Group #30594488 — subgroups

Subgroups — #30594488

Order 15616.

244 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 58 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	#21461703	C₂	1	2	yes	maximal
2	#21447432	C₂	1	1	yes	maximal, characteristic
3	D₆₁⋊#108	—	2	2	no	—
4	#16840748	—	2	2	no	—
5	#16840631	C₄	1	4	yes	—
6	C₂×#14519783	C₂²	1	1	yes	characteristic
7	C₂×#14519717	C₄	1	1	yes	characteristic
8	#16839541	—	2	2	no	—
9	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic
10	D₆₁⋊M₁₆	—	2	4	no	—
11	#14519848	—	4	2	no	—
12	#14519847	—	4	2	no	—
13	#14519838	—	4	2	no	—
14	#14519837	—	4	2	no	—
15	C₂×#11758222	D₄	1	1	yes	characteristic
16	#14519783	—	4	1	no	—
17	#14519783	—	4	1	no	—
18	C₂×#11758194	Q₈	1	1	yes	characteristic
19	C₂×#11758189	C₂×C₄	1	2	yes	—
20	C₂×C₈×D₆₁	D₄	1	2	yes	—
21	#14519717	—	4	1	no	—
22	C₂×#11758185	—	2	1	no	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	—	2	1	no	—
25	C₂×C₁₆×C₆₁	—	4	1	no	—
26	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₆₁	SD₁₆	1	2	yes	—
28	#11758222	—	4	1	no	—
29	#11758222	—	4	1	no	—
30	C₂×D₂₄₄	Q₁₆	1	1	yes	characteristic
31	C₂×C₄×D₆₁	C₂²⋊C₄	1	2	yes	—
32	C₂×Dic₁₂₂	D₈	1	1	yes	characteristic
33	#11758217	—	4	4	no	—
34	#11758213	—	4	4	no	—
35	D₆₁⋊C₈	—	4	4	no	—
36	C₂×C₈×C₆₁	—	2	1	no	—
37	C₂×C₈×C₆₁	—	2	1	no	—
38	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	#11758194	—	4	1	no	—
41	C₂×#92418	C₄⋊C₄	1	2	yes	—
42	#11758189	—	4	2	no	—
43	C₈×D₆₁	—	2	2	no	—
44	C₈×D₆₁	—	2	2	no	—
45	C₁₆×C₆₁	—	4	1	no	—
46	#11758185	—	4	1	no	—
47	C₂³×C₆₁	#106	1	1	yes	characteristic
48	C₂²×D₆₁	#103	1	2	yes	—
49	C₂×C₄×C₆₁	—	2	1	no	—
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
52	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
53	C₂²⋊D₆₁	—	4	2	no	—
54	C₂×Dic₆₁	#102	1	2	yes	—
55	D₂₄₄	—	4	1	no	—
56	C₄×D₆₁	—	2	2	no	—
57	C₄×D₆₁	—	2	2	no	—
58	Dic₁₂₂	—	4	1	no	—
59	#92420	—	8	4	no	—
60	C₈×C₆₁	—	2	1	no	—
61	C₈×C₆₁	#99	1	1	yes	characteristic
62	C₈×C₆₁	#99	1	1	yes	characteristic
63	#92418	—	2	2	no	—
64	#7229	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	4	1	no	—
67	C₂²×C₆₁	#340	1	1	yes	characteristic
68	C₂×D₆₁	—	2	2	no	—
69	C₂×D₆₁	—	4	2	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	H(3,ℤ/4)	1	1	yes	characteristic
72	C₄×C₆₁	#342	1	1	yes	characteristic
73	Dic₆₁	—	2	2	no	—
74	#2113	—	61	1	no	—
75	#1813	—	61	2	no	—
76	C₂×C₆₁	—	2	1	no	—
77	C₂×C₆₁	—	4	1	no	—
78	C₂×C₆₁	#1289	1	1	yes	characteristic
79	D₆₁	—	4	2	no	—
80	C₂×#131	—	61	1	no	—
81	C₂×M₃₂	—	61	1	no	—
82	C₂²×C₁₆	#6923	1	1	yes	characteristic
83	C₂×#108	—	122	2	no	—
84	#413	—	122	2	no	—
85	C₂²⋊C₁₆	—	122	2	no	—
86	#338	—	61	4	no	—
87	C₆₁	#7229	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	61	1	no	—
89	#131	—	244	1	no	—
90	#131	—	244	1	no	—
91	C₂×M₁₆	—	61	1	no	—
92	C₂×M₁₆	—	61	2	no	—
93	C₂×M₁₆	—	122	4	no	—
94	C₂²×C₈	—	61	2	no	—
95	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
96	M₃₂	—	244	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	4	1	no	—
99	C₂×C₁₆	—	122	1	no	—
100	#108	—	244	2	no	—
101	#108	—	244	2	no	—
102	#101	—	244	2	no	—
103	#100	—	244	2	no	—
104	#41	—	244	1	no	—
105	#41	—	244	1	no	—
106	C₂×Q₈	—	61	1	no	—
107	C₂×D₄	—	61	2	no	—
108	C₂×D₄	—	61	1	no	—
109	C₂²×C₄	—	61	2	no	—
110	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
111	M₁₆	—	244	2	no	—
112	M₁₆	—	244	1	no	—
113	M₁₆	—	244	4	no	—
114	M₁₆	—	244	4	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	61	2	no	—
118	C₂×C₈	—	122	2	no	—
119	C₂×C₈	—	122	2	no	—
120	C₂×C₈	—	244	4	no	—
121	C₂×C₈	C₄×#6923	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
123	C₁₆	—	4	1	no	—
124	C₁₆	—	244	1	no	—
125	C₂³	—	61	2	no	—
126	C₂³	#14519845	1	1	yes	characteristic
127	Q₈	—	244	1	no	—
128	D₄	—	244	1	no	—
129	D₄	—	244	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	61	2	no	—
133	C₂×C₄	—	122	2	no	—
134	C₂×C₄	—	122	2	no	—
135	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#14519846	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	122	2	no	—
139	C₈	—	488	4	no	—
140	C₈	#14519835	1	1	yes	characteristic
141	C₈	#14519836	1	1	yes	characteristic
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	122	2	no	—
145	C₂²	—	244	2	no	—
146	C₂²	#16840636	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	122	2	no	—
149	C₄	#16840630	1	1	yes	characteristic
150	C₄	#16840640	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	1	no	—
153	C₂	—	244	2	no	—
154	C₂	#21460682	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461703

