Group #21461578 — subgroups

Subgroups — #21461578

Order 7808.

284 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 86 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×#14519865	C₂	1	1	yes	maximal, characteristic
2	#16840743	C₂	1	1	yes	maximal, characteristic
3	#16840728	C₂	1	2	yes	maximal, complemented
4	#16840645	C₂	1	2	yes	maximal, complemented
5	#16839745	C₂	1	1	yes	maximal, characteristic
6	#14519865	—	2	2	no	—
7	#14519865	—	2	2	no	—
8	C₂×#11758213	C₄	1	2	yes	complemented
9	C₂×#11758213	C₂²	1	1	yes	characteristic
10	C₂×D₆₁⋊C₈	C₄	1	2	yes	complemented
11	C₂×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
12	#14519856	C₂²	1	2	yes	—
13	#14519836	C₄	1	4	yes	—
14	#14519831	—	2	2	no	—
15	C₈×#6923	—	2	2	no	—
16	C₂×#11758226	C₂²	1	1	yes	characteristic
17	M₁₆×D₆₁	C₂²	1	2	yes	—
18	#14519696	C₄	1	1	yes	characteristic, complemented
19	#14519693	C₄	1	2	yes	complemented
20	#101×C₆₁	C₄	1	1	yes	characteristic, complemented
21	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	#11758226	—	2	2	no	—
23	#11758226	—	2	2	no	—
24	C₂×D₂₄₄	C₂×C₄	1	2	yes	—
25	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₆₁	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
28	C₂²⋊#6923	C₂×C₄	1	4	yes	—
29	C₂×#92420	C₂×C₄	1	2	yes	—
30	C₂×#92420	C₂×C₄	1	1	yes	characteristic
31	C₂×#92420	C₂×C₄	1	1	yes	characteristic
32	C₄⋊#6923	—	2	2	no	—
33	C₄×#6923	—	2	2	no	—
34	#11758213	—	2	2	no	—
35	#11758213	—	2	4	no	—
36	D₆₁⋊C₈	—	2	2	no	—
37	D₆₁⋊C₈	—	2	2	no	—
38	D₆₁⋊C₈	D₄	1	2	yes	—
39	D₆₁⋊C₈	D₄	1	2	yes	—
40	M₁₆×C₆₁	C₂×C₄	1	2	yes	—
41	#11758194	C₂×C₄	1	2	yes	—
42	#11758189	—	2	2	no	—
43	C₈×D₆₁	—	2	2	no	—
44	C₂²×D₆₁	C₄²	1	2	yes	—
45	C₂²×D₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×#6923	—	2	4	no	—
47	Q₈×C₆₁	—	2	2	no	—
48	C₂×C₄×C₆₁	C₄²	1	2	yes	—
49	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
51	D₂₄₄	—	2	4	no	—
52	D₂₄₄	—	2	2	no	—
53	C₄×D₆₁	—	2	2	no	—
54	C₄×D₆₁	—	2	2	no	—
55	C₄×D₆₁	C₂×D₄	1	2	yes	—
56	C₄×D₆₁	#41	1	2	yes	—
57	#92420	—	2	2	no	—
58	#92420	C₂²⋊C₄	1	1	yes	characteristic
59	#92420	C₂²⋊C₄	1	1	yes	characteristic
60	#92420	C₂²⋊C₄	1	2	yes	—
61	C₈×C₆₁	—	2	2	no	—
62	#92418	—	2	2	no	—
63	C₂²×C₆₁	C₂×C₄²	1	1	yes	characteristic
64	C₂×D₆₁	—	2	2	no	—
65	C₂×D₆₁	—	4	1	no	—
