Group #16845568 — subgroups

Subgroups — #16845568

Order 3968.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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	2	yes	maximal
2	C₂×#11758391	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#14520804	C₂	1	2	yes	maximal
5	C₃₁×#376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
10	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
11	C₃₁×#95	C₄	1	4	yes	—
12	C₄⋊(C₃₁⋊C₈)	C₄	1	8	yes	complemented
13	#11758391	—	2	2	no	—
14	C₄×C₃₁⋊C₈	—	2	2	no	—
15	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₄²×C₃₁	—	2	2	no	—
20	C₄²×C₃₁	C₂×C₄	1	4	yes	—
21	C₄²×C₃₁	D₄	1	2	yes	—
22	C₄²×C₃₁	Q₈	1	2	yes	—
23	C₂×C₃₁⋊C₈	—	2	2	no	—
24	C₂×C₃₁⋊C₈	—	2	2	no	—
25	C₂×C₃₁⋊C₈	—	2	2	no	—
26	C₂×C₃₁⋊C₈	—	2	2	no	—
27	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
28	C₂×C₃₁⋊C₈	C₂×C₄	1	4	yes	—
29	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₃₁	—	2	4	no	—
31	C₂×C₄×C₃₁	—	2	4	no	—
32	C₂×C₄×C₃₁	—	2	2	no	—
33	C₂×C₄×C₃₁	—	2	4	no	—
34	C₂×C₄×C₃₁	C₄²	1	4	yes	—
35	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₃₁	#41	1	2	yes	—
40	C₃₁⋊C₈	—	2	4	no	—
41	C₃₁⋊C₈	—	2	4	no	—
42	#1768	—	31	1	no	maximal, Sylow
43	C₂²×C₃₁	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₃₁	#117	1	1	yes	characteristic
46	C₂²×C₃₁	#131	1	4	yes	—
47	C₄×C₃₁	—	2	4	no	—
48	C₄×C₃₁	—	2	4	no	—
49	C₄×C₃₁	—	4	2	no	—
50	C₄×C₃₁	C₄×D₄	1	2	yes	—
51	C₄×C₃₁	C₄×Q₈	1	2	yes	—
52	C₂×C₄⋊C₈	—	31	2	no	—
53	C₂×#97	—	31	1	no	—
54	C₂×C₄×C₈	—	31	1	no	—
55	#376	D₃₁	1	1	yes	characteristic
56	#336	—	31	2	no	—
57	C₂×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃₁	#405	1	2	yes	—
59	C₂×C₃₁	#424	1	2	yes	—
60	C₂×C₃₁	#433	1	2	yes	—
61	C₂²×C₈	—	31	2	no	—
62	C₂²×C₈	—	31	2	no	—
63	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
64	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
66	C₄⋊C₈	—	31	8	no	—
67	#97	—	62	2	no	—
68	C₄×C₈	—	62	2	no	—
69	#95	Dic₃₁	1	4	yes	—
70	C₃₁	#1768	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
72	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
73	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
75	C₂×C₈	—	31	4	no	—
76	C₂×C₈	—	31	4	no	—
77	C₂×C₈	—	62	2	no	—
78	C₂×C₈	—	62	2	no	—
79	C₂×C₈	—	62	2	no	—
80	C₂×C₈	—	62	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	Dic₆₂	1	2	yes	—
83	C₄²	C₄×D₃₁	1	4	yes	—
84	C₄²	D₁₂₄	1	2	yes	—
85	C₂³	C₂²×Dic₃₁	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₃₁	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
92	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
94	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
95	C₂×C₄	#92495	1	2	yes	—
96	C₈	—	62	4	no	—
97	C₈	—	62	4	no	—
98	C₂²	C₂×C₄×Dic₃₁	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
100	C₂²	#11758503	1	1	yes	characteristic
101	C₂²	#11758525	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×Dic₆₂	1	2	yes	—
106	C₄	C₄×D₁₂₄	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₃₁	1	1	yes	characteristic
108	C₂	#14521005	1	2	yes	—
109	C₂	#14521076	1	2	yes	—
110	C₂	#14521110	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₃₁⋊C₈)

C₂

yes

maximal

C₂×#11758391

C₂

yes

maximal, characteristic

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

#14520804

C₂

yes

maximal

C₃₁×#376

C₂

yes

maximal, characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

C₃₁×#95

C₄

yes

—

C₄⋊(C₃₁⋊C₈)

C₄

yes

complemented

#11758391

—

C₄×C₃₁⋊C₈

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂³×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄²

yes

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1768

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄²

yes

characteristic

C₂²×C₃₁

C₂×C₄⋊C₄

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

C₄×Q₈

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

D₃₁

yes

characteristic

#336

—

C₂×C₃₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₃₁

#405

yes

—

C₂×C₃₁

#424

yes

—

C₂×C₃₁

#433

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₂×C₄²

C₂×D₃₁

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

#95

Dic₃₁

yes

—

C₃₁

#1768

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

Dic₆₂

yes

—

C₄²

C₄×D₃₁

yes

—

C₄²

D₁₂₄

yes

—

C₂³

C₂²×Dic₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₃₁

yes

—

C₂×C₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

C₂×C₄×D₃₁

yes

—

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

#92495

yes

—

C₈

—

C₈

—

C₂²

C₂×C₄×Dic₃₁

yes

characteristic

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

100

C₂²

#11758503

yes

characteristic

101

C₂²

#11758525

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×Dic₆₂

yes

—

106

C₄

C₄×D₁₂₄

yes

—

107

C₂

C₄×C₄⋊Dic₃₁

yes

characteristic

108

C₂

#14521005

yes

—

109

C₂

#14521076

yes

—

110

C₂

#14521110

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₁
commutator	C₂²×C₃₁	C₂×C₄²
Fitting	C₃₁×#376	C₂
socle, prime radical	C₂³×C₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#376	D₃₁
hypocenter	C₃₁	#1768
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₁

commutator

C₂²×C₃₁

C₂×C₄²

Fitting

C₃₁×#376

C₂

socle, prime radical

C₂³×C₃₁

C₂²×C₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
12. C₄⋊(C₃₁⋊C₈)	C₄	semidirect
70. C₃₁	#1768	semidirect

H ≅ G/N

type

12. C₄⋊(C₃₁⋊C₈)

C₄

semidirect

70. C₃₁

#1768

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×#11758391	2
C₂×C₄×C₃₁⋊C₈	2
#14520804	2
C₃₁×#376	2
#1768	31

subgroup

index

C₂×C₄⋊(C₃₁⋊C₈)

C₂×#11758391

C₂×C₄×C₃₁⋊C₈

#14520804

C₃₁×#376

#1768

Subgroups — #16845568

Conjugacy classes of subgroups

Complements

Maximal subgroups