Group #30594215 — subgroups

Subgroups — #30594215

Order 15616.

266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 84 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	C₄×#14519687	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11758187	C₄	1	4	yes	complemented
4	C₂²⋊#11758187	—	2	2	no	—
5	#16840650	—	2	2	no	—
6	C₂×#14519687	—	2	2	no	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₄²×Dic₆₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
10	C₂×#11758187	—	2	4	no	—
11	C₂²×#92418	D₄	1	2	yes	—
12	C₂×C₄×Dic₆₁	—	2	1	no	—
13	C₂×C₄×Dic₆₁	—	2	1	no	—
14	C₂×C₄×Dic₆₁	C₈	1	2	yes	—
15	#14519687	—	2	8	no	—
16	C₄×#92418	C₂×C₄	1	2	yes	—
17	C₄×#92418	D₄	1	1	yes	characteristic
18	C₄×#92418	Q₈	1	1	yes	characteristic
19	C₂²×C₈×C₆₁	—	2	1	no	—
20	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₆₁	—	2	1	no	—
22	C₄×C₈×C₆₁	C₈	1	2	yes	—
23	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
26	C₂²×Dic₆₁	—	2	2	no	—
27	C₂×C₈×C₆₁	—	2	4	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₄²×C₆₁	C₄²	1	1	yes	characteristic
30	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
31	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
33	C₄×Dic₆₁	—	2	4	no	—
34	C₄×Dic₆₁	—	4	2	no	—
35	C₄×Dic₆₁	—	4	2	no	—
36	C₂×#92418	—	2	4	no	—
37	C₂×#92418	C₂²⋊C₄	1	2	yes	—
38	C₂×#92418	C₄⋊C₄	1	2	yes	—
39	#11758187	—	4	4	no	—
40	C₂³×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₄⋊C₈	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	4	no	—
51	C₂×Dic₆₁	—	4	2	no	—
52	C₂×Dic₆₁	—	4	2	no	—
53	C₈×C₆₁	—	2	4	no	—
54	#92418	—	2	2	no	—
55	#92418	#104	1	4	yes	—
56	#7149	—	61	1	no	maximal, Sylow
57	C₂²×C₆₁	—	2	1	no	—
58	C₂²×C₆₁	—	2	1	no	—
59	C₂²×C₆₁	#336	1	1	yes	characteristic
60	C₂²×C₆₁	#349	1	2	yes	—
61	C₄×C₆₁	—	2	1	no	—
62	C₄×C₆₁	—	2	1	no	—
63	C₄×C₆₁	—	2	1	no	—
64	C₄×C₆₁	—	2	1	no	—
65	C₄×C₆₁	#339	1	2	yes	—
66	C₄×C₆₁	#347	1	2	yes	—
67	Dic₆₁	—	4	4	no	—
68	#2163	—	61	2	no	—
69	C₄×#97	—	61	1	no	—
70	C₂×C₆₁	—	2	2	no	—
71	C₂×C₆₁	#1276	1	1	yes	characteristic
72	C₂×C₆₁	#1375	1	1	yes	characteristic
73	C₂×C₆₁	#1377	1	1	yes	characteristic
74	C₂×#97	—	122	2	no	—
75	C₂×C₄×C₈	—	61	1	no	—
76	C₂×C₄×C₈	#6923	1	1	yes	characteristic
77	C₄³	—	61	1	no	—
78	C₄⋊C₁₆	—	61	4	no	—
79	C₂²⋊C₁₆	—	122	2	no	—
80	#346	—	122	2	no	—
81	C₆₁	#7149	1	1	yes	characteristic, complemented, Sylow
82	C₂²×C₈	—	2	1	no	—
83	C₂²×C₈	—	61	2	no	—
84	C₂×C₄²	—	61	2	no	—
85	C₂×C₄²	—	122	1	no	—
86	C₂×C₄²	—	122	1	no	—
87	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
88	C₂×C₁₆	—	122	4	no	—
89	#97	—	122	8	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	—	61	2	no	—
92	C₄×C₈	—	61	1	no	—
93	C₄×C₈	—	61	1	no	—
94	C₄×C₈	#92420	1	2	yes	—
95	C₂²×C₄	—	122	2	no	—
96	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
97	C₂²×C₄	#11758213	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	61	2	no	—
102	C₂×C₈	—	61	2	no	—
103	C₂×C₈	—	122	4	no	—
104	C₄²	—	122	4	no	—
105	C₄²	—	244	2	no	—
106	C₄²	—	244	2	no	—
107	C₄²	C₄×#6923	1	1	yes	characteristic
108	C₄²	C₄⋊#6923	1	1	yes	characteristic
109	C₄²	C₂×#92420	1	1	yes	characteristic
110	C₄²	#11758217	1	1	yes	characteristic
111	C₁₆	—	244	4	no	—
112	C₂³	#14519834	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	122	4	no	—
117	C₂×C₄	—	244	2	no	—
118	C₂×C₄	—	244	2	no	—
119	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	C₄×#92420	1	1	yes	characteristic
121	C₂×C₄	#14519832	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊#92420	1	1	yes	characteristic
123	C₂×C₄	C₄⋊#92420	1	1	yes	characteristic
124	C₂×C₄	#14519844	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	61	4	no	—
127	C₈	—	122	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#16840628	1	1	yes	characteristic
131	C₂²	#16840659	1	2	yes	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	244	4	no	—
137	C₄	#16840633	1	2	yes	—
138	C₄	#16840655	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#21460647	1	1	yes	characteristic
141	C₂	#21460861	1	1	yes	characteristic
142	C₂	#21460867	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462180