66	C₂×D₆₁	—	4	2	no	—
67	C₂×D₆₁	C₄×D₄	1	2	yes	—
68	#6923	—	4	4	no	—
69	C₄×C₆₁	—	2	2	no	—
70	C₄×C₆₁	C₄×D₄	1	2	yes	—
71	Dic₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	Dic₆₁	#117	1	1	yes	characteristic
73	#1757	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	—	2	1	no	—
75	C₂×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
76	D₆₁	—	2	2	no	—
77	D₆₁	—	4	2	no	—
78	C₂×#131	—	61	1	no	—
79	#413	—	61	1	no	—
80	#410	—	61	1	no	—
81	#405	—	61	2	no	—
82	#344	—	61	2	no	—
83	C₆₁	#1757	1	1	yes	characteristic, complemented, Sylow
84	C₂×#41	—	61	1	no	—
85	#131	—	122	2	no	—
86	#131	—	122	2	no	—
87	C₂×M₁₆	—	61	2	no	—
88	C₂×M₁₆	—	61	1	no	—
89	C₂×M₁₆	—	61	2	no	—
90	C₂²×C₈	—	61	1	no	—
91	C₂²×C₈	—	61	2	no	—
92	#117	—	61	2	no	—
93	#101	—	61	1	no	—
94	#101	#6923	1	1	yes	characteristic, complemented
95	#100	—	61	2	no	—
96	#99	—	61	4	no	—
97	#97	—	122	2	no	—
98	C₄×C₈	—	122	2	no	—
99	#41	—	122	2	no	—
100	#41	—	122	2	no	—
101	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
102	C₂×D₄	—	61	2	no	—
103	C₂×D₄	—	61	1	no	—
104	C₂²×C₄	—	61	1	no	—
105	C₂²×C₄	—	61	2	no	—
106	M₁₆	—	61	2	no	—
107	M₁₆	—	122	2	no	—
108	M₁₆	—	122	4	no	—
109	M₁₆	—	122	2	no	—
110	M₁₆	C₂×#6923	1	2	yes	—
111	C₂×C₈	—	61	2	no	—
112	C₂×C₈	—	61	2	no	—
113	C₂×C₈	—	61	1	no	—
114	C₂×C₈	—	61	1	no	—
115	C₂×C₈	—	61	2	no	—
116	C₂×C₈	—	122	2	no	—
117	C₂×C₈	—	122	2	no	—
118	C₂×C₈	—	122	2	no	—
119	C₄⋊C₄	—	122	2	no	—
120	C₂²⋊C₄	—	61	4	no	—
121	C₄²	—	122	2	no	—
122	C₂³	—	61	1	no	—
123	C₂³	—	61	2	no	—
124	Q₈	—	2	2	no	—
125	D₄	—	122	4	no	—
126	D₄	—	122	2	no	—
127	C₂×C₄	—	61	1	no	—
128	C₂×C₄	—	61	2	no	—
129	C₂×C₄	—	61	2	no	—
130	C₂×C₄	—	122	2	no	—
131	C₂×C₄	—	122	2	no	—
132	C₂×C₄	—	122	4	no	—
133	C₂×C₄	C₄×#6923	1	2	yes	—
134	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	61	1	no	—
137	C₈	—	61	1	no	—
138	C₈	—	61	2	no	—
139	C₈	—	122	2	no	—
140	C₈	—	122	2	no	—
141	C₂²	—	61	2	no	—
142	C₂²	—	122	2	no	—
143	C₂²	—	244	1	no	—
144	C₂²	—	244	2	no	—
145	C₂²	C₂×C₄×#6923	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	61	1	no	—
148	C₄	—	61	1	no	—
149	C₄	—	244	4	no	—
150	C₄	D₄×#6923	1	2	yes	—
151	C₂	—	2	1	no	—
152	C₂	—	122	2	no	—
153	C₂	—	244	2	no	—
154	C₂	C₂²⋊C₄×#6923	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519865

