Group #16847870 — subgroups

Subgroups — #16847870

Order 3968.

136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 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	#14521450	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic
3	#14520877	C₂	1	1	yes	maximal, characteristic, complemented
4	#14520874	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#14520850	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#357	C₂	1	1	yes	maximal, characteristic
8	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
9	D₄×Dic₃₁	—	2	1	no	—
10	#11758467	—	2	1	no	—
11	C₂×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#107	C₂²	1	1	yes	characteristic
13	#11758420	—	2	1	no	—
14	#11758406	C₂²	1	1	yes	characteristic
15	#11758406	C₂²	1	1	yes	characteristic
16	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
18	#11758396	—	2	1	no	—
19	C₂×D₄×C₃₁	—	2	1	no	—
20	C₂²×Dic₃₁	—	4	1	no	—
21	D₈×C₃₁	—	2	1	no	—
22	D₈×C₃₁	Q₈	1	2	yes	—
23	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₃₁	—	2	1	no	—
25	C₂²⋊Dic₃₁	—	4	1	no	—
26	C₄⋊Dic₃₁	—	2	1	no	—
27	C₄⋊Dic₃₁	—	2	1	no	—
28	C₄⋊Dic₃₁	—	4	1	no	—
29	Dic₃₁⋊C₄	—	4	1	no	—
30	C₄×Dic₃₁	D₄	1	1	yes	characteristic
31	C₂×C₃₁⋊C₈	D₄	1	1	yes	characteristic
32	C₁₆×C₃₁	—	2	1	no	—
33	C₃₁⋊C₁₆	—	2	1	no	—
34	C₂³×C₃₁	—	4	1	no	—
35	D₄×C₃₁	—	2	2	no	—
36	D₄×C₃₁	—	4	1	no	—
37	C₂×C₄×C₃₁	—	4	1	no	—
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₃₁	—	4	1	no	—
40	C₂×Dic₃₁	—	4	1	no	—
41	C₂×Dic₃₁	—	8	2	no	—
42	C₂×Dic₃₁	D₈	1	1	yes	characteristic
43	C₂×Dic₃₁	D₈	1	1	yes	characteristic
44	C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₃₁	#41	1	1	yes	characteristic
46	C₃₁⋊C₈	—	2	1	no	—
47	#2228	—	31	1	no	maximal, Sylow
48	C₂²×C₃₁	—	4	2	no	—
49	C₂²×C₃₁	—	8	1	no	—
50	C₂²×C₃₁	—	8	1	no	—
51	C₂²×C₃₁	C₂×D₈	1	1	yes	characteristic
52	C₄×C₃₁	—	8	1	no	—
53	C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₃₁	#137	1	1	yes	characteristic
55	Dic₃₁	—	2	1	no	—
56	Dic₃₁	—	2	1	no	—
57	Dic₃₁	—	8	1	no	—
58	Dic₃₁	—	8	1	no	—
59	#499	—	31	1	no	—
60	C₄×D₈	—	31	1	no	—
61	#367	—	31	1	no	—
62	#366	—	31	1	no	—
63	C₄⋊C₁₆	—	31	1	no	—
64	#357	—	31	1	no	—
65	#357	D₃₁	1	1	yes	characteristic
66	C₂×C₃₁	—	8	2	no	—
67	C₂×C₃₁	#474	1	1	yes	characteristic
68	C₂×C₃₁	#508	1	1	yes	characteristic
69	C₂×C₃₁	#509	1	1	yes	characteristic
70	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
71	#125	—	62	1	no	—
72	C₄×D₄	—	62	1	no	—
73	C₂×C₁₆	—	31	1	no	—
74	C₂×C₁₆	C₂×D₃₁	1	1	yes	characteristic
75	#107	—	31	1	no	—
76	#107	—	31	1	no	—
77	#107	C₂×D₃₁	1	1	yes	characteristic
78	#106	—	62	1	no	—
79	#102	—	62	1	no	—
80	C₄×C₈	—	31	1	no	—
81	C₃₁	#2228	1	1	yes	characteristic, complemented, Sylow
82	C₂×D₄	—	2	1	no	—
83	C₂²×C₄	—	124	1	no	—
84	D₈	—	2	1	no	—
85	D₈	Dic₆₂	1	2	yes	—
86	C₂×C₈	—	31	1	no	—
87	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	62	1	no	—
90	C₄⋊C₄	—	62	1	no	—
91	C₄⋊C₄	—	124	1	no	—
92	C₄⋊C₄	—	124	1	no	—
93	C₂²⋊C₄	—	124	1	no	—
94	C₄²	—	31	1	no	—
95	C₁₆	—	2	1	no	—
96	C₁₆	—	62	1	no	—
97	C₂³	—	4	1	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	31	1	no	—
102	C₂×C₄	—	31	1	no	—
103	C₂×C₄	—	124	1	no	—
104	C₂×C₄	—	124	1	no	—
105	C₂×C₄	—	248	2	no	—
106	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
107	C₈	—	62	1	no	—
108	C₈	C₂×Dic₆₂	1	1	yes	characteristic
109	C₈	#92497	1	1	yes	characteristic
110	C₂²	—	4	2	no	—
111	C₂²	—	8	1	no	—
112	C₂²	—	8	1	no	—
113	C₂²	D₈×D₃₁	1	1	yes	characteristic
114	C₄	—	8	1	no	—
115	C₄	—	62	1	no	—
116	C₄	—	62	1	no	—
117	C₄	—	248	1	no	—
118	C₄	—	248	1	no	—
119	C₄	#11758455	1	1	yes	characteristic
120	C₄	#11758492	1	1	yes	characteristic
121	C₂	—	8	2	no	—
122	C₂	#14521327	1	1	yes	characteristic
123	C₂	#14521480	1	1	yes	characteristic
124	C₂	#14521483	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521450

