Group #30603788 — subgroups

Subgroups — #30603788

Order 15616.

228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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	#21462182	C₂	1	2	yes	maximal
2	#21460737	C₂	1	4	yes	maximal, complemented
3	#21444101	C₂	1	1	yes	maximal, characteristic
4	#16840683	C₄	1	4	yes	—
5	#16840657	C₄	1	4	yes	—
6	#16840650	C₂²	1	4	yes	—
7	#16839813	C₄	1	2	yes	—
8	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	#16839484	C₂²	1	2	yes	—
10	C₈×#92418	C₄	1	1	yes	characteristic
11	C₈²×C₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758187	C₂×C₄	1	8	yes	—
13	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
14	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
15	#14519706	—	2	2	no	—
16	#14519689	—	2	2	no	—
17	C₄×#92418	C₂×C₄	1	2	yes	—
18	C₄×#92418	C₂³	1	1	yes	characteristic
19	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₆₁	C₄²	1	1	yes	characteristic
23	C₂×C₄×D₆₁	M₁₆	1	2	yes	—
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₂×C₈×C₆₁	C₄²	1	2	yes	—
26	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
27	C₄×Dic₆₁	C₄²	1	1	yes	characteristic
28	C₄×Dic₆₁	M₁₆	1	2	yes	—
29	C₂×#92418	—	2	2	no	—
30	C₂×#92418	C₂²×C₄	1	2	yes	—
31	C₈×D₆₁	—	2	4	no	—
32	#11758187	—	2	8	no	—
33	C₂²×D₆₁	#97	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
35	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
36	C₂×Dic₆₁	#97	1	2	yes	—
37	C₂×Dic₆₁	#97	1	1	yes	characteristic
38	C₄×D₆₁	—	2	2	no	—
39	C₄×D₆₁	—	4	2	no	—
40	C₈×C₆₁	—	2	2	no	—
41	C₈×C₆₁	—	4	2	no	—
42	#92418	—	4	2	no	—
43	#92418	#131	1	4	yes	—
44	#11416	—	61	1	no	maximal, Sylow
45	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
46	C₂×D₆₁	—	2	2	no	—
47	C₂×D₆₁	—	2	1	no	—
48	C₄×C₆₁	—	2	2	no	—
49	C₄×C₆₁	#405	1	2	yes	—
50	Dic₆₁	—	2	2	no	—
51	Dic₆₁	—	2	1	no	—
52	#2163	—	61	2	no	—
53	#1450	—	61	1	no	—
54	#1313	—	61	4	no	—
55	C₂×C₆₁	#1452	1	1	yes	characteristic
56	C₂×C₆₁	#2110	1	2	yes	—
57	D₆₁	—	4	1	no	—
58	#431	—	61	2	no	—
59	C₂×C₄×C₈	—	61	1	no	—
60	C₄⋊C₁₆	—	61	4	no	—
61	C₂²⋊C₁₆	—	61	4	no	—
62	#346	—	61	4	no	—
63	#322	—	61	2	no	—
64	C₈²	—	61	1	no	—
65	C₈²	#6923	1	1	yes	characteristic
66	C₆₁	#11416	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	—	61	2	no	—
68	C₂×C₄²	—	61	1	no	—
69	C₂×C₁₆	—	61	8	no	—
70	C₄⋊C₈	—	122	2	no	—
71	C₂²⋊C₈	—	122	2	no	—
72	C₄×C₈	—	61	2	no	—
73	C₄×C₈	—	61	1	no	—
74	C₄×C₈	—	61	2	no	—
75	C₄×C₈	C₂×#6923	1	2	yes	—
76	C₄×C₈	C₂×#6923	1	1	yes	characteristic
77	C₂²×C₄	—	61	1	no	—
78	C₂²×C₄	—	61	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	61	2	no	—
81	C₂×C₈	—	122	4	no	—
82	C₂×C₈	—	122	2	no	—
83	C₂×C₈	C₄×#6923	1	2	yes	—
84	C₄²	—	61	1	no	—
85	C₄²	—	61	2	no	—
86	C₄²	C₂²×#6923	1	1	yes	characteristic
87	C₁₆	—	122	8	no	—
88	C₂³	—	61	1	no	—
89	C₂×C₄	—	61	2	no	—
90	C₂×C₄	—	61	1	no	—
91	C₂×C₄	—	122	2	no	—
92	C₂×C₄	—	244	2	no	—
93	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
94	C₂×C₄	D₆₁⋊M₁₆	1	2	yes	—
95	C₈	—	2	2	no	—
96	C₈	—	4	2	no	—
97	C₈	—	61	4	no	—
98	C₈	—	244	2	no	—
99	C₂²	—	122	2	no	—
100	C₂²	—	122	1	no	—
101	C₂²	#16840721	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	122	2	no	—
104	C₄	—	122	1	no	—
105	C₄	#16840725	1	2	yes	—
106	C₂	—	244	1	no	—
107	C₂	#21461033	1	1	yes	characteristic
108	C₂	#21462041	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21462182

C₂

yes

maximal

#21460737

C₂

yes

maximal, complemented

#21444101

C₂

yes

maximal, characteristic

#16840683

C₄

yes

—

#16840657

C₄

yes

—

#16840650

C₂²

yes

—

#16839813

C₄

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839484

C₂²

yes

—

C₈×#92418

C₄

yes

characteristic

C₈²×C₆₁

C₄

yes

characteristic

C₂×#11758187

C₂×C₄

yes

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₄²×D₆₁

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×#92418

C₂³

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₄²

yes

characteristic

C₂×C₄×D₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₄²

yes

characteristic

C₄×Dic₆₁

M₁₆

yes

—

C₂×#92418

—

C₂×#92418

C₂²×C₄

yes

—

C₈×D₆₁

—

#11758187

—

C₂²×D₆₁

#97

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂×M₁₆

yes

—

C₂×Dic₆₁

#97

yes

—

C₂×Dic₆₁

#97

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

#131

yes

—

#11416

—

maximal, Sylow

C₂²×C₆₁

C₂×#97

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#405

yes

—

Dic₆₁

—

Dic₆₁

—

#2163

—

#1450

—

#1313

—

C₂×C₆₁

#1452

yes

characteristic

C₂×C₆₁

#2110

yes

—

D₆₁

—

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₈²

—

C₈²

#6923

yes

characteristic

C₆₁

#11416

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

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

C₄×#6923

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#6923

yes

characteristic

C₁₆

—

122

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂×C₄×#6923

yes

characteristic

C₂×C₄

D₆₁⋊M₁₆

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

244

—

C₂²

—

122

—

100

C₂²

—

122

—

101

C₂²

#16840721

yes

characteristic

102

C₄

—

103

C₄

—

122

—

104

C₄

—

122

—

105

C₄

#16840725

yes

—

106

C₂

—

244

—

107

C₂

#21461033

yes

characteristic

108

C₂

#21462041

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂²×D₆₁	#97
cosocle	C₄×#92418	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#11416
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×C₄×#6923

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₄²

C₂²×#6923

Fitting

C₈²×C₆₁

C₄

socle

C₂²×C₆₁

C₂×#97

C₂²×D₆₁

C₂³

C₈²

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₈²

N	H ≅ G/N	type
2. #21460737	C₂	semidirect
66. C₆₁	#11416	semidirect

H ≅ G/N

type

2. #21460737

C₂

semidirect

66. C₆₁

#11416

semidirect

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

subgroup	index
#21462182	2
#21460737	2
#21444101	2
#11416	61

subgroup

index

Subgroups — #30603788

Conjugacy classes of subgroups

Complements

Maximal subgroups