Group #30665331 — subgroups

Subgroups — #30665331

Order 16768.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22553692	C₂	1	1	yes	maximal, characteristic
2	#22553597	C₂	1	2	yes	maximal
3	#22553065	C₂	1	1	yes	maximal, characteristic
4	#22553059	C₂	1	2	yes	maximal
5	C₈×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₂₄	—	2	1	no	—
7	C₂×Dic₅₂₄	—	2	1	no	—
8	#16862246	C₂²	1	2	yes	—
9	#16862177	—	2	2	no	—
10	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
11	#16862170	—	2	2	no	—
12	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₃₁	—	2	1	no	—
14	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₃₁	—	2	1	no	—
16	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₆₂	—	2	2	no	—
18	Dic₅₂₄	—	4	2	no	—
19	Dic₅₂₄	—	4	2	no	—
20	C₂×C₁₃₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₃₁	—	2	2	no	—
22	C₄⋊Dic₁₃₁	—	2	2	no	—
23	C₂×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₁	—	2	1	no	—
25	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₁₃₁	—	2	1	no	—
28	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
29	C₄²×C₁₃₁	—	2	1	no	—
30	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
31	Dic₂₆₂	—	4	2	no	—
32	Dic₂₆₂	—	4	2	no	—
33	C₂×Dic₁₃₁	—	4	2	no	—
34	C₁₃₁⋊C₈	—	4	2	no	—
35	Q₈×C₁₃₁	—	2	1	no	—
36	Q₈×C₁₃₁	Q₁₆	1	2	yes	complemented
37	C₂×C₄×C₁₃₁	—	2	1	no	—
38	C₂×C₄×C₁₃₁	—	2	1	no	—
39	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
42	C₈×C₁₃₁	—	2	1	no	—
43	C₈×C₁₃₁	—	4	1	no	—
44	C₈×C₁₃₁	SD₁₆	1	2	yes	—
45	C₂²×C₁₃₁	#121	1	1	yes	characteristic
46	C₄×C₁₃₁	—	2	2	no	—
47	C₄×C₁₃₁	—	2	1	no	—
48	C₄×C₁₃₁	—	4	1	no	—
49	C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
51	C₄×C₁₃₁	#135	1	1	yes	characteristic
52	C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
53	Dic₁₃₁	—	8	2	no	—
54	C₂×C₁₃₁	#460	1	1	yes	characteristic
55	C₂×C₁₃₁	#467	1	1	yes	characteristic
56	C₂×C₁₃₁	#473	1	1	yes	characteristic
57	C₁₃₁	#1670	1	1	yes	characteristic, complemented, Sylow
58	#1670	—	131	1	no	maximal, Sylow
59	#494	—	131	1	no	—
60	#475	—	131	2	no	—
61	C₈×Q₈	D₁₃₁	1	1	yes	characteristic
62	#335	—	131	1	no	—
63	#332	—	131	2	no	—
64	C₂×Q₁₆	—	262	1	no	—
65	C₂×Q₁₆	—	262	1	no	—
66	C₄⋊Q₈	—	131	2	no	—
67	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
68	#106	—	262	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	131	2	no	—
71	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
72	#103	—	262	2	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
75	C₂×Q₈	—	262	2	no	—
76	C₂×Q₈	D₅₂₄	1	1	yes	characteristic
77	Q₁₆	—	524	2	no	—
78	Q₁₆	—	524	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	262	2	no	—
81	C₂×C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	262	2	no	—
85	C₄⋊C₄	—	262	2	no	—
86	C₄⋊C₄	D₅₂₄	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	524	2	no	—
91	Q₈	—	524	2	no	—
92	Q₈	Dic₅₂₄	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	524	2	no	—
96	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
97	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
98	C₂×C₄	#14523513	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	524	2	no	—
102	C₈	#14523500	1	2	yes	—
103	C₂²	#16862220	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	1048	2	no	—
108	C₄	C₂×#14523500	1	1	yes	characteristic
109	C₄	C₂×Dic₅₂₄	1	1	yes	characteristic
110	C₄	#16862268	1	1	yes	characteristic
111	C₄	#16862271	1	1	yes	characteristic
112	C₂	#22553512	1	1	yes	characteristic
113	C₂	#22553553	1	1	yes	characteristic
114	C₂	#22553587	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553692

