Group #21454429 — subgroups

Subgroups — #21454429

Order 7808.

440 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 209 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	#16840528	C₂	1	2	yes	maximal
2	C₂×#14519785	C₂	1	1	yes	maximal, characteristic
3	#16839811	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839809	C₂	1	4	yes	maximal, complemented
5	C₂×#14519712	C₂	1	2	yes	maximal, complemented
6	#108×D₆₁	C₂	1	4	yes	maximal, complemented
7	C₆₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758222	C₂²	1	1	yes	characteristic
9	#14519785	—	2	4	no	—
10	#14519785	C₄	1	8	yes	—
11	#14519783	—	2	4	no	—
12	C₂×#11758194	C₂²	1	2	yes	—
13	C₂×#11758189	C₂²	1	2	yes	—
14	M₁₆×D₆₁	C₂²	1	8	yes	—
15	C₂²×#92418	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₆₁	C₂²	1	2	yes	—
17	#14519713	C₂²	1	4	yes	—
18	#14519712	C₂²	1	4	yes	—
19	#14519712	C₂²	1	4	yes	—
20	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
21	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
22	#108×C₆₁	C₂²	1	4	yes	—
23	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₆₁	Q₈	1	2	yes	—
25	#11758222	C₂×C₄	1	4	yes	—
26	#11758222	D₄	1	4	yes	—
27	C₂×D₂₄₄	Q₈	1	1	yes	characteristic
28	C₂×C₄×D₆₁	C₂³	1	2	yes	—
29	C₂×Dic₁₂₂	Q₈	1	1	yes	characteristic
30	M₁₆×C₆₁	—	2	2	no	—
31	M₁₆×C₆₁	C₂×C₄	1	4	yes	—
32	M₁₆×C₆₁	C₂³	1	4	yes	—
33	C₂×C₈×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	C₂³	1	2	yes	—
35	#11758194	—	2	2	no	—
36	#11758194	C₂³	1	4	yes	—
37	C₂×#92418	—	2	2	no	—
38	C₂×#92418	C₂×C₄	1	4	yes	—
39	C₂×#92418	C₂³	1	1	yes	characteristic
40	C₂×#92418	C₂³	1	1	yes	characteristic
41	#11758189	—	2	8	no	—
42	#11758189	C₂×C₄	1	8	yes	—
43	C₈×D₆₁	—	2	8	no	—
44	C₈×D₆₁	C₂×C₄	1	8	yes	—
45	C₂³×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×D₆₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
51	C₂²⋊D₆₁	C₄⋊C₄	1	8	yes	—
52	C₂×Dic₆₁	C₂×Q₈	1	2	yes	—
53	D₂₄₄	C₄⋊C₄	1	4	yes	—
54	C₄×D₆₁	C₂²×C₄	1	4	yes	—
55	C₄×D₆₁	C₂×D₄	1	4	yes	—
56	Dic₁₂₂	C₄⋊C₄	1	4	yes	—
57	C₈×C₆₁	—	2	4	no	—
58	C₈×C₆₁	C₂²×C₄	1	4	yes	—
59	#92418	—	2	4	no	—
60	#92418	C₂²×C₄	1	2	yes	—
61	#92418	C₂²×C₄	1	2	yes	—
62	C₂²×C₆₁	—	4	1	no	—
63	C₂²×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
65	C₂×D₆₁	—	4	2	no	—
66	C₂×D₆₁	C₂×C₄⋊C₄	1	4	yes	—
67	C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
