Group #16845864 — subgroups

Subgroups — #16845864

Order 3968.

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

normal subgroups only

This group has 256 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	C₂×C₄⋊(C₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₃₁⋊C₈	C₂	1	4	yes	maximal
3	#14520895	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
6	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
8	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
9	C₃₁×C₄⋊C₈	—	2	2	no	—
10	C₃₁⋊#95	C₄	1	2	yes	—
11	C₃₁⋊#95	C₄	1	2	yes	—
12	Dic₃₁⋊C₈	—	2	8	no	—
13	C₄⋊(C₃₁⋊C₈)	—	2	2	no	—
14	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₁	C₂×C₄	1	4	yes	—
18	C₂×C₈×C₃₁	—	2	4	no	—
19	C₂×C₈×C₃₁	—	2	2	no	—
20	C₄²×C₃₁	D₄	1	2	yes	—
21	C₄²×C₃₁	Q₈	1	2	yes	—
22	C₄×Dic₃₁	D₄	1	4	yes	—
23	C₄×Dic₃₁	Q₈	1	4	yes	—
24	C₂×C₃₁⋊C₈	—	2	2	no	—
25	C₂×C₃₁⋊C₈	—	2	4	no	—
26	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₃₁	—	2	2	no	—
28	C₂×C₄×C₃₁	—	2	2	no	—
29	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
35	C₂×Dic₃₁	—	2	4	no	—
36	C₂×Dic₃₁	—	2	4	no	—
37	C₂×Dic₃₁	C₄⋊C₄	1	8	yes	—
38	C₈×C₃₁	—	4	2	no	—
39	C₃₁⋊C₈	—	4	2	no	—
40	#1843	—	31	1	no	maximal, Sylow
41	C₂²×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
42	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₃₁	#131	1	2	yes	—
44	C₂²×C₃₁	#131	1	2	yes	—
45	C₄×C₃₁	—	2	2	no	—
46	C₄×C₃₁	—	2	2	no	—
47	C₄×C₃₁	#127	1	1	yes	characteristic
48	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
49	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
50	Dic₃₁	—	2	8	no	—
51	C₂×C₄⋊C₈	—	31	4	no	—
52	C₂×C₄⋊C₈	—	31	1	no	—
53	C₂×C₄⋊C₈	D₃₁	1	1	yes	characteristic
54	#376	—	31	1	no	—
55	C₂×C₃₁	#384	1	1	yes	characteristic
56	C₂×C₃₁	#424	1	4	yes	—
57	C₂×C₃₁	#424	1	1	yes	characteristic
58	C₂×C₃₁	#424	1	1	yes	characteristic
59	C₂²×C₈	—	31	2	no	—
60	C₂²×C₈	C₂×D₃₁	1	2	yes	—
61	C₂×C₄²	—	31	2	no	—
62	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	62	8	no	—
65	C₄⋊C₈	—	62	2	no	—
66	#95	—	31	2	no	—
67	#95	—	31	2	no	—
68	C₃₁	#1843	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₄	—	31	4	no	—
70	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	62	4	no	—
76	C₂×C₈	—	62	2	no	—
77	C₄²	—	31	4	no	—
78	C₄²	—	31	4	no	—
79	C₄²	Dic₆₂	1	2	yes	—
80	C₄²	C₂²⋊D₃₁	1	2	yes	—
81	C₂³	C₂×C₄×D₃₁	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	31	8	no	—
85	C₂×C₄	—	62	4	no	—
86	C₂×C₄	—	62	4	no	—
87	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
88	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
89	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
90	C₂×C₄	D₄×D₃₁	1	2	yes	—
91	C₂×C₄	Q₈×D₃₁	1	2	yes	—
92	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	124	2	no	—
95	C₂²	C₄⋊C₄×D₃₁	1	2	yes	—
96	C₂²	#11758478	1	2	yes	—
97	C₂²	#11758484	1	2	yes	—
98	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	62	8	no	—
102	C₄	#11758465	1	2	yes	—
103	C₄	#11758517	1	1	yes	characteristic
104	C₄	#11758521	1	1	yes	characteristic
105	C₂	#14520918	1	1	yes	characteristic
106	C₂	#14521074	1	4	yes	—
107	C₂	#14521075	1	1	yes	characteristic
108	C₂	#14521077	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₃₁⋊C₈

C₂

yes

maximal

#14520895

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁×C₄⋊C₈

—

C₃₁⋊#95

C₄

yes

—

C₃₁⋊#95

C₄

yes

—

Dic₃₁⋊C₈

—

C₄⋊(C₃₁⋊C₈)

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₄×Dic₃₁

D₄

yes

—

C₄×Dic₃₁

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₄⋊C₄

yes

—

C₈×C₃₁

—

C₃₁⋊C₈

—

#1843

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#131

yes

—

C₂²×C₃₁

#131

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#127

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₃₁

yes

characteristic

#376

—

C₂×C₃₁

#384

yes

characteristic

C₂×C₃₁

#424

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

#95

—

C₃₁

#1843

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₄×D₃₁

yes

characteristic

C₂²×C₄

C₄×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

Dic₆₂

yes

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₃₁⋊C₄

yes

—

C₂×C₄

Dic₃₁⋊C₄

yes

—

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

D₄×D₃₁

yes

—

C₂×C₄

Q₈×D₃₁

yes

—

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

C₈

—

C₈

—

124

—

C₂²

C₄⋊C₄×D₃₁

yes

—

C₂²

#11758478

yes

—

C₂²

#11758484

yes

—

C₂²

C₂×Dic₃₁⋊C₄

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#11758465

yes

—

103

C₄

#11758517

yes

characteristic

104

C₄

#11758521

yes

characteristic

105

C₂

#14520918

yes

characteristic

106

C₂

#14521074

yes

—

107

C₂

#14521075

yes

characteristic

108

C₂

#14521077

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₁
commutator, socle, prime radical	C₂³×C₃₁	C₂²×C₄
Fitting	C₂×C₃₁×C₄⋊C₈	C₂
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₃₁
hypocenter	C₃₁	#1843
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator, socle, prime radical

C₂³×C₃₁

C₂²×C₄

Fitting

C₂×C₃₁×C₄⋊C₈

C₂

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
68. C₃₁	#1843	semidirect

H ≅ G/N

type

68. C₃₁

#1843

semidirect

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

subgroup	index
C₂×C₄⋊(C₃₁⋊C₈)	2
C₂×Dic₃₁⋊C₈	2
#14520895	2
C₂×C₃₁×C₄⋊C₈	2
#1843	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₂×Dic₃₁⋊C₈

#14520895

C₂×C₃₁×C₄⋊C₈

#1843

Subgroups — #16845864

Conjugacy classes of subgroups

Complements

Maximal subgroups