Group #30605986 — subgroups

Subgroups — #30605986

Order 15616.

214 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 70 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	#21461414	C₂	1	2	yes	maximal
2	#21460843	C₂	1	1	yes	maximal, characteristic
3	#21460780	C₂	1	2	yes	maximal
4	#21460736	C₂	1	1	yes	maximal, characteristic
5	#21447689	C₂	1	1	yes	maximal, characteristic
6	#16840811	—	2	2	no	—
7	#16840651	C₂²	1	1	yes	characteristic
8	#16840623	C₂²	1	2	yes	—
9	#16840612	—	2	2	no	—
10	#16840603	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₈₈	C₄	1	1	yes	characteristic
12	#16839856	C₂²	1	2	yes	—
13	#16839565	C₄	1	2	yes	—
14	#16839537	C₂²	1	1	yes	characteristic
15	C₆₁×#367	C₄	1	1	yes	characteristic
16	C₂×#11758187	—	2	1	no	—
17	#14519844	—	4	2	no	—
18	C₄⋊#92420	—	2	2	no	—
19	C₄×#92420	—	4	2	no	—
20	C₄×#92420	D₄	1	2	yes	—
21	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
22	#14519742	—	2	2	no	—
23	Dic₄₈₈	C₈	1	4	yes	complemented
24	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
25	#14519700	—	2	2	no	—
26	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
28	#107×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₂₂	—	2	2	no	—
30	C₂×#92420	—	2	2	no	—
31	C₂×#92420	—	4	2	no	—
32	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₆₁	—	2	2	no	—
34	#11758196	—	4	2	no	—
35	C₄×Dic₆₁	—	4	2	no	—
36	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂×#92418	#41	1	1	yes	characteristic
38	Dic₂₄₄	—	2	2	no	—
39	Dic₂₄₄	C₂×C₈	1	4	yes	—
40	#11758187	—	4	1	no	—
41	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
42	#11758185	—	2	1	no	—
43	C₂×C₄×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
45	C₂×Dic₆₁	—	4	2	no	—
46	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
47	Dic₁₂₂	—	2	4	no	—
48	Dic₁₂₂	—	4	2	no	—
49	#92420	—	4	2	no	—
50	#92420	—	8	2	no	—
51	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₆₁	#131	1	1	yes	characteristic
53	#92418	—	2	1	no	—
54	#12066	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	#442	1	1	yes	characteristic
56	C₄×C₆₁	—	8	2	no	—
57	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
58	C₄×C₆₁	#444	1	1	yes	characteristic
59	Dic₆₁	—	2	2	no	—
60	Dic₆₁	—	4	4	no	—
61	#2176	—	61	1	no	—
62	#1579	—	61	2	no	—
63	#1368	—	61	1	no	—
64	#1334	—	61	2	no	—
65	#1313	—	61	1	no	—
66	C₂×C₆₁	#1580	1	1	yes	characteristic
67	C₂×C₆₁	#2180	1	2	yes	—
68	C₂×Q₃₂	—	61	1	no	—
69	C₈×Q₈	—	122	2	no	—
70	C₄×Q₁₆	—	61	2	no	—
71	#367	#6923	1	1	yes	characteristic
72	#358	—	61	2	no	—
73	#346	—	61	1	no	—
74	#346	—	61	1	no	—
75	#334	—	61	2	no	—
76	Q₈⋊C₈	—	122	2	no	—
77	#322	—	61	1	no	—
78	C₆₁	#12066	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	61	2	no	—
80	C₄×Q₈	—	122	2	no	—
81	Q₃₂	—	61	4	no	—
82	C₂×C₁₆	—	61	1	no	—
83	C₂×C₁₆	—	122	1	no	—
84	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
85	#107	C₂×#6923	1	2	yes	—
86	C₄⋊C₈	—	122	2	no	—
87	C₄⋊C₈	—	244	2	no	—
88	#103	—	122	2	no	—
89	C₄×C₈	—	61	1	no	—
90	C₄×C₈	—	61	2	no	—
91	C₄×C₈	—	244	2	no	—
92	C₂×Q₈	—	122	2	no	—
93	Q₁₆	—	61	4	no	—
94	Q₁₆	—	122	2	no	—
95	C₂×C₈	—	61	1	no	—
96	C₂×C₈	—	122	2	no	—
97	C₂×C₈	—	244	2	no	—
98	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	244	2	no	—
101	C₄²	—	61	1	no	—
102	C₄²	—	244	2	no	—
103	C₁₆	—	122	1	no	—
104	C₁₆	—	244	1	no	—
105	C₁₆	D₆₁⋊C₈	1	2	yes	—
106	Q₈	—	122	4	no	—
107	Q₈	—	244	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	61	2	no	—
110	C₂×C₄	—	244	2	no	—
111	C₂×C₄	D₄×#6923	1	1	yes	characteristic
112	C₈	—	122	1	no	—
113	C₈	—	488	2	no	—
114	C₈	—	244	2	no	—
115	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
116	C₈	#14519865	1	1	yes	characteristic
117	C₂²	#16840803	1	1	yes	characteristic
118	C₄	—	8	2	no	—
119	C₄	—	122	2	no	—
120	C₄	—	244	4	no	—
121	C₄	#16840790	1	1	yes	characteristic
122	C₄	#16840808	1	1	yes	characteristic
123	C₂	#21461417	1	1	yes	characteristic
124	C₂	#21462225	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461414

