Group #16848297 — subgroups

Subgroups — #16848297

Order 3968.

784 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 309 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×#11758503	C₂	1	2	yes	maximal
2	#14520956	C₂	1	8	yes	maximal, complemented
3	#14520912	C₂	1	4	yes	maximal
4	C₂³×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₁	—	2	8	no	—
6	C₂²×Q₈×C₃₁	C₄	1	8	yes	complemented
7	C₂²×Q₈×C₃₁	C₂²	1	4	yes	—
8	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
9	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	—
11	#11758503	—	2	8	no	—
12	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
13	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
14	C₃₁⋊#95	C₂²	1	16	yes	—
15	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
16	C₂×Q₈×C₃₁	—	2	32	no	—
17	C₂×Q₈×C₃₁	—	2	8	no	—
18	C₂×Q₈×C₃₁	—	2	8	no	—
19	C₂×Q₈×C₃₁	C₂×C₄	1	16	yes	—
20	C₂²×C₄×C₃₁	—	2	8	no	—
21	C₂²×C₄×C₃₁	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₃₁	D₄	1	8	yes	—
24	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
25	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
26	C₂²×Dic₃₁	C₂³	1	8	yes	—
27	C₂²⋊Dic₃₁	—	2	8	no	—
28	C₄⋊Dic₃₁	—	2	8	no	—
29	C₄×Dic₃₁	—	2	8	no	—
30	C₂³×C₃₁	—	2	4	no	—
31	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
32	C₂³×C₃₁	C₂×D₄	1	4	yes	—
33	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
34	Q₈×C₃₁	—	2	32	no	—
35	C₂×C₄×C₃₁	—	2	16	no	—
36	C₂×C₄×C₃₁	—	2	8	no	—
37	C₂×C₄×C₃₁	C₂²⋊C₄	1	16	yes	—
38	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
42	C₂×Dic₃₁	—	2	8	no	—
43	C₂×Dic₃₁	—	2	16	no	—
44	#2310	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	4	no	—
46	C₂²×C₃₁	—	2	8	no	—
47	C₂²×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
48	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
49	C₂²×C₃₁	C₂²×D₄	1	2	yes	—
50	C₂²×C₃₁	2₋¹⁺⁴	1	4	yes	—
51	C₄×C₃₁	—	2	8	no	—
52	C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
53	Dic₃₁	—	4	8	no	—
54	C₂³×Q₈	D₃₁	1	1	yes	characteristic
55	C₂×#117	—	31	2	no	—
56	#391	—	31	8	no	—
57	#381	—	31	4	no	—
58	C₂×C₃₁	—	2	4	no	—
59	C₂×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₃₁	#519	1	2	yes	—
61	C₂×C₃₁	#536	1	4	yes	—
62	C₂²×Q₈	—	2	8	no	—
63	C₂²×Q₈	Dic₃₁	1	8	yes	complemented
64	C₂²×Q₈	C₂×D₃₁	1	4	yes	—
65	C₂³×C₄	C₂×D₃₁	1	2	yes	—
66	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
67	#117	—	62	8	no	—
68	C₂×C₄⋊C₄	—	31	4	no	—
69	C₂×C₂²⋊C₄	—	31	4	no	—
70	C₂×C₄²	—	31	4	no	—
71	#95	—	31	16	no	—
72	C₃₁	#2310	1	1	yes	characteristic, complemented, Sylow
73	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	8	no	—
77	C₂×Q₈	C₂×Dic₃₁	1	16	yes	—
78	C₂²×C₄	—	2	8	no	—
79	C₂²×C₄	—	31	8	no	—
80	C₂²×C₄	C₂×Dic₃₁	1	8	yes	—
81	C₂²×C₄	C₂×Dic₃₁	1	4	yes	—
82	C₂²×C₄	C₂²⋊D₃₁	1	8	yes	—
83	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
84	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
85	C₄⋊C₄	—	62	8	no	—
86	C₂²⋊C₄	—	62	8	no	—
87	C₄²	—	62	8	no	—
88	C₂³	—	2	4	no	—
89	C₂³	C₂²×Dic₃₁	1	2	yes	—
90	C₂³	C₂×C₂²⋊D₃₁	1	4	yes	—
91	C₂³	C₂³×D₃₁	1	1	yes	characteristic
92	Q₈	—	2	32	no	—
93	C₂×C₄	—	2	16	no	—
94	C₂×C₄	—	2	8	no	—
95	C₂×C₄	—	62	16	no	—
96	C₂×C₄	—	62	8	no	—
97	C₂×C₄	C₂²⋊Dic₃₁	1	16	yes	—
98	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
99	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₃₁	1	4	yes	—
102	C₂²	—	2	8	no	—
103	C₂²	—	2	4	no	—
104	C₂²	C₂×C₂²⋊Dic₃₁	1	4	yes	—
105	C₂²	#11758564	1	4	yes	—
106	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
107	C₂²	C₂²×C₂²⋊D₃₁	1	2	yes	—
108	C₄	—	2	8	no	—
109	C₄	—	124	8	no	—
110	C₄	C₂×C₂²⋊Dic₃₁	1	8	yes	—
111	C₂	—	2	4	no	—
112	C₂	C₂²×C₂²⋊Dic₃₁	1	1	yes	characteristic
113	C₂	#14521527	1	2	yes	—
114	C₂	#14521623	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758503

