂²	semidirect
18. #11758393	C₂²	semidirect
32. Dic₁₂₄	D₄	semidirect
33. D₂₄₈	D₄	semidirect
34. #92470	D₄	semidirect
62. #493	D₃₁	semidirect
76. C₂×D₈	C₂×D₃₁	semidirect
78. #127	C₂×D₃₁	semidirect
82. C₄×C₈	C₂×D₃₁	semidirect
83. C₃₁	#3292	semidirect

H ≅ G/N

type

1. #14521850

C₂

semidirect

2. #14521316

C₂

semidirect

3. #14521216

C₂

semidirect

4. #14521167

C₂

semidirect

5. C₃₁×#493

C₂

semidirect

6. #11758561

C₂²

semidirect

7. C₂×D₈×C₃₁

C₂²

semidirect

8. C₃₁×#127

C₂²

semidirect

9. #11758509

C₂²

semidirect

14. #11758478

C₂²

semidirect

15. C₄×C₈×C₃₁

C₂²

semidirect

16. #11758421

C₂²

semidirect

17. #11758407

C₂²

semidirect

18. #11758393

C₂²

semidirect

32. Dic₁₂₄

D₄

semidirect

33. D₂₄₈

D₄

semidirect

34. #92470

D₄

semidirect

62. #493

D₃₁

semidirect

76. C₂×D₈

C₂×D₃₁

semidirect

78. #127

C₂×D₃₁

semidirect

82. C₄×C₈

C₂×D₃₁

semidirect

83. C₃₁

#3292

semidirect

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

subgroup	index
#14521850	2
#14521316	2
#14521216	2
#14521167	2
C₃₁×#493	2
#3292	31

subgroup

index

C₃₁×#493

Subgroups — #16857510

Conjugacy classes of subgroups

Complements

Maximal subgroups