C₂

yes

maximal, characteristic

#22553597

C₂

yes

maximal

#22553065

C₂

yes

maximal, characteristic

#22553059

C₂

yes

maximal

C₈×Q₈×C₁₃₁

C₂

yes

maximal, characteristic

C₂×Dic₅₂₄

—

C₂×Dic₅₂₄

—

#16862246

C₂²

yes

—

#16862177

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

#16862170

—

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

—

C₄⋊C₈×C₁₃₁

C₂²

yes

characteristic

C₄×C₈×C₁₃₁

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

—

Dic₅₂₄

—

Dic₅₂₄

—

C₂×C₁₃₁⋊C₈

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₂×Q₈×C₁₃₁

D₄

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

D₄

yes

characteristic

C₂×C₈×C₁₃₁

D₄

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

D₄

yes

characteristic

C₄²×C₁₃₁

—

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

Q₁₆

yes

complemented

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

#41

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

SD₁₆

yes

—

C₂²×C₁₃₁

#121

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₃₁

C₂×Q₁₆

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1670

—

131

maximal, Sylow

#494

—

131

—

#475

—

131

—

C₈×Q₈

D₁₃₁

yes

characteristic

#335

—

131

—

#332

—

131

—

C₂×Q₁₆

—

262

—

C₂×Q₁₆

—

262

—

C₄⋊Q₈

—

131

—

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

#106

—

262

—

C₄⋊C₈

—

C₄⋊C₈

—

131

—

C₄⋊C₈

C₂×D₁₃₁

yes

characteristic

#103

—

262

—

C₄×C₈

—

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₂×Q₈

—

262

—

C₂×Q₈

D₅₂₄

yes

characteristic

Q₁₆

—

524

—

Q₁₆

—

524

—

C₂×C₈

—

C₂×C₈

—

262

—

C₂×C₈

C₂²⋊D₁₃₁

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

262

—

C₄⋊C₄

D₅₂₄

yes

characteristic

C₄²

—

C₄²

C₂²×D₁₃₁

yes

characteristic

Q₈

—

Q₈

—

524

—

Q₈

—

524

—

Q₈

Dic₅₂₄

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

524

—

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

C₂×C₄

C₂×D₅₂₄

yes

characteristic

C₂×C₄

#14523513

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

524

—

102

C₈

#14523500

yes

—

103

C₂²

#16862220

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

1048

—

108

C₄

C₂×#14523500

yes

characteristic

109

C₄

C₂×Dic₅₂₄

yes

characteristic

110

C₄

#16862268

yes

characteristic

111

C₄

#16862271

yes

characteristic

112

C₂

#22553512

yes

characteristic

113

C₂

#22553553

yes

characteristic

114

C₂

#22553587

yes

characteristic

	subgroup	G/H
center	C₂²	#16862220
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	C₈×Q₈×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#121
hypercenter	C₈×Q₈	D₁₃₁
hypocenter	C₁₃₁	#1670
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#16862220

commutator, cosocle

C₄²×C₁₃₁

C₂³

Frattini

C₄²

C₂²×D₁₃₁

Fitting

C₈×Q₈×C₁₃₁

C₂

socle, prime radical

C₂²×C₁₃₁

#121

hypercenter

C₈×Q₈

D₁₃₁

hypocenter

C₁₃₁

#1670

LCS

G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
36. Q₈×C₁₃₁	Q₁₆	semidirect
57. C₁₃₁	#1670	semidirect
92. Q₈	Dic₅₂₄	semidirect

H ≅ G/N

type

36. Q₈×C₁₃₁

Q₁₆

semidirect

57. C₁₃₁

#1670

semidirect

92. Q₈

Dic₅₂₄

semidirect

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

subgroup	index
#22553692	2
#22553597	2
#22553065	2
#22553059	2
C₈×Q₈×C₁₃₁	2
#1670	131

subgroup

index

C₈×Q₈×C₁₃₁

131

Subgroups — #30665331

Conjugacy classes of subgroups

Complements

Maximal subgroups