Group #21461203 — subgroups

Subgroups — #21461203

Order 7808.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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	#16840781	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#6923	C₂	1	1	yes	maximal, characteristic
3	#16840616	C₂	1	4	yes	maximal
4	#16840317	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#6923	—	2	1	no	—
6	C₂×C₄×#6923	—	2	1	no	—
7	C₄⋊#92420	—	2	1	no	—
8	#14519834	C₂²	1	2	yes	—
9	C₄×#92420	—	2	1	no	—
10	(C₂×#6923)⋊C₄	C₂²	1	2	yes	—
11	#14519777	C₄	1	1	yes	characteristic, complemented
12	#14519768	—	2	1	no	—
13	#14519757	—	2	1	no	—
14	#14519753	C₂²	1	2	yes	—
15	Q₈×Dic₆₁	C₄	1	2	yes	complemented
16	#14519742	—	2	1	no	—
17	#14519735	—	2	1	no	—
18	C₄²×D₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic, complemented
20	C₂²×#6923	—	2	2	no	—
21	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
23	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
24	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₂₂	—	2	1	no	—
26	C₂×#92420	—	2	2	no	—
27	C₄⋊#6923	—	2	4	no	—
28	C₄×#6923	—	4	2	no	—
29	C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	#11758198	—	2	2	no	—
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	2	no	—
35	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
36	#11758196	—	2	2	no	—
37	C₄×Dic₆₁	—	2	2	no	—
38	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₆₁	D₄	1	1	yes	characteristic
40	C₄×Dic₆₁	D₄	1	1	yes	characteristic
41	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×#6923	—	2	4	no	—
43	C₂×#6923	—	4	2	no	—
44	C₂×#6923	—	4	2	no	—
45	Q₈×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	—	2	2	no	—
47	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	2	no	—
51	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
54	C₄×D₆₁	—	2	2	no	—
55	C₄×D₆₁	—	4	1	no	—
56	C₄×D₆₁	SD₁₆	1	2	yes	—
57	C₄×D₆₁	Q₁₆	1	2	yes	—
58	Dic₁₂₂	—	4	2	no	—
59	#92420	—	4	2	no	—
60	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×D₆₁	—	2	1	no	—
62	C₂×D₆₁	—	2	1	no	—
63	C₂×D₆₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×D₆₁	C₂×Q₁₆	1	1	yes	characteristic
65	#6923	—	4	4	no	—
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	—	4	2	no	—
68	C₄×C₆₁	#103	1	2	yes	—
69	C₄×C₆₁	#104	1	2	yes	—
70	Dic₆₁	—	2	1	no	—
71	Dic₆₁	—	2	1	no	—
72	Dic₆₁	—	4	2	no	—
73	Dic₆₁	#103	1	2	yes	—
74	#1503	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	#410	1	1	yes	characteristic
76	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
77	C₂×C₆₁	C₂×#104	1	1	yes	characteristic
78	D₆₁	—	2	1	no	—
79	D₆₁	—	2	1	no	—
80	#533	—	61	1	no	—
81	#431	—	61	1	no	—
82	C₄×C₄⋊C₄	—	61	1	no	—
83	#328	—	61	4	no	—
84	C₆₁	#1503	1	1	yes	characteristic, complemented, Sylow
85	C₄⋊Q₈	—	61	1	no	—
86	C₄⋊Q₈	#6923	1	1	yes	characteristic, complemented
87	#125	—	122	1	no	—
88	C₂²⋊Q₈	—	61	2	no	—
89	C₂²⋊Q₈	—	122	1	no	—
90	C₄×Q₈	—	61	2	no	—
91	C₄×Q₈	—	122	1	no	—
92	#117	—	122	1	no	—
93	C₂×C₄⋊C₄	—	122	1	no	—
94	C₂×C₄²	—	61	1	no	—
95	C₂×C₄²	—	122	1	no	—
96	C₄⋊C₈	—	122	1	no	—
97	C₂²⋊C₈	—	61	2	no	—
98	C₄×C₈	—	122	1	no	—
99	#95	—	61	2	no	—
100	C₂×Q₈	—	122	1	no	—
101	C₂×Q₈	C₂×#6923	1	2	yes	—
102	C₂²×C₄	—	61	1	no	—
103	C₂²×C₄	—	61	1	no	—
104	C₂²×C₄	—	61	1	no	—
105	C₂²×C₄	—	122	2	no	—
106	C₂×C₈	—	122	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	61	2	no	—
110	C₄⋊C₄	—	122	2	no	—
111	C₄⋊C₄	—	122	2	no	—
112	C₄⋊C₄	—	122	4	no	—
113	C₄⋊C₄	—	122	1	no	—
114	C₂²⋊C₄	—	122	2	no	—
115	C₄²	—	61	1	no	—
116	C₄²	—	61	1	no	—
117	C₄²	—	61	1	no	—
118	C₄²	—	122	2	no	—
119	C₄²	—	244	2	no	—
120	C₄²	C₂×#6923	1	1	yes	characteristic
121	C₂³	—	61	1	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	244	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	61	1	no	—
126	C₂×C₄	—	61	2	no	—
127	C₂×C₄	—	61	2	no	—
128	C₂×C₄	—	61	1	no	—
129	C₂×C₄	—	61	1	no	—
130	C₂×C₄	—	122	2	no	—
131	C₂×C₄	—	122	4	no	—
132	C₂×C₄	—	122	2	no	—
133	C₂×C₄	—	244	1	no	—
134	C₂×C₄	—	244	2	no	—
135	C₂×C₄	—	244	2	no	—
136	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
137	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
138	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
139	C₈	—	244	2	no	—
140	C₂²	—	61	1	no	—
141	C₂²	—	61	1	no	—
142	C₂²	—	122	1	no	—
143	C₂²	—	122	1	no	—
144	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	4	2	no	—
147	C₄	—	61	2	no	—
148	C₄	—	122	1	no	—
149	C₄	—	122	1	no	—
150	C₄	—	244	4	no	—
151	C₄	—	244	2	no	—
152	C₄	#14519840	1	2	yes	—
153	C₄	#14519842	1	2	yes	—
154	C₂	—	122	1	no	—
155	C₂	—	122	1	no	—
156	C₂	#16840742	1	1	yes	characteristic
157	C₂	C₂×#14519840	1	1	yes	characteristic
158	C₂	C₂×#14519842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840781

