Group #30599477 — subgroups

Subgroups — #30599477

Order 15616.

228 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 76 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	#21461382	C₂	1	2	yes	maximal
2	#21460676	C₂	1	4	yes	maximal
3	#21444101	C₂	1	1	yes	maximal, characteristic
4	#16840784	C₂²	1	4	yes	—
5	#16840626	—	2	2	no	—
6	#16840624	—	2	2	no	—
7	#16839813	C₂²	1	2	yes	—
8	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	#16839484	C₄	1	2	yes	—
10	C₈×#92418	C₄	1	1	yes	characteristic
11	C₈²×C₆₁	C₄	1	1	yes	characteristic
12	C₄⋊#92420	—	2	4	no	—
13	#14519834	—	2	4	no	—
14	#14519832	—	2	4	no	—
15	C₂×C₈×D₆₁	Q₈	1	2	yes	—
16	C₄²×D₆₁	C₂³	1	1	yes	characteristic
17	#14519706	—	2	2	no	—
18	#14519689	—	2	2	no	—
19	C₄×#92418	C₂×C₄	1	2	yes	—
20	C₄×#92418	C₂×C₄	1	1	yes	characteristic
21	C₈×Dic₆₁	D₄	1	2	yes	—
22	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₆₁	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×D₆₁	#41	1	2	yes	—
26	C₂×#92420	—	4	4	no	—
27	C₂×C₈×C₆₁	—	2	2	no	—
28	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
29	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
31	C₄×Dic₆₁	#41	1	2	yes	—
32	C₂×#92418	—	2	2	no	—
33	C₂×#92418	C₄⋊C₄	1	2	yes	—
34	C₈×D₆₁	—	2	4	no	—
35	C₂²×D₆₁	#125	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	#117	1	2	yes	—
38	C₂×Dic₆₁	#124	1	2	yes	—
39	C₂×Dic₆₁	#125	1	1	yes	characteristic
40	C₄×D₆₁	—	2	2	no	—
41	C₄×D₆₁	—	4	2	no	—
42	#92420	—	8	4	no	—
43	C₈×C₆₁	—	2	2	no	—
44	C₈×C₆₁	—	4	2	no	—
45	#92418	—	4	2	no	—
46	#92418	#108	1	4	yes	—
47	#8797	—	61	1	no	maximal, Sylow
48	C₂²×C₆₁	#382	1	1	yes	characteristic
49	C₂×D₆₁	—	2	2	no	—
50	C₂×D₆₁	—	2	1	no	—
51	C₄×C₆₁	—	2	2	no	—
52	C₄×C₆₁	C₂×#108	1	2	yes	—
53	Dic₆₁	—	2	2	no	—
54	Dic₆₁	—	2	1	no	—
55	#1568	—	61	2	no	—
56	#1450	—	61	1	no	—
57	#1287	—	61	4	no	—
58	C₂×C₆₁	#1833	1	2	yes	—
59	C₂×C₆₁	#1837	1	1	yes	characteristic
60	D₆₁	—	4	1	no	—
61	#432	—	61	4	no	—
62	#431	—	61	2	no	—
63	C₂×C₄×C₈	—	61	1	no	—
64	#335	—	122	2	no	—
65	#334	—	122	2	no	—
66	#322	—	61	2	no	—
67	C₈²	—	61	1	no	—
68	C₈²	#6923	1	1	yes	characteristic
69	C₆₁	#8797	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	61	2	no	—
71	C₂×C₄²	—	61	1	no	—
72	C₄⋊C₈	—	122	2	no	—
73	C₄⋊C₈	—	122	4	no	—
74	C₂²⋊C₈	—	122	2	no	—
75	C₂²⋊C₈	—	122	4	no	—
76	#97	—	122	4	no	—
77	C₄×C₈	—	61	2	no	—
78	C₄×C₈	—	61	1	no	—
79	C₄×C₈	—	61	2	no	—
80	C₄×C₈	C₂×#6923	1	2	yes	—
81	C₄×C₈	C₂×#6923	1	1	yes	characteristic
82	C₂²×C₄	—	61	1	no	—
83	C₂²×C₄	—	61	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	61	2	no	—
86	C₂×C₈	—	122	4	no	—
87	C₂×C₈	—	122	2	no	—
88	C₂×C₈	—	244	4	no	—
89	C₂×C₈	C₄⋊#6923	1	2	yes	—
90	C₄²	—	61	1	no	—
91	C₄²	—	61	2	no	—
92	C₄²	C₂²×#6923	1	1	yes	characteristic
93	C₂³	—	61	1	no	—
94	C₂×C₄	—	61	2	no	—
95	C₂×C₄	—	61	1	no	—
96	C₂×C₄	—	122	2	no	—
97	C₂×C₄	—	244	2	no	—
98	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
99	C₂×C₄	#14519856	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	4	2	no	—
102	C₈	—	61	4	no	—
103	C₈	—	488	4	no	—
104	C₈	—	244	2	no	—
105	C₂²	—	122	2	no	—
106	C₂²	—	122	1	no	—
107	C₂²	#16840706	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	122	2	no	—
110	C₄	—	122	1	no	—
111	C₄	D₆₁⋊#108	1	2	yes	—
112	C₂	—	244	1	no	—
113	C₂	#21461738	1	2	yes	—
114	C₂	#21461744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461382

