Group #16844500 — subgroups

Subgroups — #16844500

Order 3968.

158 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14521102	C₂	1	1	yes	maximal, characteristic
2	#14521073	C₂	1	1	yes	maximal, characteristic
3	#14520802	C₂	1	2	yes	maximal, complemented
4	#14520800	C₂	1	2	yes	maximal, complemented
5	C₃₁×#431	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×#92474	—	2	1	no	—
8	C₃₁×C₄⋊C₈	C₄	1	2	yes	—
9	C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
10	C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂²⋊(C₃₁⋊C₈)	—	2	1	no	—
12	C₄⋊(C₃₁⋊C₈)	—	2	1	no	—
13	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
16	C₄×C₃₁⋊C₈	—	2	1	no	—
17	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	Q₈	1	1	yes	characteristic
20	C₂×C₈×C₃₁	C₂×C₄	1	4	yes	—
21	C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₃₁	D₄	1	1	yes	characteristic
23	C₄²×C₃₁	Q₈	1	1	yes	characteristic
24	C₄²×C₃₁	C₂³	1	1	yes	characteristic
25	#92474	—	4	2	no	—
26	C₂×C₃₁⋊C₈	—	2	1	no	—
27	C₂×C₃₁⋊C₈	—	2	1	no	—
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂³×C₃₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₃₁	—	2	2	no	—
31	C₂×C₄×C₃₁	—	2	2	no	—
32	C₂×C₄×C₃₁	—	2	1	no	—
33	C₂×C₄×C₃₁	—	2	1	no	—
34	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₃₁	—	2	4	no	—
41	C₃₁⋊C₈	—	4	1	no	—
42	C₃₁⋊C₈	—	4	2	no	—
43	C₃₁⋊C₈	—	4	1	no	—
44	#1574	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	1	no	—
46	C₂²×C₃₁	—	2	1	no	—
47	C₂²×C₃₁	—	2	1	no	—
48	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₃₁	—	2	1	no	—
50	C₄×C₃₁	—	2	1	no	—
51	C₄×C₃₁	—	2	1	no	—
52	C₄×C₃₁	#108	1	2	yes	—
53	C₄×C₃₁	#131	1	2	yes	—
54	C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₄×C₃₁	#137	1	1	yes	characteristic
56	#431	—	31	1	no	—
57	#431	D₃₁	1	1	yes	characteristic
58	#423	—	31	1	no	—
59	#335	—	31	2	no	—
60	#334	—	31	2	no	—
61	C₂×C₃₁	—	4	1	no	—
62	C₂×C₃₁	#424	1	1	yes	characteristic
63	C₂×C₃₁	#428	1	1	yes	characteristic
64	C₂×C₃₁	C₂×#108	1	1	yes	characteristic
65	C₂×M₁₆	—	62	1	no	—
66	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
67	C₄⋊C₈	—	31	1	no	—
68	C₄⋊C₈	—	31	2	no	—
69	C₄⋊C₈	—	31	1	no	—
70	C₄⋊C₈	—	62	1	no	—
71	C₄⋊C₈	Dic₃₁	1	2	yes	—
72	C₂²⋊C₈	—	62	1	no	—
73	C₂²⋊C₈	Dic₃₁	1	2	yes	—
74	C₄×C₈	—	62	1	no	—
75	C₄×C₈	C₂×D₃₁	1	2	yes	—
76	C₃₁	#1574	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₄	Dic₆₂	1	1	yes	characteristic
78	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₃₁	1	1	yes	characteristic
80	M₁₆	—	124	2	no	—
81	C₂×C₈	—	62	1	no	—
82	C₂×C₈	—	62	1	no	—
83	C₂×C₈	—	62	2	no	—
84	C₂×C₈	C₂×Dic₃₁	1	4	yes	—
85	C₄²	Dic₆₂	1	1	yes	characteristic
86	C₄²	D₁₂₄	1	1	yes	characteristic
87	C₄²	C₂×Dic₃₁	1	1	yes	characteristic
88	C₄²	C₂²×D₃₁	1	1	yes	characteristic
89	C₂³	C₄⋊Dic₃₁	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₃₁	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
98	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
99	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
100	C₈	—	2	4	no	—
101	C₈	—	124	1	no	—
102	C₈	—	124	2	no	—
103	C₈	—	124	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	#11758407	1	2	yes	—
112	C₄	#11758485	1	1	yes	characteristic
113	C₄	#11758486	1	1	yes	characteristic
114	C₄	#11758525	1	2	yes	—
115	C₂	—	4	1	no	—
116	C₂	#14521076	1	1	yes	characteristic
117	C₂	#14521091	1	1	yes	characteristic
118	C₂	C₂×#11758407	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521102

C₂

yes

maximal, characteristic

#14521073

C₂

yes

maximal, characteristic

#14520802

C₂

yes

maximal, complemented

#14520800

C₂

yes

maximal, complemented

C₃₁×#431

C₂

yes

maximal, characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×#92474

—

C₃₁×C₄⋊C₈

C₄

yes

—

C₃₁×C₂²⋊C₈

C₄

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

C₂²⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄×C₃₁⋊C₈

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

Q₈

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

Q₈

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1574

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#108

yes

—

C₄×C₃₁

#131

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

—

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×M₁₆

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₃₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₃₁

yes

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

—

C₃₁

#1574

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₆₂

yes

characteristic

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

characteristic

M₁₆

—

124

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄²

Dic₆₂

yes

characteristic

C₄²

D₁₂₄

yes

characteristic

C₄²

C₂×Dic₃₁

yes

characteristic

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

C₄⋊Dic₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

C₂×C₄

C₄⋊Dic₃₁

yes

characteristic

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

100

C₈

—

101

C₈

—

124

—

102

C₈

—

124

—

103

C₈

—

124

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#11758407

yes

—

112

C₄

#11758485

yes

characteristic

113

C₄

#11758486

yes

characteristic

114

C₄

#11758525

yes

—

115

C₂

—

116

C₂

#14521076

yes

characteristic

117

C₂

#14521091

yes

characteristic

118

C₂

C₂×#11758407

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#431	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁	C₄⋊C₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	#431	D₃₁
hypocenter	C₃₁	#1574
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#431

C₂

socle

C₂²×C₃₁

C₂×C₄⋊C₄

prime radical

C₂³×C₃₁

C₄⋊C₄

cosocle

C₄²×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #14520802	C₂	semidirect
4. #14520800	C₂	semidirect
76. C₃₁	#1574	semidirect

H ≅ G/N

type

3. #14520802

C₂

semidirect

4. #14520800

C₂

semidirect

76. C₃₁

#1574

semidirect

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

subgroup	index
#14521102	2
#14521073	2
#14520802	2
#14520800	2
C₃₁×#431	2
#1574	31

subgroup

index

C₃₁×#431

Subgroups — #16844500

Conjugacy classes of subgroups

Complements

Maximal subgroups