Group #21445622 — subgroups

Subgroups — #21445622

Order 7808.

284 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14519784	C₂	1	1	yes	maximal, characteristic
2	#16839736	C₂	1	1	yes	maximal, characteristic
3	#16839720	C₂	1	2	yes	maximal, complemented
4	#16839534	C₂	1	2	yes	maximal, complemented
5	C₆₁×#413	C₂	1	1	yes	maximal, characteristic
6	#14519784	—	2	2	no	—
7	#14519784	—	2	2	no	—
8	C₂×#11758194	C₄	1	2	yes	complemented
9	C₂×#11758194	C₂²	1	1	yes	characteristic
10	C₂²×#92418	C₄	1	2	yes	complemented
11	C₂²×#92418	C₂²	1	1	yes	characteristic
12	#14519736	C₂²	1	2	yes	—
13	#14519692	C₄	1	4	yes	—
14	#14519687	—	2	2	no	—
15	C₈×Dic₆₁	—	2	2	no	—
16	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
18	#101×C₆₁	C₄	1	2	yes	complemented
19	#100×C₆₁	C₄	1	2	yes	complemented
20	#41×C₆₁	—	2	2	no	—
21	#41×C₆₁	—	2	2	no	—
22	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₆₁	—	2	2	no	—
28	M₁₆×C₆₁	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₂²⋊Dic₆₁	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₆₁	—	2	2	no	—
32	C₄×Dic₆₁	—	2	2	no	—
33	#11758194	—	2	2	no	—
34	#11758194	—	2	4	no	—
35	C₂×#92418	—	2	2	no	—
36	C₂×#92418	—	2	2	no	—
37	C₂×#92418	C₂×C₄	1	2	yes	—
38	C₂×#92418	C₂×C₄	1	1	yes	characteristic
39	C₂×#92418	C₂×C₄	1	1	yes	characteristic
40	C₂×#92418	D₄	1	2	yes	—
41	C₂×#92418	D₄	1	2	yes	—
42	C₂³×C₆₁	C₄²	1	2	yes	—
43	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₆₁	—	2	2	no	—
45	D₄×C₆₁	—	2	4	no	—
46	D₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	—	2	2	no	—
49	C₂×C₄×C₆₁	C₄²	1	2	yes	—
50	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₆₁	#41	1	2	yes	—
54	C₂×Dic₆₁	—	2	4	no	—
55	C₈×C₆₁	—	2	4	no	—
56	#92418	—	2	2	no	—
57	#92418	C₂²⋊C₄	1	1	yes	characteristic
58	#92418	C₂²⋊C₄	1	1	yes	characteristic
59	#92418	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₆₁	—	2	2	no	—
61	C₂²×C₆₁	—	4	1	no	—
62	C₂²×C₆₁	—	4	2	no	—
63	C₂²×C₆₁	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₆₁	C₄×D₄	1	2	yes	—
65	C₄×C₆₁	—	2	2	no	—
66	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₆₁	#117	1	1	yes	characteristic
68	C₄×C₆₁	C₄×D₄	1	2	yes	—
69	Dic₆₁	—	4	4	no	—
70	#1757	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	—	2	2	no	—
72	C₂×C₆₁	—	2	1	no	—
73	C₂×C₆₁	—	4	2	no	—
74	C₂×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×#131	—	61	1	no	—
76	#413	D₆₁	1	1	yes	characteristic
77	#410	—	61	1	no	—
78	#405	—	61	2	no	—
79	#344	—	61	2	no	—
80	C₆₁	#1757	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
82	#131	—	122	2	no	—
83	#131	—	122	2	no	—
84	C₂×M₁₆	—	61	2	no	—
85	C₂×M₁₆	—	61	1	no	—
86	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
87	C₂²×C₈	—	61	1	no	—
88	C₂²×C₈	—	61	2	no	—
89	#117	—	61	2	no	—
90	#101	Dic₆₁	1	2	yes	complemented
91	#100	Dic₆₁	1	2	yes	complemented
92	#99	—	61	4	no	—
93	#97	—	122	2	no	—
94	C₄×C₈	—	122	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
98	C₂×D₄	C₄×D₆₁	1	2	yes	—
99	C₂×D₄	C₂×Dic₆₁	1	1	yes	characteristic
100	C₂²×C₄	C₄×D₆₁	1	2	yes	—
101	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
102	M₁₆	—	2	2	no	—
103	M₁₆	—	122	4	no	—
104	M₁₆	—	122	2	no	—
105	M₁₆	C₂×Dic₆₁	1	4	yes	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	61	2	no	—
110	C₂×C₈	—	61	2	no	—
111	C₂×C₈	—	61	2	no	—
112	C₂×C₈	—	122	2	no	—
113	C₂×C₈	—	122	2	no	—
114	C₄⋊C₄	—	122	2	no	—
115	C₂²⋊C₄	—	61	4	no	—
116	C₄²	—	122	2	no	—
117	C₂³	C₄×Dic₆₁	1	2	yes	—
118	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
119	Q₈	—	2	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	122	4	no	—
125	C₂×C₄	C₄×Dic₆₁	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
127	C₂×C₄	D₄×D₆₁	1	2	yes	—
128	C₂×C₄	#11758224	1	2	yes	—
129	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₈	—	61	1	no	—
132	C₈	—	61	1	no	—
133	C₈	—	61	2	no	—
134	C₈	—	122	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₂×C₄×Dic₆₁	1	1	yes	characteristic
139	C₂²	#14519737	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	244	4	no	—
142	C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
143	C₄	#14519733	1	1	yes	characteristic
144	C₄	D₄×Dic₆₁	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	C₂²⋊C₄×Dic₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519784

