Group #30604067 — subgroups

Subgroups — #30604067

Order 15616.

288 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 98 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	#21462180	C₂	1	2	yes	maximal
2	#21460757	C₂	1	4	yes	maximal
3	#21454754	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11758187	—	2	2	no	—
5	C₂²⋊#11758187	C₂²	1	4	yes	—
6	#16840650	—	2	2	no	—
7	#16839893	—	2	2	no	—
8	Q₈×#92418	—	2	2	no	—
9	#16839789	C₂²	1	2	yes	—
10	#16839786	C₄	1	1	yes	characteristic
11	#16839718	C₄	1	2	yes	—
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	C₆₁×#533	C₄	1	1	yes	characteristic
14	C₂×#11758187	—	2	4	no	—
15	C₂×#11758194	C₂×C₄	1	2	yes	—
16	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
17	C₂²×#92418	C₂³	1	1	yes	characteristic
18	C₄⋊#92418	—	2	2	no	—
19	C₄⋊#92418	—	2	4	no	—
20	#14519688	—	2	2	no	—
21	C₄×#92418	—	2	2	no	—
22	C₄×#92418	D₄	1	2	yes	—
23	C₄×#92418	D₄	1	2	yes	—
24	C₄⋊Q₈×C₆₁	C₈	1	2	yes	—
25	#125×C₆₁	C₈	1	2	yes	—
26	C₂²⋊Q₈×C₆₁	—	2	2	no	—
27	C₄×Q₈×C₆₁	—	2	2	no	—
28	#117×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₆₁	—	2	2	no	—
31	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
32	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₆₁	—	2	4	no	—
34	C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₄⋊C₄×C₆₁	C₂×C₈	1	4	yes	—
36	C₂²⋊C₄×C₆₁	—	2	2	no	—
37	C₄²×C₆₁	—	2	2	no	—
38	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
41	#11758194	—	4	2	no	—
42	C₂×#92418	—	2	2	no	—
43	C₂×#92418	—	2	2	no	—
44	C₂×#92418	—	4	1	no	—
45	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#92418	C₂×D₄	1	2	yes	—
48	#11758187	—	4	4	no	—
49	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
50	Q₈×C₆₁	—	4	4	no	—
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	—	2	4	no	—
54	C₂×C₄×C₆₁	—	4	2	no	—
55	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
59	#92418	—	2	2	no	—
60	#92418	—	2	1	no	—
61	#92418	—	4	2	no	—
62	#92418	#101	1	2	yes	—
63	#11497	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	—	2	2	no	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	—	2	2	no	—
69	C₄×C₆₁	—	2	1	no	—
70	C₄×C₆₁	—	4	4	no	—
71	C₄×C₆₁	#410	1	1	yes	characteristic
72	C₄×C₆₁	C₂×#101	1	1	yes	characteristic
73	#2960	—	61	1	no	—
74	#2163	—	61	2	no	—
75	#1323	—	61	4	no	—
76	C₂×C₆₁	—	4	1	no	—
77	C₂×C₆₁	#1461	1	1	yes	characteristic
78	C₂×C₆₁	#2117	1	2	yes	—
79	#533	#6923	1	1	yes	characteristic
80	#446	—	122	2	no	—
81	C₈×Q₈	—	122	2	no	—
82	#424	—	61	2	no	—
83	#423	—	61	1	no	—
84	#405	—	61	2	no	—
85	C₂×C₄×C₈	—	61	1	no	—
86	C₄⋊C₁₆	—	122	2	no	—
87	C₂²⋊C₁₆	—	61	4	no	—
88	#346	—	122	2	no	—
89	C₆₁	#11497	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₁₆	—	61	2	no	—
91	C₂²×C₈	—	61	1	no	—
92	C₂²×C₈	—	61	1	no	—
93	C₄⋊Q₈	#92420	1	2	yes	—
94	#125	#92420	1	2	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	C₂×#6923	1	2	yes	—
98	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
99	C₂×C₁₆	—	122	4	no	—
100	C₄⋊C₈	—	122	4	no	—
101	C₄⋊C₈	—	122	2	no	—
102	#97	—	122	2	no	—
103	C₄×C₈	—	61	2	no	—
104	C₄×C₈	—	61	2	no	—
105	C₄×C₈	—	122	2	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	#11758217	1	2	yes	—
108	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
109	M₁₆	—	244	2	no	—
110	C₂×C₈	—	61	1	no	—
111	C₂×C₈	—	61	1	no	—
112	C₂×C₈	—	61	2	no	—
113	C₂×C₈	—	122	2	no	—
114	C₂×C₈	—	122	2	no	—
115	C₂×C₈	—	244	1	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×#92420	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂×#92420	1	1	yes	characteristic
122	C₄²	C₂×#92420	1	1	yes	characteristic
123	C₄²	C₂²⋊#6923	1	2	yes	—
124	C₁₆	—	244	4	no	—
125	C₂³	C₂×#11758217	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂²⋊#92420	1	2	yes	—
132	C₂×C₄	C₂²⋊#92420	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
134	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
135	C₈	—	61	2	no	—
136	C₈	—	122	2	no	—
137	C₈	—	122	1	no	—
138	C₈	—	244	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	4	no	—
146	C₄	#16840743	1	1	yes	characteristic
147	C₄	D₆₁⋊#101	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#21461062	1	1	yes	characteristic
150	C₂	#21462061	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21462180

