Group #30599530 — subgroups

Subgroups — #30599530

Order 15616.

354 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 144 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	#21461760	C₂	1	2	yes	maximal
2	#21461377	C₂	1	4	yes	maximal
3	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519848	—	2	4	no	—
5	#16840766	C₂²	1	4	yes	—
6	#16840625	—	2	8	no	—
7	C₄×C₈×D₆₁	C₂²	1	2	yes	—
8	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	C₈×#92418	C₄	1	1	yes	characteristic
10	C₈×#92418	C₄	1	2	yes	—
11	C₈²×C₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758213	—	2	4	no	—
13	#14519848	—	4	8	no	—
14	C₄⋊#92420	—	2	8	no	—
15	C₂×C₈×D₆₁	D₄	1	2	yes	—
16	C₂×C₈×D₆₁	Q₈	1	4	yes	—
17	C₄²×D₆₁	C₂³	1	1	yes	characteristic
18	C₄×#92418	C₂×C₄	1	2	yes	—
19	C₄×#92418	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₆₁	D₄	1	4	yes	—
21	C₈×Dic₆₁	Q₈	1	2	yes	—
22	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	C₂×Q₈	1	2	yes	—
26	C₂×#92420	—	4	4	no	—
27	#11758213	—	4	8	no	—
28	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
30	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
32	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×#92418	C₄⋊C₄	1	4	yes	—
34	C₂×#92418	C₄⋊C₄	1	2	yes	—
35	C₈×D₆₁	—	2	8	no	—
36	C₈×D₆₁	D₈	1	4	yes	—
37	C₈×D₆₁	Q₁₆	1	4	yes	—
38	C₂²×D₆₁	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
41	C₂×Dic₆₁	#127	1	1	yes	characteristic
42	C₂×Dic₆₁	C₄⋊Q₈	1	2	yes	—
43	C₄×D₆₁	—	2	4	no	—
44	C₄×D₆₁	C₂×D₈	1	2	yes	—
45	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
46	#92420	—	8	4	no	—
47	C₈×C₆₁	#107	1	4	yes	—
48	C₈×C₆₁	#108	1	8	yes	complemented
49	#92418	—	2	4	no	—
50	#92418	#107	1	4	yes	—
51	#8823	—	61	1	no	maximal, Sylow
52	C₂²×C₆₁	#384	1	1	yes	characteristic
53	C₂×D₆₁	—	2	2	no	—
54	C₂×D₆₁	#500	1	2	yes	—
55	C₄×C₆₁	C₂×#107	1	2	yes	—
56	C₄×C₆₁	C₂×#108	1	4	yes	—
57	Dic₆₁	—	2	2	no	—
58	Dic₆₁	#493	1	1	yes	characteristic
59	Dic₆₁	#494	1	1	yes	characteristic
60	#1848	—	61	2	no	—
61	#1567	—	61	4	no	—
62	C₂×C₈²	—	61	1	no	—
63	C₂×C₆₁	#1845	1	1	yes	characteristic
64	C₂×C₆₁	#1848	1	2	yes	—
65	D₆₁	—	2	2	no	—
66	C₂×#108	—	122	4	no	—
67	#423	—	61	4	no	—
68	C₂×C₄×C₈	—	61	2	no	—
69	C₂×C₄×C₈	—	61	1	no	—
70	#335	—	122	8	no	—
71	C₈²	—	61	2	no	—
72	C₈²	—	61	1	no	—
73	C₈²	#6923	1	1	yes	characteristic
74	C₆₁	#8823	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	122	4	no	—
76	C₂²×C₈	—	61	2	no	—
77	C₂²×C₈	—	61	4	no	—
78	C₂×C₄²	—	61	1	no	—
79	#108	—	244	8	no	—
80	C₄⋊C₈	—	122	8	no	—
81	C₄×C₈	—	61	4	no	—
82	C₄×C₈	—	61	2	no	—
83	C₄×C₈	—	61	1	no	—
84	C₄×C₈	—	61	2	no	—
85	C₄×C₈	C₂×#6923	1	1	yes	characteristic
86	C₄×C₈	C₂×#6923	1	2	yes	—
87	C₂²×C₄	—	61	1	no	—
88	C₂²×C₄	—	61	2	no	—
89	M₁₆	—	244	8	no	—
90	C₂×C₈	—	61	2	no	—
91	C₂×C₈	—	61	4	no	—
92	C₂×C₈	—	61	4	no	—
93	C₂×C₈	—	61	4	no	—
94	C₂×C₈	—	122	8	no	—
95	C₂×C₈	—	244	4	no	—
96	C₂×C₈	C₄⋊#6923	1	2	yes	—
97	C₂×C₈	C₄⋊#6923	1	4	yes	—
98	C₄²	—	61	2	no	—
99	C₄²	—	61	1	no	—
100	C₄²	C₂²×#6923	1	1	yes	characteristic
101	C₂³	—	61	1	no	—
102	C₂×C₄	—	61	1	no	—
103	C₂×C₄	—	61	2	no	—
104	C₂×C₄	—	61	2	no	—
105	C₂×C₄	—	61	2	no	—
106	C₂×C₄	—	122	4	no	—
107	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
109	C₈	—	61	4	no	—
110	C₈	—	122	4	no	—
111	C₈	—	488	4	no	—
112	C₈	#14519846	1	4	yes	—
113	C₈	#14519848	1	8	yes	complemented
114	C₂²	—	61	2	no	—
115	C₂²	—	122	2	no	—
116	C₂²	#16840708	1	1	yes	characteristic
117	C₄	—	61	1	no	—
118	C₄	—	61	1	no	—
119	C₄	—	122	2	no	—
120	C₄	C₂×#14519846	1	2	yes	—
121	C₄	C₂×#14519848	1	4	yes	—
122	C₂	—	122	2	no	—
123	C₂	#21461755	1	1	yes	characteristic
124	C₂	#21461760	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461760