C₂

yes

maximal, characteristic

#16839736

C₂

yes

maximal, characteristic

#16839720

C₂

yes

maximal, complemented

#16839534

C₂

yes

maximal, complemented

C₆₁×#413

C₂

yes

maximal, characteristic

#14519784

—

#14519784

—

C₂×#11758194

C₄

yes

complemented

C₂×#11758194

C₂²

yes

characteristic

C₂²×#92418

C₄

yes

complemented

C₂²×#92418

C₂²

yes

characteristic

#14519736

C₂²

yes

—

#14519692

C₄

yes

—

#14519687

—

C₈×Dic₆₁

—

C₂×#41×C₆₁

C₂²

yes

characteristic

C₂×M₁₆×C₆₁

C₂²

yes

—

#101×C₆₁

C₄

yes

complemented

#100×C₆₁

C₄

yes

complemented

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

M₁₆×C₆₁

—

M₁₆×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₂²⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

—

—

—

—

C₂×C₄

yes

—

C₂×#92418

C₂×C₄

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

C₂×#92418

D₄

yes

—

C₂×#92418

D₄

yes

—

C₂³×C₆₁

C₄²

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄²

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₈×C₆₁

—

#92418

—

#92418

C₂²⋊C₄

yes

characteristic

#92418

C₂²⋊C₄

yes

characteristic

#92418

C₂²⋊C₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×C₄²

yes

characteristic

C₂²×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

#117

yes

characteristic

C₄×C₆₁

C₄×D₄

yes

—

Dic₆₁

—

#1757

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₄×C₂²⋊C₄

yes

characteristic

C₂×#131

—

#413

D₆₁

yes

—

—

—

C₆₁

#1757

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

characteristic

#131

—

122

—

#131

—

122

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#101

Dic₆₁

yes

complemented

#100

Dic₆₁

yes

complemented

#99

—

#97

—

122

—

C₄×C₈

—

122

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₆₁

yes

characteristic

C₂×D₄

C₄×D₆₁

yes

—

C₂×D₄

C₂×Dic₆₁

yes

characteristic

100

C₂²×C₄

C₄×D₆₁

yes

—

101

C₂²×C₄

C₂²×D₆₁

yes

characteristic

102

M₁₆

—

103

M₁₆

—

122

—

104

M₁₆

—

122

—

105

M₁₆

C₂×Dic₆₁

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

122

—

113

C₂×C₈

—

122

—

114

C₄⋊C₄

—

122

—

115

C₂²⋊C₄

—

116

C₄²

—

122

—

117

C₂³

C₄×Dic₆₁

yes

—

118

C₂³

C₂×C₄×D₆₁

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

122

—

125

C₂×C₄

C₄×Dic₆₁

yes

—

126

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

127

C₂×C₄

D₄×D₆₁

yes

—

128

C₂×C₄

#11758224

yes

—

129

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

122

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×Dic₆₁

yes

characteristic

139

C₂²

#14519737

yes

—

140

C₄

—

141

C₄

—

244

—

142

C₄

C₂²⋊C₄×D₆₁

yes

characteristic

143

C₄

#14519733

yes

characteristic

144

C₄

D₄×Dic₆₁

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₂²⋊C₄×Dic₆₁

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₆₁
commutator	C₂²×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#413	C₂
socle	C₂×C₆₁	C₄×C₂²⋊C₄
hypercenter	#413	D₆₁
hypocenter	C₆₁	#1757
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

C₂²⋊C₄×D₆₁

commutator

C₂²×C₆₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#413

C₂

socle

C₂×C₆₁

C₄×C₂²⋊C₄

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #16839720	C₂	semidirect
4. #16839534	C₂	semidirect
8. C₂×#11758194	C₄	semidirect
10. C₂²×#92418	C₄	semidirect
18. #101×C₆₁	C₄	semidirect
19. #100×C₆₁	C₄	semidirect
80. C₆₁	#1757	semidirect
90. #101	Dic₆₁	semidirect
91. #100	Dic₆₁	semidirect

H ≅ G/N

type

3. #16839720

C₂

semidirect

4. #16839534

C₂

semidirect

8. C₂×#11758194

C₄

semidirect

10. C₂²×#92418

C₄

semidirect

18. #101×C₆₁

C₄

semidirect

19. #100×C₆₁

C₄

semidirect

80. C₆₁

#1757

semidirect

90. #101

Dic₆₁

semidirect

91. #100

Dic₆₁

semidirect

Subgroups — #21445622

Conjugacy classes of subgroups

Complements