Group #29567227 — subgroups

Subgroups — #29567227

Order 15616.

298 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₄×#14519696	C₂	1	2	yes	maximal
2	C₆₁×#1985	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519696	—	2	2	no	—
4	C₄×#11758194	—	2	2	no	—
5	#16839531	C₄	1	4	yes	—
6	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₆₁	C₄	1	1	yes	characteristic
8	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
9	C₆₁×#336	—	2	2	no	—
10	C₂×#11758194	—	2	4	no	—
11	#14519696	—	2	8	no	—
12	#14519688	—	4	2	no	—
13	C₄×#92418	—	4	2	no	—
14	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₆₁	—	2	1	no	—
16	C₂²×C₈×C₆₁	—	2	1	no	—
17	C₄×Q₈×C₆₁	—	4	1	no	—
18	C₄×Q₈×C₆₁	—	4	1	no	—
19	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₁	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₆₁	D₄	1	2	yes	—
23	C₄⋊C₈×C₆₁	—	4	1	no	—
24	#97×C₆₁	—	4	1	no	—
25	C₂×Q₈×C₆₁	—	2	4	no	—
26	C₂×Q₈×C₆₁	—	4	1	no	—
27	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	—	4	1	no	—
34	C₂×C₈×C₆₁	—	4	1	no	—
35	C₂×C₈×C₆₁	—	4	2	no	—
36	C₂×C₈×C₆₁	—	4	1	no	—
37	C₄⋊C₄×C₆₁	—	4	2	no	—
38	C₄⋊C₄×C₆₁	—	4	1	no	—
39	C₄²×C₆₁	—	2	1	no	—
40	C₄²×C₆₁	—	2	1	no	—
41	C₄²×C₆₁	—	2	2	no	—
42	C₄²×C₆₁	—	2	2	no	—
43	#11758194	—	2	8	no	—
44	#11758194	—	4	4	no	—
45	C₂×#92418	—	4	4	no	—
46	C₂³×C₆₁	#95	1	1	yes	characteristic
47	Q₈×C₆₁	—	4	4	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	—	2	2	no	—
50	C₂×C₄×C₆₁	—	2	1	no	—
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	2	2	no	—
53	C₂×C₄×C₆₁	—	2	1	no	—
54	C₂×C₄×C₆₁	—	2	2	no	—
55	C₂×C₄×C₆₁	—	2	1	no	—
56	C₂×C₄×C₆₁	—	4	1	no	—
57	C₂×C₄×C₆₁	—	4	2	no	—
58	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
59	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#104	1	4	yes	—
61	C₈×C₆₁	—	4	2	no	—
62	C₈×C₆₁	—	8	1	no	—
63	#92418	—	4	8	no	—
64	#7284	—	61	1	no	maximal, Sylow
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	2	1	no	—
67	C₂²×C₆₁	#339	1	2	yes	—
68	C₂²×C₆₁	#342	1	1	yes	characteristic
69	C₄×C₆₁	—	2	4	no	—
70	C₄×C₆₁	—	2	1	no	—
71	C₄×C₆₁	—	4	1	no	—
72	C₄×C₆₁	—	4	1	no	—
73	C₄×C₆₁	—	4	2	no	—
74	C₄×C₆₁	#355	1	1	yes	characteristic
75	C₄×C₆₁	#356	1	1	yes	characteristic
76	#1985	D₆₁	1	1	yes	characteristic
77	C₄×#101	—	61	2	no	—
78	C₂×C₆₁	—	2	2	no	—
79	C₂×C₆₁	#1297	1	1	yes	characteristic
80	C₂×C₆₁	#1392	1	1	yes	characteristic
81	C₂×C₆₁	#1394	1	1	yes	characteristic
82	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
83	C₂×C₄⋊C₈	Dic₆₁	1	1	yes	characteristic
84	C₂×#101	—	122	2	no	—
85	C₄×M₁₆	—	122	2	no	—
86	C₂×#97	Dic₆₁	1	1	yes	characteristic
87	#343	—	61	4	no	—
88	#336	—	2	2	no	—
89	C₆₁	#7284	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	D₂₄₄	1	1	yes	characteristic
91	C₂×M₁₆	—	122	4	no	—
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₄×Q₈	—	4	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	C₂×C₄⋊C₄	Dic₁₂₂	1	1	yes	characteristic
97	C₂×C₄⋊C₄	C₄×D₆₁	1	2	yes	—
98	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
100	C₄⋊C₈	—	4	1	no	—
101	#101	—	122	8	no	—
102	#97	—	4	1	no	—
103	#97	—	244	2	no	—
104	C₄×C₈	—	244	2	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	1	no	—
107	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
108	C₂²×C₄	#11758196	1	2	yes	—
109	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
110	C₂²×C₄	#11758198	1	2	yes	—
111	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
112	M₁₆	—	122	8	no	—
113	M₁₆	—	244	4	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	4	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	244	4	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	1	no	—
126	C₂³	#14519684	1	1	yes	characteristic
127	Q₈	—	4	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	#14519692	1	1	yes	characteristic
139	C₂×C₄	#14519692	1	1	yes	characteristic
140	C₂×C₄	#14519703	1	4	yes	—
141	C₈	—	4	2	no	—
142	C₈	—	8	1	no	—
143	C₈	—	244	8	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	#16839523	1	2	yes	—
147	C₂²	#16839529	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	4	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	2	no	—
152	C₄	—	4	1	no	—
153	C₄	#16839559	1	1	yes	characteristic
154	C₄	#16839561	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#21443704	1	1	yes	characteristic
157	C₂	#21443942	1	1	yes	characteristic
158	C₂	#21443946	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519696