C₂

yes

maximal

#21460676

C₂

yes

maximal

#21444101

C₂

yes

maximal, characteristic

#16840784

C₂²

yes

—

#16840626

—

#16840624

—

#16839813

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839484

C₄

yes

—

C₈×#92418

C₄

yes

characteristic

C₈²×C₆₁

C₄

yes

—

—

—

C₂×C₈×D₆₁

Q₈

yes

—

C₄²×D₆₁

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×D₆₁

#41

yes

—

C₂×#92420

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×D₄

yes

characteristic

C₄×Dic₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

—

C₈×D₆₁

—

C₂²×D₆₁

#125

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×Dic₆₁

#124

yes

—

C₂×Dic₆₁

#125

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

#108

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#108

yes

—

Dic₆₁

—

Dic₆₁

—

#1568

—

#1450

—

#1287

—

C₂×C₆₁

#1833

yes

—

C₂×C₆₁

#1837

yes

characteristic

D₆₁

—

#432

—

#431

—

C₂×C₄×C₈

—

#335

—

122

—

#334

—

122

—

#322

—

C₈²

—

C₈²

#6923

yes

characteristic

C₆₁

#8797

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6923

yes

—

C₄×C₈

C₂×#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₄⋊#6923

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#6923

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

C₂×C₄

#14519856

yes

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₈

—

488

—

104

C₈

—

244

—

105

C₂²

—

122

—

106

C₂²

—

122

—

107

C₂²

#16840706

yes

characteristic

108

C₄

—

109

C₄

—

122

—

110

C₄

—

122

—

111

C₄

D₆₁⋊#108

yes

—

112

C₂

—

244

—

113

C₂

#21461738

yes

—

114

C₂

#21461744

yes

characteristic

	subgroup	G/H
center	C₂²	#16840706
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	#382
prime radical	C₂²×D₆₁	#125
cosocle	C₄²×D₆₁	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#8797
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#16840706

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×#6923

Fitting

C₈²×C₆₁

C₄

C₂²×C₆₁

C₂²×D₆₁

C₄²×D₆₁

C₂³

C₈²

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
69. C₆₁	#8797	semidirect

H ≅ G/N

type

69. C₆₁

#8797

semidirect

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

subgroup	index
#21461382	2
#21460676	2
#21444101	2
#8797	61

subgroup

index

Subgroups — #30599477

Conjugacy classes of subgroups

Complements

Maximal subgroups