C₂

yes

maximal

#21460757

C₂

yes

maximal

#21454754

C₂

yes

maximal, characteristic

C₄⋊#11758187

—

C₂²⋊#11758187

C₂²

yes

—

—

—

—

C₂²

yes

—

#16839786

C₄

yes

characteristic

#16839718

C₄

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₆₁×#533

C₄

yes

characteristic

C₂×#11758187

—

C₂×#11758194

C₂×C₄

yes

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×#92418

D₄

yes

—

C₄⋊Q₈×C₆₁

C₈

yes

—

#125×C₆₁

C₈

yes

—

C₂²⋊Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

#117×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₈

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₂²⋊C₄

yes

characteristic

C₂×#92418

C₂×D₄

yes

—

#11758187

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

—

—

—

yes

—

#11497

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1461

yes

characteristic

C₂×C₆₁

#2117

yes

—

yes

—

122

—

C₈×Q₈

—

122

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

122

—

C₂²⋊C₁₆

—

#346

—

122

—

C₆₁

#11497

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

#92420

yes

—

#125

#92420

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×#6923

yes

—

C₂×C₄²

C₂×#6923

yes

characteristic

C₂×C₁₆

—

122

—

100

C₄⋊C₈

—

122

—

101

C₄⋊C₈

—

122

—

102

#97

—

122

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

122

—

106

C₂×Q₈

—

107

C₂²×C₄

#11758217

yes

—

108

C₂²×C₄

C₂²×#6923

yes

characteristic

109

M₁₆

—

244

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

122

—

114

C₂×C₈

—

122

—

115

C₂×C₈

—

244

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×#92420

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂×#92420

yes

characteristic

122

C₄²

C₂×#92420

yes

characteristic

123

C₄²

C₂²⋊#6923

yes

—

124

C₁₆

—

244

—

125

C₂³

C₂×#11758217

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊#92420

yes

—

132

C₂×C₄

C₂²⋊#92420

yes

—

133

C₂×C₄

C₂×C₂²⋊#6923

yes

characteristic

134

C₂×C₄

C₂²×#92420

yes

characteristic

135

C₈

—

136

C₈

—

122

—

137

C₈

—

122

—

138

C₈

—

244

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊#92420

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#16840743

yes

characteristic

147

C₄

D₆₁⋊#101

yes

characteristic

148

C₂

—

149

C₂

#21461062

yes

characteristic

150

C₂

#21462061

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#533	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×#92418	C₂³
hypercenter	#533	#6923
hypocenter	C₆₁	#11497
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₆₁×#533

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₈

prime radical

C₂³×C₆₁

C₂×M₁₆

cosocle

C₂²×#92418

C₂³

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
89. C₆₁	#11497	semidirect

H ≅ G/N

type

89. C₆₁

#11497

semidirect

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

subgroup	index
#21462180	2
#21460757	2
#21454754	2
#11497	61

subgroup

index

Subgroups — #30604067

Conjugacy classes of subgroups

Complements

Maximal subgroups