Group #16845229 — subgroups

Subgroups — #16845229

Order 3968.

152 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14521454	C₂	1	1	yes	maximal, characteristic
2	#14521259	C₂	1	2	yes	maximal
3	#14520800	C₂	1	1	yes	maximal, characteristic
4	#14520794	C₂	1	2	yes	maximal
5	C₃₁×#446	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×#92482	—	2	2	no	—
8	#11758446	C₂²	1	2	yes	—
9	C₃₁×C₄⋊C₈	—	2	1	no	—
10	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₃₁×#97	—	2	1	no	—
12	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	#11758406	—	2	1	no	—
14	#11758406	—	2	1	no	—
15	#11758398	—	2	2	no	—
16	C₄⋊(C₃₁⋊C₈)	C₂²	1	2	yes	—
17	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
18	C₂×Dic₆₂	—	2	2	no	—
19	C₂×C₈×C₃₁	—	2	1	no	—
20	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₃₁	—	2	1	no	—
23	C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₄²×C₃₁	—	2	1	no	—
25	C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	#92482	—	4	4	no	—
27	C₄⋊Dic₃₁	—	2	2	no	—
28	C₄⋊Dic₃₁	—	2	2	no	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	Q₈×C₃₁	—	2	2	no	—
31	C₂×C₄×C₃₁	—	2	1	no	—
32	C₂×C₄×C₃₁	—	2	1	no	—
33	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
36	C₂×Dic₃₁	—	4	2	no	—
37	Dic₆₂	—	4	2	no	—
38	Dic₆₂	—	4	2	no	—
39	C₈×C₃₁	—	2	1	no	—
40	C₈×C₃₁	—	4	1	no	—
41	C₈×C₃₁	Q₁₆	1	2	yes	complemented
42	C₃₁⋊C₈	—	4	2	no	—
43	#1694	—	31	1	no	maximal, Sylow
44	C₂²×C₃₁	#121	1	1	yes	characteristic
45	C₄×C₃₁	—	2	1	no	—
46	C₄×C₃₁	—	4	1	no	—
47	C₄×C₃₁	—	4	1	no	—
48	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
50	C₄×C₃₁	#137	1	1	yes	characteristic
51	Dic₃₁	—	8	2	no	—
52	#500	—	31	1	no	—
53	#462	—	31	2	no	—
54	#446	D₃₁	1	1	yes	characteristic
55	#334	—	31	1	no	—
56	#332	—	31	2	no	—
57	C₂×C₃₁	#462	1	1	yes	characteristic
58	C₂×C₃₁	#469	1	1	yes	characteristic
59	C₂×C₃₁	#473	1	1	yes	characteristic
60	C₂×Q₁₆	—	62	2	no	—
61	C₄⋊Q₈	—	31	2	no	—
62	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
63	#107	—	62	1	no	—
64	#107	—	62	1	no	—
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	31	2	no	—
67	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
68	#103	—	62	2	no	—
69	#97	—	2	1	no	—
70	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
71	C₃₁	#1694	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₈	—	62	2	no	—
73	C₂×Q₈	D₁₂₄	1	1	yes	characteristic
74	Q₁₆	—	124	4	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	62	2	no	—
77	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	62	2	no	—
81	C₄⋊C₄	—	62	2	no	—
82	C₄⋊C₄	D₁₂₄	1	1	yes	characteristic
83	C₄²	—	2	1	no	—
84	C₄²	C₂²×D₃₁	1	1	yes	characteristic
85	Q₈	—	2	2	no	—
86	Q₈	—	124	2	no	—
87	Q₈	—	124	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	124	2	no	—
91	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
92	C₂×C₄	#92495	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	1	no	—
96	C₈	—	124	2	no	—
97	C₈	#92482	1	2	yes	complemented
98	C₂²	#11758507	1	1	yes	characteristic
99	C₄	—	2	1	no	—
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	248	2	no	—
103	C₄	#11758485	1	2	yes	—
104	C₄	C₂×#92482	1	1	yes	characteristic
105	C₄	#11758528	1	1	yes	characteristic
106	C₂	#14521259	1	1	yes	characteristic
107	C₂	#14521302	1	1	yes	characteristic
108	C₂	#14521324	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521454

C₂

yes

maximal, characteristic

#14521259

C₂

yes

maximal

#14520800

C₂

yes

maximal, characteristic

#14520794

C₂

yes

maximal

C₃₁×#446

C₂

yes

maximal, characteristic

C₄×Q₈×C₃₁

C₂²

yes

characteristic

C₂×#92482

—

#11758446

C₂²

yes

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

C₂²

yes

characteristic

C₃₁×#97

—

C₄×C₈×C₃₁

C₂²

yes

—

—

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×Dic₆₂

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92482

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

Q₁₆

yes

complemented

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×Q₁₆

yes

characteristic

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₄×C₃₁

#137

yes

characteristic

Dic₃₁

—

#500

—

#462

—

#446

D₃₁

yes

characteristic

#334

—

#332

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₃₁

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

characteristic

#103

—

#97

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1694

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

D₁₂₄

yes

characteristic

Q₁₆

—

124

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₂₄

yes

characteristic

C₄²

—

C₄²

C₂²×D₃₁

yes

characteristic

Q₈

—

Q₈

—

124

—

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

#92495

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

C₈

—

C₈

—

C₈

—

124

—

C₈

yes

C₂²

yes

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

248

—

103

C₄

#11758485

yes

—

104

C₄

C₂×#92482

yes

characteristic

105

C₄

#11758528

yes

characteristic

106

C₂

#14521259

yes

characteristic

107

C₂

#14521302

yes

characteristic

108

C₂

#14521324

yes

characteristic

	subgroup	G/H
center	C₂²	#11758507
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#446	C₂
socle, prime radical	C₂²×C₃₁	#121
hypercenter	#446	D₃₁
hypocenter	C₃₁	#1694
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#11758507

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#446

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
41. C₈×C₃₁	Q₁₆	semidirect
71. C₃₁	#1694	semidirect
97. C₈	#92482	semidirect

H ≅ G/N

type

41. C₈×C₃₁

Q₁₆

semidirect

71. C₃₁

#1694

semidirect

97. C₈

#92482

semidirect

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

subgroup	index
#14521454	2
#14521259	2
#14520800	2
#14520794	2
C₃₁×#446	2
#1694	31

subgroup

index

C₃₁×#446

Subgroups — #16845229

Conjugacy classes of subgroups

Complements

Maximal subgroups