Group #30606072 — subgroups

Subgroups — #30606072

Order 15616.

210 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 68 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	#21461459	C₂	1	2	yes	maximal
2	#21460850	C₂	1	1	yes	maximal, characteristic
3	#21460775	C₂	1	2	yes	maximal
4	C₁₆×#92420	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840813	—	2	2	no	—
7	#16840648	C₂²	1	1	yes	characteristic
8	#16840625	C₂²	1	2	yes	—
9	Q₈⋊#92420	—	2	2	no	—
10	C₈×#92420	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
12	#16839588	C₄	1	1	yes	characteristic
13	#16839566	C₄	1	2	yes	—
14	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₆₁	C₄	1	1	yes	characteristic
16	C₂×#11758187	—	2	1	no	—
17	#14519844	—	4	2	no	—
18	C₄⋊#92420	—	2	2	no	—
19	#14519832	—	4	2	no	—
20	C₄×#92420	D₄	1	1	yes	characteristic
21	C₄×#92420	D₄	1	1	yes	characteristic
22	Q₈×Dic₆₁	—	2	2	no	—
23	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
24	#14519711	C₂×C₄	1	2	yes	—
25	#14519701	—	2	2	no	—
26	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
28	Q₃₂×C₆₁	—	2	2	no	—
29	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₆₁	—	2	2	no	—
31	C₂×#92420	—	2	1	no	—
32	C₂×#92420	—	4	2	no	—
33	C₂×#92420	D₈	1	1	yes	characteristic
34	C₂×#92420	D₈	1	1	yes	characteristic
35	Q₁₆×C₆₁	—	2	2	no	—
36	Q₁₆×C₆₁	—	2	2	no	—
37	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₄⋊Dic₆₁	—	2	2	no	—
39	C₄⋊Dic₆₁	—	4	2	no	—
40	C₄×Dic₆₁	—	4	2	no	—
41	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×#92418	#41	1	1	yes	characteristic
43	#11758187	—	2	2	no	—
44	C₁₆×C₆₁	M₁₆	1	2	yes	—
45	#11758185	—	2	1	no	—
46	Q₈×C₆₁	—	4	2	no	—
47	Q₈×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	—	4	2	no	—
49	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
50	C₂×Dic₆₁	—	4	2	no	—
51	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
52	C₂×Dic₆₁	#135	1	1	yes	characteristic
53	#92420	—	2	2	no	—
54	#92420	—	2	1	no	—
55	#92420	—	8	2	no	—
56	#92420	Q₃₂	1	2	yes	complemented
57	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₆₁	#131	1	1	yes	characteristic
59	#92418	—	2	1	no	—
60	#12083	—	61	1	no	maximal, Sylow
61	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
62	C₄×C₆₁	—	8	2	no	—
63	C₄×C₆₁	#436	1	1	yes	characteristic
64	C₄×C₆₁	#443	1	1	yes	characteristic
65	Dic₆₁	—	2	1	no	—
66	Dic₆₁	—	8	2	no	—
67	Dic₆₁	C₂×Q₃₂	1	1	yes	characteristic
68	Dic₆₁	#508	1	1	yes	characteristic
69	C₄×Q₃₂	—	61	1	no	—
70	#1591	—	61	2	no	—
71	#1371	—	61	1	no	—
72	#1332	—	61	2	no	—
73	C₈×C₁₆	—	61	1	no	—
74	C₂×C₆₁	#1589	1	1	yes	characteristic
75	C₂×C₆₁	C₄×Q₃₂	1	1	yes	characteristic
76	C₂×C₆₁	#2178	1	1	yes	characteristic
77	C₂×Q₃₂	#6923	1	1	yes	characteristic
78	#446	—	122	2	no	—
79	C₄×Q₁₆	—	61	2	no	—
80	#366	—	61	1	no	—
81	#358	—	61	2	no	—
82	C₄×C₁₆	—	61	1	no	—
83	C₄×C₁₆	—	61	1	no	—
84	#335	—	61	2	no	—
85	Q₈⋊C₈	—	122	2	no	—
86	C₈²	—	61	1	no	—
87	C₆₁	#12083	1	1	yes	characteristic, complemented, Sylow
88	C₂×Q₁₆	C₂×#6923	1	2	yes	—
89	C₄×Q₈	—	122	2	no	—
90	Q₃₂	—	2	2	no	—
91	C₂×C₁₆	—	61	1	no	—
92	C₂×C₁₆	—	122	1	no	—
93	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
94	#107	—	61	2	no	—
95	C₄⋊C₈	—	122	2	no	—
96	C₄⋊C₈	—	244	2	no	—
97	#103	—	122	2	no	—
98	#97	—	244	2	no	—
99	C₄×C₈	—	61	1	no	—
100	C₄×C₈	—	61	1	no	—
101	C₄×C₈	—	61	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	Q₁₆	—	2	2	no	—
104	Q₁₆	—	2	2	no	—
105	C₂×C₈	—	61	1	no	—
106	C₂×C₈	—	61	1	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	122	1	no	—
109	C₂×C₈	—	244	2	no	—
110	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
111	C₄⋊C₄	—	122	2	no	—
112	C₄⋊C₄	—	244	2	no	—
113	C₄²	—	61	1	no	—
114	C₄²	—	244	2	no	—
115	C₁₆	—	122	2	no	—
116	C₁₆	—	122	1	no	—
117	C₁₆	#11758217	1	2	yes	—
118	Q₈	—	4	2	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	61	1	no	—
122	C₂×C₄	—	61	1	no	—
123	C₂×C₄	—	244	2	no	—
124	C₂×C₄	D₄×#6923	1	1	yes	characteristic
125	C₈	—	61	2	no	—
126	C₈	—	122	1	no	—
127	C₈	—	122	2	no	—
128	C₈	—	122	1	no	—
129	C₈	—	488	2	no	—
130	C₈	C₂×#11758217	1	1	yes	characteristic
131	C₈	#14519864	1	1	yes	characteristic
132	C₂²	D₈×#6923	1	1	yes	characteristic
133	C₄	—	8	2	no	—
134	C₄	—	61	1	no	—
135	C₄	—	61	1	no	—
136	C₄	—	122	1	no	—
137	C₄	—	488	2	no	—
138	C₄	#16840797	1	1	yes	characteristic
139	C₄	#16840804	1	1	yes	characteristic
140	C₂	#21461452	1	1	yes	characteristic
141	C₂	Q₃₂×#6923	1	1	yes	characteristic
142	C₂	#21462220	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461459