C₂

yes

maximal

#21461377

C₂

yes

maximal

C₈²×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519848

—

#16840766

C₂²

yes

—

#16840625

—

C₄×C₈×D₆₁

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₈×#92418

C₄

yes

characteristic

C₈×#92418

C₄

yes

—

C₈²×C₆₁

C₄

yes

—

—

—

C₂×C₈×D₆₁

D₄

yes

—

C₂×C₈×D₆₁

Q₈

yes

—

C₄²×D₆₁

C₂³

yes

characteristic

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×D₄

yes

characteristic

C₂×C₄×D₆₁

C₂×Q₈

yes

—

C₂×#92420

—

#11758213

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×D₄

yes

—

C₄×Dic₆₁

C₂×Q₈

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

D₈

yes

—

C₈×D₆₁

Q₁₆

yes

—

C₂²×D₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×Dic₆₁

#127

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

—

C₄×D₆₁

—

C₄×D₆₁

C₂×D₈

yes

—

C₄×D₆₁

C₂×Q₁₆

yes

—

#92420

—

C₈×C₆₁

#107

yes

—

C₈×C₆₁

yes

—

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

#500

yes

—

C₄×C₆₁

C₂×#107

yes

—

C₄×C₆₁

C₂×#108

yes

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

—

—

C₂×C₈²

—

C₂×C₆₁

#1845

yes

characteristic

C₂×C₆₁

#1848

yes

—

D₆₁

—

C₂×#108

—

122

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#335

—

122

—

C₈²

—

C₈²

—

C₈²

#6923

yes

characteristic

C₆₁

#8823

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

244

—

C₄⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6923

yes

characteristic

C₄×C₈

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₄⋊#6923

yes

—

C₂×C₈

C₄⋊#6923

yes

—

C₄²

—

C₄²

—

100

C₄²

C₂²×#6923

yes

characteristic

101

C₂³

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

122

—

107

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

108

C₂×C₄

C₂×C₄⋊#6923

yes

—

109

C₈

—

110

C₈

—

122

—

111

C₈

—

488

—

112

C₈

#14519846

yes

—

113

C₈

#14519848

yes

complemented

114

C₂²

—

115

C₂²

—

122

—

116

C₂²

#16840708

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

122

—

120

C₄

C₂×#14519846

yes

—

121

C₄

C₂×#14519848

yes

—

122

C₂

—

122

—

123

C₂

#21461755

yes

characteristic

124

C₂

#21461760

yes

—

	subgroup	G/H
center	C₂²	#16840708
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	#384
prime radical	C₂²×D₆₁	C₄⋊Q₈
cosocle	C₄²×D₆₁	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#8823
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#16840708

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×#6923

Fitting

C₈²×C₆₁

C₄

socle

C₂²×C₆₁

#384

prime radical

C₂²×D₆₁

C₄⋊Q₈

cosocle

C₄²×D₆₁

C₂³

C₈²

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
48. C₈×C₆₁	#108	semidirect
74. C₆₁	#8823	semidirect
113. C₈	#14519848	semidirect

H ≅ G/N

type

48. C₈×C₆₁

#108

semidirect

74. C₆₁

#8823

semidirect

113. C₈

#14519848

semidirect

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

subgroup	index
#21461760	2
#21461377	2
C₈²×D₆₁	2
#8823	61

subgroup

index

#21461760

#21461377

C₈²×D₆₁

#8823

Subgroups — #30599530

Conjugacy classes of subgroups

Complements

Maximal subgroups