C₂

yes

maximal

#21460843

C₂

yes

maximal, characteristic

#21460780

C₂

yes

maximal

#21460736

C₂

yes

maximal, characteristic

#21447689

C₂

yes

maximal, characteristic

#16840811

—

#16840651

C₂²

yes

characteristic

#16840623

C₂²

yes

—

#16840612

—

#16840603

C₂²

yes

characteristic

C₂×Dic₄₈₈

C₄

yes

characteristic

#16839856

C₂²

yes

—

#16839565

C₄

yes

—

#16839537

C₂²

yes

characteristic

C₆₁×#367

C₄

yes

—

—

—

—

D₄

yes

—

C₂×Dic₂₄₄

C₂×C₄

yes

—

#14519742

—

Dic₄₈₈

C₈

yes

complemented

C₂×#11758185

C₂×C₄

yes

characteristic

#14519700

—

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

#107×C₆₁

C₂×C₄

yes

—

C₂×Dic₁₂₂

—

C₂×#92420

—

C₂×#92420

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×D₄

yes

yes

Dic₂₄₄

—

Dic₂₄₄

C₂×C₈

yes

—

#11758187

—

C₁₆×C₆₁

C₂×C₈

yes

—

#11758185

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

AGL(1,ℤ/8)

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

#92420

—

C₈×C₆₁

C₂²×C₈

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

#444

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

—

—

—

—

C₂×C₆₁

#1580

yes

characteristic

C₂×C₆₁

#2180

yes

—

C₂×Q₃₂

—

C₈×Q₈

—

122

—

C₄×Q₁₆

—

yes

—

—

—

—

Q₈⋊C₈

—

122

—

#322

—

C₆₁

#12066

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

122

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

#103

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

244

—

C₂×Q₈

—

122

—

Q₁₆

—

Q₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₂²×#6923

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

244

—

101

C₄²

—

102

C₄²

—

244

—

103

C₁₆

—

122

—

104

C₁₆

—

244

—

105

C₁₆

D₆₁⋊C₈

yes

—

106

Q₈

—

122

—

107

Q₈

—

244

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

244

—

111

C₂×C₄

D₄×#6923

yes

characteristic

112

C₈

—

122

—

113

C₈

—

488

—

114

C₈

—

244

—

115

C₈

C₂×D₆₁⋊C₈

yes

characteristic

116

C₈

#14519865

yes

characteristic

117

C₂²

#16840803

yes

characteristic

118

C₄

—

119

C₄

—

122

—

120

C₄

—

244

—

121

C₄

#16840790

yes

characteristic

122

C₄

#16840808

yes

characteristic

123

C₂

#21461417

yes

characteristic

124

C₂

#21462225

yes

—

	subgroup	G/H
center	C₂²	#16840803
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#367	C₄
socle, prime radical	C₂²×C₆₁	#442
cosocle	C₈×Dic₆₁	C₂³
hypercenter	#367	#6923
hypocenter	C₆₁	#12066
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

#16840803

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₆₁×#367

C₄

socle, prime radical

C₂²×C₆₁

#442

cosocle

C₈×Dic₆₁

C₂³

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
23. Dic₄₈₈	C₈	semidirect
78. C₆₁	#12066	semidirect

H ≅ G/N

type

23. Dic₄₈₈

C₈

semidirect

78. C₆₁

#12066

semidirect

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

subgroup	index
#21461414	2
#21460843	2
#21460780	2
#21460736	2
#21447689	2
#12066	61

subgroup

index

Subgroups — #30605986

Conjugacy classes of subgroups

Complements

Maximal subgroups