68	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
70	Dic₆₁	C₂×C₄⋊C₄	1	4	yes	—
71	#2916	—	61	1	no	maximal, Sylow
72	C₂×C₆₁	—	2	1	no	—
73	C₂×C₆₁	—	2	2	no	—
74	C₂×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	D₆₁	—	2	4	no	—
76	#568	—	61	2	no	—
77	C₂×#131	—	61	1	no	—
78	#430	—	61	1	no	—
79	#430	—	61	4	no	—
80	#430	D₆₁	1	1	yes	characteristic, complemented
81	C₂×#108	—	61	2	no	—
82	C₂×#108	—	61	4	no	—
83	C₆₁	#2916	1	1	yes	characteristic, complemented, Sylow
84	C₂×#41	—	61	1	no	—
85	#131	—	61	8	no	—
86	#131	—	122	4	no	—
87	#131	—	122	4	no	—
88	C₂×M₁₆	—	61	2	no	—
89	C₂×M₁₆	—	61	2	no	—
90	C₂×M₁₆	—	61	8	no	—
91	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
92	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
93	C₂²×C₈	—	61	1	no	—
94	C₂²×C₈	—	61	2	no	—
95	#108	—	61	4	no	—
96	#108	—	61	4	no	—
97	#108	—	61	4	no	—
98	#108	C₂×D₆₁	1	4	yes	—
99	#41	—	61	4	no	—
100	#41	—	61	4	no	—
101	C₂×Q₈	—	61	1	no	—
102	C₂×D₄	—	61	2	no	—
103	C₂×D₄	—	61	1	no	—
104	C₂²×C₄	—	61	2	no	—
105	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
106	M₁₆	—	2	2	no	—
107	M₁₆	—	61	8	no	—
108	M₁₆	—	61	4	no	—
109	M₁₆	—	122	8	no	—
110	M₁₆	—	122	2	no	—
111	M₁₆	C₄×D₆₁	1	4	yes	—
112	M₁₆	C₂²×D₆₁	1	4	yes	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	61	4	no	—
115	C₂×C₈	—	61	1	no	—
116	C₂×C₈	—	61	1	no	—
117	C₂×C₈	—	61	8	no	—
118	C₂×C₈	—	122	8	no	—
119	C₂×C₈	—	122	2	no	—
120	C₂×C₈	C₂²×D₆₁	1	2	yes	—
121	C₂³	—	61	2	no	—
122	C₂³	Q₈×D₆₁	1	1	yes	characteristic
123	Q₈	—	61	4	no	—
124	D₄	—	61	8	no	—
125	D₄	—	61	4	no	—
126	C₂×C₄	—	61	2	no	—
127	C₂×C₄	—	61	4	no	—
128	C₂×C₄	—	61	4	no	—
129	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
130	C₂×C₄	D₄×D₆₁	1	2	yes	—
131	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
132	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
133	C₈	—	2	4	no	—
134	C₈	—	61	2	no	—
135	C₈	—	61	2	no	—
136	C₈	—	122	4	no	—
137	C₈	C₂×C₄×D₆₁	1	4	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	—	61	4	no	—
140	C₂²	—	244	2	no	—
141	C₂²	C₄⋊C₄×D₆₁	1	2	yes	—
142	C₂²	C₂×Q₈×D₆₁	1	1	yes	characteristic
143	C₄	—	61	4	no	—
144	C₄	C₄⋊C₄×D₆₁	1	2	yes	—
145	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
146	C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	2	no	—
149	C₂	—	122	4	no	—
150	C₂	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840528