C₂

yes

maximal

#21447432

C₂

yes

maximal, characteristic

D₆₁⋊#108

—

#16840748

—

#16840631

C₄

yes

—

C₂×#14519783

C₂²

yes

characteristic

C₂×#14519717

C₄

yes

characteristic

#16839541

—

C₂²×C₁₆×C₆₁

C₄

yes

characteristic

D₆₁⋊M₁₆

—

—

—

—

—

D₄

yes

—

—

Q₈

yes

characteristic

C₂×#11758189

C₂×C₄

yes

—

C₂×C₈×D₆₁

D₄

yes

—

#14519717

—

C₂×#11758185

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₂²⋊D₆₁

SD₁₆

yes

—

#11758222

—

#11758222

—

C₂×D₂₄₄

Q₁₆

yes

characteristic

C₂×C₄×D₆₁

C₂²⋊C₄

yes

—

C₂×Dic₁₂₂

D₈

yes

characteristic

#11758217

—

#11758213

—

D₆₁⋊C₈

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

#11758194

—

C₂×#92418

C₄⋊C₄

yes

—

#11758189

—

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

#106

yes

characteristic

C₂²×D₆₁

#103

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂²⋊D₆₁

—

C₂×Dic₆₁

#102

yes

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#340

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

#2113

—

#1813

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#1289

yes

characteristic

D₆₁

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

#6923

yes

characteristic

C₂×#108

—

122

—

#413

—

122

—

C₂²⋊C₁₆

—

122

—

#338

—

C₆₁

#7229

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

244

—

#131

—

244

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

characteristic

M₃₂

—

244

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

100

#108

—

244

—

101

#108

—

244

—

102

#101

—

244

—

103

#100

—

244

—

104

#41

—

244

—

105

#41

—

244

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄⋊#6923

yes

characteristic

111

M₁₆

—

244

—

112

M₁₆

—

244

—

113

M₁₆

—

244

—

114

M₁₆

—

244

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

122

—

120

C₂×C₈

—

244

—

121

C₂×C₈

C₄×#6923

yes

characteristic

122

C₂×C₈

C₂²⋊#6923

yes

characteristic

123

C₁₆

—

124

C₁₆

—

244

—

125

C₂³

—

126

C₂³

#14519845

yes

characteristic

127

Q₈

—

244

—

128

D₄

—

244

—

129

D₄

—

244

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

122

—

134

C₂×C₄

—

122

—

135

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

136

C₂×C₄

#14519846

yes

characteristic

137

C₈

—

138

C₈

—

122

—

139

C₈

—

488

—

140

C₈

#14519835

yes

characteristic

141

C₈

#14519836

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

122

—

145

C₂²

—

244

—

146

C₂²

#16840636

yes

characteristic

147

C₄

—

148

C₄

—

122

—

149

C₄

#16840630

yes

characteristic

150

C₄

#16840640

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

244

—

154

C₂

#21460682

yes

characteristic

	subgroup	G/H
center	C₄	#16840630
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂²×C₈	C₂×#6923
Fitting	C₂²×C₁₆×C₆₁	C₄
socle	C₂×C₆₁	#1289
prime radical	C₂×#11758222	D₄
cosocle	C₂×#14519783	C₂²
hypercenter	C₂²×C₁₆	#6923
hypocenter	C₆₁	#7229
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₄

#16840630

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂²×C₈

C₂×#6923

Fitting

C₂²×C₁₆×C₆₁

C₄

C₂×C₆₁

D₄

C₂²

C₂²×C₁₆

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
87. C₆₁	#7229	semidirect

H ≅ G/N

type

87. C₆₁

#7229

semidirect

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

subgroup	index
#21461703	2
#21447432	2
#7229	61

subgroup

index

#21461703

#21447432

#7229

Subgroups — #30594488

Conjugacy classes of subgroups

Complements

Maximal subgroups