Group #16845106 — subgroups

Subgroups — #16845106

Order 3968.

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

normal subgroups only

This group has 156 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	#14521260	C₂	1	2	yes	maximal
3	C₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic
4	#14520796	C₂	1	2	yes	maximal
5	C₃₁×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₂₄	—	2	2	no	—
7	#11758446	C₂²	1	2	yes	—
8	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
9	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	#11758406	—	2	1	no	—
12	#11758406	—	2	1	no	—
13	#11758404	—	2	2	no	—
14	Dic₃₁⋊C₈	—	2	1	no	—
15	C₈×Dic₃₁	—	2	1	no	—
16	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×Dic₆₂	—	2	2	no	—
18	C₂×Dic₆₂	D₄	1	1	yes	characteristic
19	C₂×C₈×C₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₄²×C₃₁	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₃₁	—	2	2	no	—
24	C₄⋊Dic₃₁	—	2	2	no	—
25	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
26	Dic₃₁⋊C₄	—	2	1	no	—
27	C₄×Dic₃₁	—	2	1	no	—
28	C₂×C₃₁⋊C₈	—	2	1	no	—
29	Dic₁₂₄	—	4	4	no	—
30	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
33	C₂×Dic₃₁	—	2	1	no	—
34	C₂×Dic₃₁	—	2	1	no	—
35	C₂×Dic₃₁	—	4	2	no	—
36	Dic₆₂	—	2	1	no	—
37	Dic₆₂	—	4	2	no	—
38	Dic₆₂	—	4	2	no	—
39	Dic₆₂	D₈	1	2	yes	—
40	C₈×C₃₁	—	2	1	no	—
41	C₈×C₃₁	—	4	2	no	—
42	C₈×C₃₁	Q₁₆	1	2	yes	complemented
43	C₃₁⋊C₈	—	4	1	no	—
44	#1674	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	#121	1	1	yes	characteristic
46	C₄×C₃₁	—	2	1	no	—
47	C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
48	C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×C₃₁	#135	1	1	yes	characteristic
50	C₄×C₃₁	#137	1	1	yes	characteristic
51	Dic₃₁	—	2	2	no	—
52	Dic₃₁	—	4	1	no	—
53	Dic₃₁	—	8	2	no	—
54	#500	—	31	1	no	—
55	#462	—	31	2	no	—
56	C₈×Q₈	—	31	1	no	—
57	#335	D₃₁	1	1	yes	characteristic
58	#332	—	31	2	no	—
59	C₂×C₃₁	#462	1	1	yes	characteristic
60	C₂×C₃₁	#466	1	1	yes	characteristic
61	C₂×C₃₁	#473	1	1	yes	characteristic
62	C₂×Q₁₆	—	62	2	no	—
63	C₄⋊Q₈	—	31	2	no	—
64	C₄×Q₈	—	31	1	no	—
65	#107	—	62	1	no	—
66	#107	—	62	1	no	—
67	C₄⋊C₈	—	31	1	no	—
68	C₄⋊C₈	—	62	1	no	—
69	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
70	#103	—	62	2	no	—
71	C₄×C₈	—	62	1	no	—
72	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
73	C₃₁	#1674	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	31	1	no	—
75	C₂×Q₈	—	62	2	no	—
76	Q₁₆	—	124	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	62	1	no	—
79	C₂×C₈	D₁₂₄	1	1	yes	characteristic
80	C₂×C₈	D₁₂₄	1	1	yes	characteristic
81	C₄⋊C₄	—	31	1	no	—
82	C₄⋊C₄	—	62	1	no	—
83	C₄⋊C₄	—	62	2	no	—
84	C₄⋊C₄	—	62	2	no	—
85	C₄²	—	62	1	no	—
86	C₄²	C₂²×D₃₁	1	1	yes	characteristic
87	Q₈	—	31	2	no	—
88	Q₈	—	62	1	no	—
89	Q₈	—	124	2	no	—
90	Q₈	—	124	2	no	—
91	C₂×C₄	—	62	1	no	—
92	C₂×C₄	—	62	1	no	—
93	C₂×C₄	—	124	2	no	—
94	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
95	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
96	C₂×C₄	#92499	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	4	2	no	—
99	C₈	—	124	1	no	—
100	C₈	Dic₁₂₄	1	2	yes	complemented
101	C₂²	#11758472	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	62	2	no	—
104	C₄	—	124	1	no	—
105	C₄	—	248	2	no	—
106	C₄	C₂×Dic₁₂₄	1	1	yes	characteristic
107	C₄	#11758486	1	1	yes	characteristic
108	C₄	D₈×D₃₁	1	1	yes	characteristic
109	C₄	#11758496	1	1	yes	characteristic
110	C₂	#14521260	1	1	yes	characteristic
111	C₂	#14521286	1	1	yes	characteristic
112	C₂	#14521325	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521454

C₂

yes

maximal, characteristic

#14521260

C₂

yes

maximal

C₈×Dic₆₂

C₂

yes

maximal, characteristic

#14520796

C₂

yes

maximal

C₃₁×#335

C₂

yes

maximal, characteristic

C₂×Dic₁₂₄

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

C₂²

yes

characteristic

C₃₁×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

—

—

Dic₃₁⋊C₈

—

C₈×Dic₃₁

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

characteristic

C₂×Dic₆₂

—

C₂×Dic₆₂

D₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

D₄

yes

characteristic

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

—

Dic₁₂₄

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

D₈

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

Q₁₆

yes

complemented

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

C₂×D₈

yes

characteristic

C₄×C₃₁

C₂×Q₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

#500

—

#462

—

C₈×Q₈

—

#335

D₃₁

yes

characteristic

#332

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

—

#103

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1674

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

124

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂₄

yes

characteristic

C₂×C₈

D₁₂₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₃₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

124

—

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

D₄×D₃₁

yes

characteristic

C₂×C₄

#92499

yes

characteristic

C₈

—

C₈

—

C₈

—

124

—

100

C₈

Dic₁₂₄

yes

complemented

101

C₂²

#11758472

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

124

—

105

C₄

—

248

—

106

C₄

C₂×Dic₁₂₄

yes

characteristic

107

C₄

#11758486

yes

characteristic

108

C₄

D₈×D₃₁

yes

characteristic

109

C₄

#11758496

yes

characteristic

110

C₂

#14521260

yes

characteristic

111

C₂

#14521286

yes

characteristic

112

C₂

#14521325

yes

characteristic

	subgroup	G/H
center	C₂²	#11758472
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#335	C₂
socle, prime radical	C₂²×C₃₁	#121
hypercenter	#335	D₃₁
hypocenter	C₃₁	#1674
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#11758472

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#335

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
42. C₈×C₃₁	Q₁₆	semidirect
73. C₃₁	#1674	semidirect
100. C₈	Dic₁₂₄	semidirect

H ≅ G/N

type

42. C₈×C₃₁

Q₁₆

semidirect

73. C₃₁

#1674

semidirect

100. C₈

Dic₁₂₄

semidirect

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

subgroup	index
#14521454	2
#14521260	2
C₈×Dic₆₂	2
#14520796	2
C₃₁×#335	2
#1674	31

subgroup

index

C₈×Dic₆₂

C₃₁×#335

Subgroups — #16845106

Conjugacy classes of subgroups

Complements

Maximal subgroups