C₂

yes

maximal

C₆₁×#1985

C₂

yes

maximal, characteristic

C₂×#14519696

—

C₄×#11758194

—

#16839531

C₄

yes

—

C₂×C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₆₁

C₄

yes

characteristic

C₂×#97×C₆₁

C₄

yes

characteristic

C₆₁×#336

—

—

—

—

—

C₂²×Q₈×C₆₁

D₄

yes

characteristic

C₂²×C₈×C₆₁

—

C₂²×C₈×C₆₁

—

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₁

Q₈

yes

characteristic

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

—

C₄⋊C₈×C₆₁

—

#97×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₄²

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

#11758194

—

#11758194

—

C₂×#92418

—

C₂³×C₆₁

#95

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#7284

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#339

yes

—

C₂²×C₆₁

#342

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

D₆₁

yes

characteristic

C₄×#101

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄×Q₈

C₂×D₆₁

yes

characteristic

C₂×C₄⋊C₈

Dic₆₁

yes

characteristic

C₂×#101

—

122

—

C₄×M₁₆

—

122

—

C₂×#97

Dic₆₁

yes

characteristic

#343

—

#336

—

C₆₁

#7284

yes

characteristic, complemented, Sylow

C₂²×Q₈

D₂₄₄

yes

characteristic

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₂₂

yes

characteristic

C₂×C₄⋊C₄

C₄×D₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

C₂×C₄²

C₂²⋊D₆₁

yes

—

100

C₄⋊C₈

—

101

#101

—

122

—

102

#97

—

103

#97

—

244

—

104

C₄×C₈

—

244

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂²×C₄

C₄×Dic₆₁

yes

characteristic

108

C₂²×C₄

#11758196

yes

—

109

C₂²×C₄

C₄⋊Dic₆₁

yes

characteristic

110

C₂²×C₄

#11758198

yes

—

111

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

112

M₁₆

—

122

—

113

M₁₆

—

244

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

244

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₂³

#14519684

yes

characteristic

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#14519692

yes

characteristic

139

C₂×C₄

#14519692

yes

characteristic

140

C₂×C₄

#14519703

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

244

—

144

C₂²

—

145

C₂²

—

146

C₂²

#16839523

yes

—

147

C₂²

#16839529

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#16839559

yes

characteristic

154

C₄

#16839561

yes

characteristic

155

C₂

—

156

C₂

#21443704

yes

characteristic

157

C₂

#21443942

yes

characteristic

158

C₂

#21443946

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519692
commutator	C₂²×C₄×C₆₁	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₆₁
Fitting	C₆₁×#1985	C₂
socle	C₂²×C₆₁	#342
prime radical	C₂³×C₆₁	#95
cosocle	C₂×C₄×Q₈×C₆₁	C₂²
hypercenter	#1985	D₆₁
hypocenter	C₆₁	#7284
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

subgroup

G/H

center

C₂×C₄

#14519692

commutator

C₂²×C₄×C₆₁

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₆₁

Fitting

C₆₁×#1985

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄×Q₈×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂³×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

N	H ≅ G/N	type
89. C₆₁	#7284	semidirect

H ≅ G/N

type

89. C₆₁

#7284

semidirect

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

subgroup	index
C₄×#14519696	2
C₆₁×#1985	2
#7284	61

subgroup

index

C₄×#14519696

C₆₁×#1985

#7284

Subgroups — #29567227

Conjugacy classes of subgroups

Complements

Maximal subgroups