C₂

yes

maximal

C₄×#14519687

C₂

yes

maximal, characteristic

C₄⋊#11758187

C₄

yes

—

—

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₄²×Dic₆₁

C₄

yes

characteristic

C₂×C₄×C₈×C₆₁

C₄

yes

characteristic

C₂×#11758187

—

C₂²×#92418

D₄

yes

—

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

—

C₂×C₄×Dic₆₁

C₈

yes

—

#14519687

—

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

D₄

yes

characteristic

C₄×#92418

Q₈

yes

characteristic

C₂²×C₈×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₈

yes

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₆₁

M₁₆

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄²×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

characteristic

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

C₂²⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

#11758187

—

C₂³×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#95

yes

characteristic

C₂×C₄×C₆₁

C₄×C₈

yes

characteristic

C₂×C₄×C₆₁

#97

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₈

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

#92418

—

#92418

#104

yes

—

#7149

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#336

yes

characteristic

C₂²×C₆₁

#349

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#339

yes

—

C₄×C₆₁

#347

yes

—

Dic₆₁

—

#2163

—

C₄×#97

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×#97

—

122

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6923

yes

characteristic

C₄³

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

122

—

#346

—

122

—

C₆₁

#7149

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

122

—

C₂×C₄²

—

122

—

C₂×C₄²

C₂×#6923

yes

characteristic

C₂×C₁₆

—

122

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#92420

yes

—

C₂²×C₄

—

122

—

C₂²×C₄

D₆₁⋊C₈

yes

characteristic

C₂²×C₄

#11758213

yes

characteristic

C₂²×C₄

C₂²⋊#6923

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

122

—

104

C₄²

—

122

—

105

C₄²

—

244

—

106

C₄²

—

244

—

107

C₄²

C₄×#6923

yes

characteristic

108

C₄²

C₄⋊#6923

yes

characteristic

109

C₄²

C₂×#92420

yes

characteristic

110

C₄²

#11758217

yes

characteristic

111

C₁₆

—

244

—

112

C₂³

#14519834

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

122

—

117

C₂×C₄

—

244

—

118

C₂×C₄

—

244

—

119

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

120

C₂×C₄

C₄×#92420

yes

characteristic

121

C₂×C₄

#14519832

yes

characteristic

122

C₂×C₄

C₂²⋊#92420

yes

characteristic

123

C₂×C₄

C₄⋊#92420

yes

characteristic

124

C₂×C₄

#14519844

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

122

—

128

C₂²

—

129

C₂²

—

130

C₂²

#16840628

yes

characteristic

131

C₂²

#16840659

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

244

—

137

C₄

#16840633

yes

—

138

C₄

#16840655

yes

—

139

C₂

—

140

C₂

#21460647

yes

characteristic

141

C₂

#21460861

yes

characteristic

142

C₂

#21460867

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#6923)⋊C₄
commutator	C₂×C₄×C₆₁	C₄×C₈
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle	C₂²×C₆₁	#336
prime radical	C₂³×C₆₁	C₂²⋊C₈
cosocle	C₂×C₄×#92418	C₂²
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#7149
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

(C₂×#6923)⋊C₄

commutator

C₂×C₄×C₆₁

C₄×C₈

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₂×C₄×C₈×C₆₁

C₄

socle

C₂²×C₆₁

#336

prime radical

C₂³×C₆₁

C₂²⋊C₈

cosocle

C₂×C₄×#92418

C₂²

C₂×C₄×C₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. C₄⋊#11758187	C₄	semidirect
81. C₆₁	#7149	semidirect

H ≅ G/N

type

3. C₄⋊#11758187

C₄

semidirect

81. C₆₁

#7149

semidirect

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

subgroup	index
#21462180	2
C₄×#14519687	2
#7149	61

subgroup

index

#21462180

C₄×#14519687

#7149

Subgroups — #30594215

Conjugacy classes of subgroups

Complements

Maximal subgroups