Group #21454538 — subgroups

Subgroups — #21454538

Order 7808.

484 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂×#14519783	C₂	1	2	yes	maximal, complemented
2	#16840312	C₂	1	1	yes	maximal, characteristic
3	#16839835	C₂	1	8	yes	maximal, complemented
4	#16839786	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839728	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11758222	C₂²	1	2	yes	—
8	#14519783	—	2	8	no	—
9	C₂×#11758194	C₂²	1	2	yes	—
10	C₂×#11758189	C₂²	1	4	yes	—
11	C₂×C₈×D₆₁	C₂²	1	4	yes	—
12	#14519777	C₄	1	1	yes	characteristic
13	#14519774	C₄	1	1	yes	characteristic
14	#14519765	C₄	1	2	yes	—
15	#14519751	C₄	1	4	yes	—
16	C₄×D₂₄₄	C₂²	1	4	yes	—
17	C₄⋊#92418	C₂²	1	4	yes	—
18	#14519689	C₂²	1	8	yes	—
19	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	#97×C₆₁	C₂²	1	4	yes	—
23	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₆₁	C₂×C₄	1	4	yes	—
26	#11758222	—	2	4	no	—
27	#11758222	—	2	4	no	—
28	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
29	C₂×D₂₄₄	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₆₁	C₂³	1	4	yes	—
31	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	D₄	1	4	yes	complemented
34	C₂×C₈×C₆₁	C₂³	1	2	yes	—
35	C₂×C₈×C₆₁	C₂³	1	2	yes	—
36	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₆₁	C₂³	1	2	yes	—
39	#11758198	C₂×C₄	1	8	yes	—
40	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
41	#11758194	—	2	4	no	—
42	C₂×#92418	C₂³	1	4	yes	—
43	#11758189	—	2	8	no	—
44	C₈×D₆₁	—	2	8	no	—
45	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₆₁	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₆₁	—	4	2	no	—
48	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₆₁	#41	1	2	yes	—
53	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₆₁	—	2	8	no	—
55	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
56	D₂₄₄	—	2	4	no	—
57	D₂₄₄	—	4	4	no	—
58	C₄×D₆₁	—	2	4	no	—
59	C₄×D₆₁	—	2	4	no	—
60	Dic₁₂₂	—	2	4	no	—
61	C₈×C₆₁	—	2	2	no	—
62	C₈×C₆₁	C₂×D₄	1	4	yes	—
63	#92418	—	2	4	no	—
64	C₂²×C₆₁	—	2	2	no	—
65	C₂²×C₆₁	C₄×D₄	1	2	yes	—
66	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₆₁	—	2	4	no	—
68	C₂×D₆₁	—	2	8	no	—
69	C₄×C₆₁	—	2	2	no	—
70	C₄×C₆₁	—	2	2	no	—
71	C₄×C₆₁	C₄×D₄	1	2	yes	—
72	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
74	Dic₆₁	—	2	4	no	—
75	#2932	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	2	2	no	—
77	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₆₁	#568	1	2	yes	—
79	D₆₁	—	4	4	no	—
80	C₂×#131	—	61	2	no	—
81	#532	—	61	1	no	—
82	#435	—	61	8	no	—
83	#423	—	61	1	no	—
84	#408	—	61	2	no	—
85	C₂×#97	D₆₁	1	1	yes	characteristic, complemented
86	C₆₁	#2932	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	61	2	no	—
88	#131	—	122	8	no	—
89	C₂×M₁₆	—	61	2	no	—
90	C₂×M₁₆	—	61	4	no	—
91	C₂²×C₈	—	61	4	no	—
92	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
94	C₄⋊Q₈	—	61	1	no	—
95	#127	—	61	1	no	—
96	#124	—	61	2	no	—
97	#121	—	61	4	no	—
98	C₄×D₄	—	61	4	no	—
99	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
100	C₄⋊C₈	—	61	4	no	—
101	C₂²⋊C₈	—	61	8	no	—
102	#97	C₂×D₆₁	1	4	yes	—
103	#41	—	122	4	no	—
104	#41	—	122	4	no	—
105	C₂×Q₈	—	61	2	no	—
106	C₂×D₄	—	61	4	no	—
107	C₂×D₄	—	61	2	no	—
108	C₂×D₄	—	61	4	no	—
109	C₂²×C₄	—	61	4	no	—
110	C₂²×C₄	C₄×D₆₁	1	2	yes	—
111	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
112	M₁₆	—	122	4	no	—
113	M₁₆	—	122	8	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	61	4	no	—
116	C₂×C₈	—	122	8	no	—
117	C₂×C₈	D₂₄₄	1	4	yes	complemented
118	C₂×C₈	C₂²×D₆₁	1	2	yes	—
119	C₂×C₈	C₂²×D₆₁	1	2	yes	—
120	C₄⋊C₄	—	61	4	no	—
121	C₂²⋊C₄	—	61	8	no	—
122	C₄²	C₄×D₆₁	1	1	yes	characteristic
123	C₄²	C₄×D₆₁	1	1	yes	characteristic
124	C₄²	C₂²×D₆₁	1	2	yes	—
125	C₂³	—	61	4	no	—
126	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
127	Q₈	—	122	4	no	—
128	D₄	—	122	8	no	—
129	D₄	—	122	4	no	—
130	D₄	—	244	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	61	4	no	—
133	C₂×C₄	—	122	4	no	—
134	C₂×C₄	—	122	4	no	—
135	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
139	C₂×C₄	#11758222	1	2	yes	—
140	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
141	C₈	—	2	2	no	—
142	C₈	—	122	4	no	—
143	C₈	C₂×D₂₄₄	1	4	yes	—
144	C₂²	—	2	2	no	—
145	C₂²	—	122	4	no	—
146	C₂²	—	122	8	no	—
147	C₂²	C₄×D₂₄₄	1	2	yes	—
148	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	122	4	no	—
152	C₄	C₄×D₂₄₄	1	2	yes	—
153	C₄	C₂²×D₂₄₄	1	1	yes	characteristic
154	C₄	C₂×#11758222	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	244	4	no	—
157	C₂	C₂×C₄×D₂₄₄	1	1	yes	characteristic
158	C₂	#16840528	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519783