C₂

yes

maximal

C₂×#14519785

C₂

yes

maximal, characteristic

#16839811

C₂

yes

maximal, characteristic, complemented

#16839809

C₂

yes

maximal, complemented

C₂×#14519712

C₂

yes

maximal, complemented

#108×D₆₁

C₂

yes

maximal, complemented

C₆₁×#430

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

C₂²

yes

characteristic

#14519785

—

#14519785

C₄

yes

—

#14519783

—

C₂×#11758194

C₂²

yes

—

C₂×#11758189

C₂²

yes

—

M₁₆×D₆₁

C₂²

yes

—

C₂²×#92418

C₂²

yes

characteristic

C₂×C₈×D₆₁

C₂²

yes

—

#14519713

C₂²

yes

—

#14519712

C₂²

yes

—

#14519712

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

characteristic

#108×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₂²⋊D₆₁

Q₈

yes

—

#11758222

C₂×C₄

yes

—

#11758222

D₄

yes

—

C₂×D₂₄₄

Q₈

yes

characteristic

C₂×C₄×D₆₁

C₂³

yes

—

C₂×Dic₁₂₂

Q₈

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

C₂×C₄

yes

—

M₁₆×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

#11758194

—

#11758194

C₂³

yes

—

C₂×#92418

—

C₂×#92418

C₂×C₄

yes

—

C₂×#92418

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

characteristic

#11758189

—

#11758189

C₂×C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

C₂×C₄

yes

—

C₂³×C₆₁

C₂×Q₈

yes

characteristic

C₂²×D₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂²⋊D₆₁

C₄⋊C₄

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

—

D₂₄₄

C₄⋊C₄

yes

—

C₄×D₆₁

C₂²×C₄

yes

—

C₄×D₆₁

C₂×D₄

yes

—

Dic₁₂₂

C₄⋊C₄

yes

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₄

yes

—

#92418

—

#92418

C₂²×C₄

yes

—

#92418

C₂²×C₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄⋊C₄

yes

—

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

C₂×C₄⋊C₄

yes

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₄×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

C₂²×D₄

yes

characteristic

Dic₆₁

C₂×C₄⋊C₄

yes

—

#2916

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂²×C₄⋊C₄

yes

characteristic

D₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

C₂×#108

—

C₂×#108

—

C₆₁

#2916

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

122

—

#131

—

122

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

—

C₂×M₁₆

C₂×D₆₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

—

—

—

C₂×D₆₁

yes

—

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₆₁

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

122

—

110

M₁₆

—

122

—

111

M₁₆

C₄×D₆₁

yes

—

112

M₁₆

C₂²×D₆₁

yes

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

122

—

119

C₂×C₈

—

122

—

120

C₂×C₈

C₂²×D₆₁

yes

—

121

C₂³

—

122

C₂³

Q₈×D₆₁

yes

characteristic

123

Q₈

—

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₆₁

yes

—

130

C₂×C₄

D₄×D₆₁

yes

—

131

C₂×C₄

Q₈×D₆₁

yes

characteristic

132

C₂×C₄

C₂³×D₆₁

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

122

—

137

C₈

C₂×C₄×D₆₁

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

244

—

141

C₂²

C₄⋊C₄×D₆₁

yes

—

142

C₂²

C₂×Q₈×D₆₁

yes

characteristic

143

C₄

—

144

C₄

C₄⋊C₄×D₆₁

yes

—

145

C₄

C₂²×C₄×D₆₁

yes

characteristic

146

C₄

C₂×D₄×D₆₁

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

122

—

150

C₂

C₂×C₄⋊C₄×D₆₁

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#430	C₂
socle	C₂×C₆₁	C₂²×C₄⋊C₄
hypercenter	#430	D₆₁
hypocenter	C₆₁	#2916
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

C₂×D₄×D₆₁

commutator

C₄×C₆₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#430

C₂

socle

C₂×C₆₁

C₂²×C₄⋊C₄

D₆₁

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
3. #16839811	C₂	semidirect
4. #16839809	C₂	semidirect
5. C₂×#14519712	C₂	semidirect
6. #108×D₆₁	C₂	semidirect
7. C₆₁×#430	C₂	semidirect
80. #430	D₆₁	semidirect
83. C₆₁	#2916	semidirect

H ≅ G/N

type

3. #16839811

C₂

semidirect

4. #16839809

C₂

semidirect

5. C₂×#14519712

C₂

semidirect

6. #108×D₆₁

C₂

semidirect

7. C₆₁×#430

C₂

semidirect

80. #430

D₆₁

semidirect

83. C₆₁

#2916

semidirect

Subgroups — #21454429

Conjugacy classes of subgroups

Complements