C₂

yes

maximal

#14520956

C₂

yes

maximal, complemented

#14520912

C₂

yes

maximal

C₂³×Q₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

C₄

yes

complemented

C₂²×Q₈×C₃₁

C₂²

yes

—

C₂³×C₄×C₃₁

C₂²

yes

—

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊Dic₃₁

C₂²

yes

—

#11758503

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂⁴×C₃₁

C₂³

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₂²×C₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

—

C₂³×C₃₁

C₂⁴

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

#2310

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₃₁

C₂³×C₄

yes

characteristic

C₂²×C₃₁

C₂²×D₄

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

—

Dic₃₁

—

C₂³×Q₈

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₃₁

#519

yes

—

C₂×C₃₁

#536

yes

—

C₂²×Q₈

—

C₂²×Q₈

Dic₃₁

yes

complemented

C₂²×Q₈

C₂×D₃₁

yes

—

C₂³×C₄

C₂×D₃₁

yes

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

C₃₁

#2310

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×D₃₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×Dic₃₁

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

—

C₂³

C₂³×D₃₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₃₁

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

—

100

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₂×C₂²⋊Dic₃₁

yes

—

105

C₂²

#11758564

yes

—

106

C₂²

C₂³×Dic₃₁

yes

characteristic

107

C₂²

C₂²×C₂²⋊D₃₁

yes

—

108

C₄

—

109

C₄

—

124

—

110

C₄

C₂×C₂²⋊Dic₃₁

yes

—

111

C₂

—

112

C₂

C₂²×C₂²⋊Dic₃₁

yes

characteristic

113

C₂

#14521527

yes

—

114

C₂

#14521623

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂³×Q₈×C₃₁	C₂
socle, cosocle	C₂³×C₃₁	C₂⁴
prime radical	C₂⁴×C₃₁	C₂³
hypercenter	C₂³×Q₈	D₃₁
hypocenter	C₃₁	#2310
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator

C₂²×C₃₁

C₂³×C₄

Fitting

C₂³×Q₈×C₃₁

C₂

socle, cosocle

C₂³×C₃₁

C₂⁴

prime radical

C₂⁴×C₃₁

C₂³

hypercenter

C₂³×Q₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #14520956	C₂	semidirect
6. C₂²×Q₈×C₃₁	C₄	semidirect
63. C₂²×Q₈	Dic₃₁	semidirect
72. C₃₁	#2310	semidirect

H ≅ G/N

type

2. #14520956

C₂

semidirect

6. C₂²×Q₈×C₃₁

C₄

semidirect

63. C₂²×Q₈

Dic₃₁

semidirect

72. C₃₁

#2310

semidirect

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

subgroup	index
C₂×#11758503	2
#14520956	2
#14520912	2
C₂³×Q₈×C₃₁	2
#2310	31

subgroup

index

C₂³×Q₈×C₃₁

Subgroups — #16848297

Conjugacy classes of subgroups

Complements

Maximal subgroups