C₂

yes

maximal, characteristic

#16840743

C₂

yes

maximal, characteristic

#16840728

C₂

yes

maximal, complemented

#16840645

C₂

yes

maximal, complemented

#16839745

C₂

yes

maximal, characteristic

#14519865

—

#14519865

—

C₂×#11758213

C₄

yes

complemented

C₂×#11758213

C₂²

yes

characteristic

C₂×D₆₁⋊C₈

C₄

yes

complemented

C₂×D₆₁⋊C₈

C₂²

yes

characteristic

#14519856

C₂²

yes

—

#14519836

C₄

yes

—

#14519831

—

C₈×#6923

—

C₂×#11758226

C₂²

yes

characteristic

M₁₆×D₆₁

C₂²

yes

—

#14519696

C₄

yes

characteristic, complemented

#14519693

C₄

yes

complemented

#101×C₆₁

C₄

yes

characteristic, complemented

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

#11758226

—

#11758226

—

C₂×D₂₄₄

C₂×C₄

yes

—

C₂×D₂₄₄

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂²⋊#6923

C₂×C₄

yes

—

C₂×#92420

C₂×C₄

yes

—

C₂×#92420

C₂×C₄

yes

characteristic

C₂×#92420

C₂×C₄

yes

—

—

—

—

D₆₁⋊C₈

—

D₆₁⋊C₈

—

D₆₁⋊C₈

D₄

yes

—

D₆₁⋊C₈

D₄

yes

—

M₁₆×C₆₁

C₂×C₄

yes

—

#11758194

C₂×C₄

yes

—

#11758189

—

C₈×D₆₁

—

C₂²×D₆₁

C₄²

yes

—

C₂²×D₆₁

C₂²×C₄

yes

characteristic

C₂×#6923

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

C₄²

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×D₄

yes

—

C₄×D₆₁

#41

yes

—

#92420

—

#92420

C₂²⋊C₄

yes

characteristic

#92420

C₂²⋊C₄

yes

characteristic

#92420

C₂²⋊C₄

yes

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₂×C₄²

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₄×D₄

yes

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

Dic₆₁

C₂×C₂²⋊C₄

yes

Dic₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

C₄×C₂²⋊C₄

yes

characteristic

D₆₁

—

D₆₁

—

—

—

—

—

—

C₆₁

#1757

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

122

—

#131

—

122

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

—

—

yes

characteristic, complemented

#100

—

#99

—

#97

—

122

—

C₄×C₈

—

122

—

#41

—

122

—

100

#41

—

122

—

101

C₂×Q₈

C₂×#6923

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

M₁₆

—

107

M₁₆

—

122

—

108

M₁₆

—

122

—

109

M₁₆

—

122

—

110

M₁₆

C₂×#6923

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

122

—

117

C₂×C₈

—

122

—

118

C₂×C₈

—

122

—

119

C₄⋊C₄

—

122

—

120

C₂²⋊C₄

—

121

C₄²

—

122

—

122

C₂³

—

123

C₂³

—

124

Q₈

—

125

D₄

—

122

—

126

D₄

—

122

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

122

—

131

C₂×C₄

—

122

—

132

C₂×C₄

—

122

—

133

C₂×C₄

C₄×#6923

yes

—

134

C₂×C₄

C₂²×#6923

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

122

—

140

C₈

—

122

—

141

C₂²

—

142

C₂²

—

122

—

143

C₂²

—

244

—

144

C₂²

—

244

—

145

C₂²

C₂×C₄×#6923

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

244

—

150

C₄

D₄×#6923

yes

—

151

C₂

—

152

C₂

—

122

—

153

C₂

—

244

—

154

C₂

C₂²⋊C₄×#6923

yes

characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×#6923
commutator	C₂²×C₆₁	C₂×C₄²
Frattini	C₂×C₄	C₂²×#6923
Fitting	#101×C₆₁	C₄
socle	C₂×C₆₁	C₄×C₂²⋊C₄
hypercenter	#101	#6923
hypocenter	C₆₁	#1757
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

C₂²⋊C₄×#6923

commutator

C₂²×C₆₁

C₂×C₄²

Frattini

C₂×C₄

C₂²×#6923

Fitting

#101×C₆₁

C₄

socle

C₂×C₆₁

C₄×C₂²⋊C₄

C₆₁

G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
3. #16840728	C₂	semidirect
4. #16840645	C₂	semidirect
8. C₂×#11758213	C₄	semidirect
10. C₂×D₆₁⋊C₈	C₄	semidirect
18. #14519696	C₄	semidirect
19. #14519693	C₄	semidirect
20. #101×C₆₁	C₄	semidirect
83. C₆₁	#1757	semidirect
94. #101	#6923	semidirect

H ≅ G/N

type

3. #16840728

C₂

semidirect

4. #16840645

C₂

semidirect

8. C₂×#11758213

C₄

semidirect

10. C₂×D₆₁⋊C₈

C₄

semidirect

18. #14519696

C₄

semidirect

19. #14519693

C₄

semidirect

20. #101×C₆₁

C₄

semidirect

83. C₆₁

#1757

semidirect

94. #101

#6923

semidirect

Subgroups — #21461578

Conjugacy classes of subgroups

Complements