C₂

yes

maximal, characteristic, complemented

D₈×Dic₃₁

C₂

yes

maximal, characteristic

#14520877

C₂

yes

maximal, characteristic, complemented

#14520874

C₂

yes

maximal, characteristic, complemented

Dic₃₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#14520850

C₂

yes

maximal, characteristic

C₃₁×#357

C₂

yes

maximal, characteristic

C₂×D₈×C₃₁

C₂²

yes

characteristic

D₄×Dic₃₁

—

#11758467

—

C₂×C₁₆×C₃₁

C₂²

yes

characteristic

C₃₁×#107

C₂²

yes

characteristic

#11758420

—

#11758406

C₂²

yes

characteristic

#11758406

C₂²

yes

characteristic

C₈×Dic₃₁

C₂²

yes

characteristic

C₂×C₃₁⋊C₁₆

C₂²

yes

characteristic

#11758396

—

C₂×D₄×C₃₁

—

C₂²×Dic₃₁

—

D₈×C₃₁

—

D₈×C₃₁

Q₈

yes

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

D₄

yes

characteristic

C₂×C₃₁⋊C₈

D₄

yes

characteristic

C₁₆×C₃₁

—

C₃₁⋊C₁₆

—

C₂³×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

D₈

yes

characteristic

C₂×Dic₃₁

D₈

yes

characteristic

C₈×C₃₁

C₂×Q₈

yes

characteristic

C₈×C₃₁

#41

yes

characteristic

C₃₁⋊C₈

—

#2228

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×D₈

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂²⋊Q₈

yes

characteristic

C₄×C₃₁

#137

yes

characteristic

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

#499

—

C₄×D₈

—

#367

—

#366

—

C₄⋊C₁₆

—

#357

—

#357

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×D₈

C₂×D₃₁

yes

characteristic

#125

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₁

yes

—

—

C₂×D₃₁

yes

characteristic

#106

—

#102

—

C₄×C₈

—

C₃₁

#2228

yes

characteristic, complemented, Sylow

C₂×D₄

—

C₂²×C₄

—

124

—

D₈

—

D₈

Dic₆₂

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₂²⋊C₄

—

124

—

C₄²

—

C₁₆

—

C₁₆

—

C₂³

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

124

—

104

C₂×C₄

—

124

—

105

C₂×C₄

—

248

—

106

C₂×C₄

D₄×D₃₁

yes

characteristic

107

C₈

—

108

C₈

C₂×Dic₆₂

yes

characteristic

109

C₈

#92497

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

D₈×D₃₁

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

248

—

118

C₄

—

248

—

119

C₄

#11758455

yes

characteristic

120

C₄

#11758492

yes

characteristic

121

C₂

—

122

C₂

#14521327

yes

characteristic

123

C₂

#14521480

yes

characteristic

124

C₂

#14521483

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₃₁
commutator, cosocle	C₂×C₈×C₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₃₁
Fitting	C₃₁×#357	C₂
socle	C₂²×C₃₁	C₂×D₈
prime radical	C₂×D₈×C₃₁	C₂²
hypercenter	#357	D₃₁
hypocenter	C₃₁	#2228
LCS	G ▷ C₂×C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₃₁

commutator, cosocle

C₂×C₈×C₃₁

C₂³

Frattini

C₂×C₈

C₂²×D₃₁

Fitting

C₃₁×#357

C₂

socle

C₂²×C₃₁

C₂×D₈

prime radical

C₂×D₈×C₃₁

C₂²

D₃₁

C₃₁

G ▷ C₂×C₈×C₃₁ ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #14521450	C₂	semidirect
3. #14520877	C₂	semidirect
4. #14520874	C₂	semidirect
5. Dic₃₁⋊C₁₆	C₂	semidirect
81. C₃₁	#2228	semidirect

H ≅ G/N

type

1. #14521450

C₂

semidirect

3. #14520877

C₂

semidirect

4. #14520874

C₂

semidirect

5. Dic₃₁⋊C₁₆

C₂

semidirect

81. C₃₁

#2228

semidirect

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

subgroup	index
#14521450	2
D₈×Dic₃₁	2
#14520877	2
#14520874	2
Dic₃₁⋊C₁₆	2
#14520850	2
C₃₁×#357	2
#2228	31

subgroup

index

D₈×Dic₃₁

Dic₃₁⋊C₁₆

C₃₁×#357

Subgroups — #16847870

Conjugacy classes of subgroups

Complements

Maximal subgroups