Group #30599474 — subgroups

Subgroups — #30599474

Order 15616.

228 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 80 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	#21461380	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	#16840625	—	2	2	no	—
6	#16840623	—	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	1	yes	characteristic
16	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
17	C₄²×D₆₁	C₂³	1	1	yes	characteristic
18	#14519706	—	2	2	no	—
19	#14519689	—	2	2	no	—
20	C₄×#92418	C₂×C₄	1	2	yes	—
21	C₄×#92418	C₂×C₄	1	1	yes	characteristic
22	C₈×Dic₆₁	D₄	1	1	yes	characteristic
23	C₈×Dic₆₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₆₁	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×D₆₁	#41	1	2	yes	—
28	C₂×#92420	—	4	4	no	—
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
32	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
34	C₄×Dic₆₁	#41	1	2	yes	—
35	C₂×#92418	—	2	2	no	—
36	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
38	C₈×D₆₁	—	2	2	no	—
39	C₈×D₆₁	—	2	2	no	—
40	C₂²×D₆₁	#125	1	1	yes	characteristic
41	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	#117	1	2	yes	—
43	C₂×Dic₆₁	#124	1	2	yes	—
44	C₂×Dic₆₁	#125	1	1	yes	characteristic
45	C₄×D₆₁	—	2	1	no	—
46	C₄×D₆₁	—	2	1	no	—
47	C₄×D₆₁	—	4	2	no	—
48	#92420	—	8	4	no	—
49	C₈×C₆₁	—	4	2	no	—
50	C₈×C₆₁	#108	1	2	yes	—
51	C₈×C₆₁	#108	1	2	yes	—
52	#92418	—	2	1	no	—
53	#92418	—	2	1	no	—
54	#92418	—	4	2	no	—
55	#8796	—	61	1	no	maximal, Sylow
56	C₂²×C₆₁	#382	1	1	yes	characteristic
57	C₂×D₆₁	—	2	2	no	—
58	C₂×D₆₁	—	2	1	no	—
59	C₄×C₆₁	—	2	2	no	—
60	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
61	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
62	Dic₆₁	—	2	2	no	—
63	Dic₆₁	—	2	1	no	—
64	#1568	—	61	2	no	—
65	#1450	—	61	1	no	—
66	#1287	—	61	4	no	—
67	C₂×C₆₁	#1833	1	2	yes	—
68	C₂×C₆₁	#1837	1	1	yes	characteristic
69	D₆₁	—	4	1	no	—
70	#432	—	61	4	no	—
71	#431	—	61	2	no	—
72	C₂×C₄×C₈	—	61	1	no	—
73	#335	—	122	2	no	—
74	#334	—	122	2	no	—
75	#322	—	61	2	no	—
76	C₈²	—	61	1	no	—
77	C₈²	#6923	1	1	yes	characteristic
78	C₆₁	#8796	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	61	1	no	—
80	C₂²×C₈	—	61	1	no	—
81	C₂×C₄²	—	61	1	no	—
82	C₄⋊C₈	—	122	2	no	—
83	C₄⋊C₈	—	122	4	no	—
84	C₂²⋊C₈	—	122	2	no	—
85	C₂²⋊C₈	—	122	4	no	—
86	#97	—	122	4	no	—
87	C₄×C₈	—	61	1	no	—
88	C₄×C₈	—	61	1	no	—
89	C₄×C₈	—	61	1	no	—
90	C₄×C₈	—	61	2	no	—
91	C₄×C₈	C₂×#6923	1	2	yes	—
92	C₄×C₈	C₂×#6923	1	1	yes	characteristic
93	C₂²×C₄	—	61	1	no	—
94	C₂²×C₄	—	61	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	61	1	no	—
97	C₂×C₈	—	61	1	no	—
98	C₂×C₈	—	122	2	no	—
99	C₂×C₈	—	122	2	no	—
100	C₂×C₈	—	122	2	no	—
101	C₂×C₈	—	244	4	no	—
102	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
103	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
104	C₄²	—	61	1	no	—
105	C₄²	—	61	2	no	—
106	C₄²	C₂²×#6923	1	1	yes	characteristic
107	C₂³	—	61	1	no	—
108	C₂×C₄	—	61	2	no	—
109	C₂×C₄	—	61	1	no	—
110	C₂×C₄	—	122	1	no	—
111	C₂×C₄	—	122	1	no	—
112	C₂×C₄	—	244	2	no	—
113	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
114	C₂×C₄	#14519856	1	2	yes	—
115	C₈	—	4	2	no	—
116	C₈	—	122	1	no	—
117	C₈	—	122	1	no	—
118	C₈	—	488	4	no	—
119	C₈	—	244	2	no	—
120	C₈	#14519847	1	2	yes	—
121	C₈	#14519848	1	2	yes	—
122	C₂²	—	122	2	no	—
123	C₂²	—	122	1	no	—
124	C₂²	#16840706	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	122	2	no	—
127	C₄	—	122	1	no	—
128	C₄	C₂×#14519847	1	1	yes	characteristic
129	C₄	C₂×#14519848	1	1	yes	characteristic
130	C₂	—	244	1	no	—
131	C₂	#21461738	1	2	yes	—
132	C₂	#21461743	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461380

C₂

yes

maximal

#21460676

C₂

yes

maximal

#21444101

C₂

yes

maximal, characteristic

#16840784

C₂²

yes

—

#16840625

—

#16840623

—

#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

characteristic

C₂×C₈×D₆₁

Q₈

yes

characteristic

C₄²×D₆₁

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

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

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×D₄

yes

characteristic

C₄×Dic₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₈×D₆₁

—

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

—

C₄×D₆₁

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

#108

yes

—

C₈×C₆₁

#108

yes

—

—

—

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

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

#8796

yes

characteristic, complemented, Sylow

C₂²×C₈

—

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

—

C₄×C₈

C₂×#6923

yes

—

C₄×C₈

C₂×#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

100

C₂×C₈

—

122

—

101

C₂×C₈

—

244

—

102

C₂×C₈

C₄⋊#6923

yes

characteristic

103

C₂×C₈

C₄⋊#6923

yes

characteristic

104

C₄²

—

105

C₄²

—

106

C₄²

C₂²×#6923

yes

characteristic

107

C₂³

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

122

—

111

C₂×C₄

—

122

—

112

C₂×C₄

—

244

—

113

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

114

C₂×C₄

#14519856

yes

—

115

C₈

—

116

C₈

—

122

—

117

C₈

—

122

—

118

C₈

—

488

—

119

C₈

—

244

—

120

C₈

#14519847

yes

—

121

C₈

#14519848

yes

—

122

C₂²

—

122

—

123

C₂²

—

122

—

124

C₂²

#16840706

yes

characteristic

125

C₄

—

126

C₄

—

122

—

127

C₄

—

122

—

128

C₄

C₂×#14519847

yes

characteristic

129

C₄

C₂×#14519848

yes

characteristic

130

C₂

—

244

—

131

C₂

#21461738

yes

—

132

C₂

#21461743

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	#8796
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
78. C₆₁	#8796	semidirect

H ≅ G/N

type

78. C₆₁

#8796

semidirect

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

subgroup	index
#21461380	2
#21460676	2
#21444101	2
#8796	61

subgroup

index

Subgroups — #30599474

Conjugacy classes of subgroups

Complements

Maximal subgroups