C₂

yes

maximal

#21460850

C₂

yes

maximal, characteristic

#21460775

C₂

yes

maximal

C₁₆×#92420

C₂

yes

maximal, characteristic

Q₃₂×Dic₆₁

C₂

yes

maximal, characteristic

#16840813

—

#16840648

C₂²

yes

characteristic

#16840625

C₂²

yes

—

Q₈⋊#92420

—

C₈×#92420

C₂²

yes

characteristic

Q₁₆×Dic₆₁

C₂²

yes

—

#16839588

C₄

yes

characteristic

#16839566

C₄

yes

—

C₁₆×Dic₆₁

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#92420

D₄

yes

characteristic

Q₈×Dic₆₁

—

C₂×#11758185

C₂×C₄

yes

characteristic

#14519711

C₂×C₄

yes

—

#14519701

—

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×Q₁₆×C₆₁

C₂×C₄

yes

—

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×#92420

—

C₂×#92420

—

C₂×#92420

D₈

yes

characteristic

C₂×#92420

D₈

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×D₄

yes

yes

—

C₁₆×C₆₁

M₁₆

yes

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₈

yes

C₂×Dic₆₁

yes

—

—

—

Q₃₂

yes

complemented

C₈×C₆₁

C₂×M₁₆

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

C₄×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×Q₃₂

yes

characteristic

Dic₆₁

#508

yes

characteristic

C₄×Q₃₂

—

#1591

—

#1371

—

#1332

—

C₈×C₁₆

—

C₂×C₆₁

#1589

yes

characteristic

C₂×C₆₁

C₄×Q₃₂

yes

C₂×C₆₁

yes

C₂×Q₃₂

yes

—

122

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

122

—

C₈²

—

C₆₁

#12083

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×#6923

yes

—

C₄×Q₈

—

122

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×#6923

yes

characteristic

#107

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

#103

—

122

—

#97

—

244

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

Q₁₆

—

104

Q₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

122

—

109

C₂×C₈

—

244

—

110

C₂×C₈

C₂²×#6923

yes

characteristic

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

244

—

113

C₄²

—

114

C₄²

—

244

—

115

C₁₆

—

122

—

116

C₁₆

—

122

—

117

C₁₆

#11758217

yes

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

244

—

124

C₂×C₄

D₄×#6923

yes

characteristic

125

C₈

—

126

C₈

—

122

—

127

C₈

—

122

—

128

C₈

—

122

—

129

C₈

—

488

—

130

C₈

C₂×#11758217

yes

characteristic

131

C₈

#14519864

yes

characteristic

132

C₂²

D₈×#6923

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

122

—

137

C₄

—

488

—

138

C₄

#16840797

yes

characteristic

139

C₄

#16840804

yes

characteristic

140

C₂

#21461452

yes

characteristic

141

C₂

Q₃₂×#6923

yes

characteristic

142

C₂

#21462220

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#6923
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₂×Q₃₂×C₆₁	C₄
socle, prime radical	C₂²×C₆₁	C₄×D₈
cosocle	C₈×Dic₆₁	C₂³
hypercenter	C₂×Q₃₂	#6923
hypocenter	C₆₁	#12083
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×#6923

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₂×Q₃₂×C₆₁

C₄

socle, prime radical

C₂²×C₆₁

C₄×D₈

cosocle

C₈×Dic₆₁

C₂³

C₂×Q₃₂

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
56. #92420	Q₃₂	semidirect
87. C₆₁	#12083	semidirect

H ≅ G/N

type

56. #92420

Q₃₂

semidirect

87. C₆₁

#12083

semidirect

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

subgroup	index
#21461459	2
#21460850	2
#21460775	2
C₁₆×#92420	2
Q₃₂×Dic₆₁	2
#12083	61

subgroup

index

#21461459

#21460850

#21460775

C₁₆×#92420

Q₃₂×Dic₆₁

#12083

Subgroups — #30606072

Conjugacy classes of subgroups

Complements

Maximal subgroups