C₂

yes

maximal, complemented

#16840312

C₂

yes

maximal, characteristic

#16839835

C₂

yes

maximal, complemented

#16839786

C₂

yes

maximal, characteristic, complemented

#16839728

C₂

yes

maximal, complemented

C₂×#97×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

C₂²

yes

—

#14519783

—

C₂×#11758194

C₂²

yes

—

C₂×#11758189

C₂²

yes

—

C₂×C₈×D₆₁

C₂²

yes

—

#14519777

C₄

yes

characteristic

#14519774

C₄

yes

characteristic

#14519765

C₄

yes

—

#14519751

C₄

yes

—

C₄×D₂₄₄

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519689

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#97×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₂²⋊D₆₁

C₂×C₄

yes

—

#11758222

—

#11758222

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

#11758198

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758194

—

C₂×#92418

C₂³

yes

—

#11758189

—

C₈×D₆₁

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂²×D₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

C₂²×C₄

yes

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×D₄

yes

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

#2932

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×D₄

yes

characteristic

C₂×C₆₁

#568

yes

—

D₆₁

—

—

—

—

—

—

C₂×#97

D₆₁

yes

characteristic, complemented

C₆₁

#2932

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

122

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

100

C₄⋊C₈

—

101

C₂²⋊C₈

—

102

#97

C₂×D₆₁

yes

—

103

#41

—

122

—

104

#41

—

122

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₆₁

yes

—

111

C₂²×C₄

C₂²×D₆₁

yes

characteristic

112

M₁₆

—

122

—

113

M₁₆

—

122

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

122

—

117

C₂×C₈

D₂₄₄

yes

complemented

118

C₂×C₈

C₂²×D₆₁

yes

—

119

C₂×C₈

C₂²×D₆₁

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₆₁

yes

characteristic

123

C₄²

C₄×D₆₁

yes

characteristic

124

C₄²

C₂²×D₆₁

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₆₁

yes

characteristic

127

Q₈

—

122

—

128

D₄

—

122

—

129

D₄

—

122

—

130

D₄

—

244

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

122

—

134

C₂×C₄

—

122

—

135

C₂×C₄

C₂×C₄×D₆₁

yes

—

136

C₂×C₄

C₂×C₄×D₆₁

yes

—

137

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

138

C₂×C₄

C₂×D₂₄₄

yes

—

139

C₂×C₄

#11758222

yes

—

140

C₂×C₄

C₂³×D₆₁

yes

characteristic

141

C₈

—

142

C₈

—

122

—

143

C₈

C₂×D₂₄₄

yes

—

144

C₂²

—

145

C₂²

—

122

—

146

C₂²

—

122

—

147

C₂²

C₄×D₂₄₄

yes

—

148

C₂²

C₂²×C₄×D₆₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

122

—

152

C₄

C₄×D₂₄₄

yes

—

153

C₄

C₂²×D₂₄₄

yes

characteristic

154

C₄

C₂×#11758222

yes

characteristic

155

C₂

—

156

C₂

—

244

—

157

C₂

C₂×C₄×D₂₄₄

yes

characteristic

158

C₂

#16840528

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₂×#97×C₆₁	C₂
hypercenter	C₂×#97	D₆₁
hypocenter	C₆₁	#2932
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₂×#97×C₆₁

C₂

hypercenter

C₂×#97

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
1. C₂×#14519783	C₂	semidirect
3. #16839835	C₂	semidirect
4. #16839786	C₂	semidirect
5. #16839728	C₂	semidirect
6. C₂×#97×C₆₁	C₂	semidirect
33. C₂×C₈×C₆₁	D₄	semidirect
85. C₂×#97	D₆₁	semidirect
86. C₆₁	#2932	semidirect
117. C₂×C₈	D₂₄₄	semidirect

H ≅ G/N

type

1. C₂×#14519783

C₂

semidirect

3. #16839835

C₂

semidirect

4. #16839786

C₂

semidirect

5. #16839728

C₂

semidirect

6. C₂×#97×C₆₁

C₂

semidirect

33. C₂×C₈×C₆₁

D₄

semidirect

85. C₂×#97

D₆₁

semidirect

86. C₆₁

#2932

semidirect

117. C₂×C₈

D₂₄₄

semidirect

Subgroups — #21454538

Conjugacy classes of subgroups

Complements