C₂

yes

maximal, characteristic

C₄×C₄⋊#6923

C₂

yes

maximal, characteristic

#16840616

C₂

yes

maximal

#16840317

C₂

yes

maximal, characteristic

C₂×C₄⋊#6923

—

C₂×C₄×#6923

—

C₄⋊#92420

—

#14519834

C₂²

yes

—

C₄×#92420

—

(C₂×#6923)⋊C₄

C₂²

yes

—

#14519777

C₄

yes

characteristic, complemented

#14519768

—

#14519757

—

#14519753

C₂²

yes

—

Q₈×Dic₆₁

C₄

yes

complemented

#14519742

—

#14519735

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic, complemented

C₂²×#6923

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×#92420

—

C₄⋊#6923

—

C₄×#6923

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₂²×D₆₁

C₂×D₄

yes

—

—

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

SD₁₆

yes

—

C₄×D₆₁

Q₁₆

yes

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂×SD₁₆

yes

characteristic

C₂×D₆₁

C₂×Q₁₆

yes

characteristic

#6923

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#103

yes

—

C₄×C₆₁

#104

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#103

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

D₆₁

—

#533

—

#431

—

C₄×C₄⋊C₄

—

#328

—

C₆₁

#1503

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

#6923

yes

characteristic, complemented

#125

—

122

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₂×C₄²

—

122

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

C₄×C₈

—

122

—

#95

—

100

C₂×Q₈

—

122

—

101

C₂×Q₈

C₂×#6923

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

122

—

106

C₂×C₈

—

122

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

122

—

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

122

—

113

C₄⋊C₄

—

122

—

114

C₂²⋊C₄

—

122

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

122

—

119

C₄²

—

244

—

120

C₄²

C₂×#6923

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

244

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

122

—

131

C₂×C₄

—

122

—

132

C₂×C₄

—

122

—

133

C₂×C₄

—

244

—

134

C₂×C₄

—

244

—

135

C₂×C₄

—

244

—

136

C₂×C₄

C₂²⋊#6923

yes

characteristic

137

C₂×C₄

C₂²⋊#6923

yes

characteristic

138

C₂×C₄

C₂²×#6923

yes

characteristic

139

C₈

—

244

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

122

—

143

C₂²

—

122

—

144

C₂²

C₂×C₂²⋊#6923

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

122

—

149

C₄

—

122

—

150

C₄

—

244

—

151

C₄

—

244

—

152

C₄

#14519840

yes

—

153

C₄

#14519842

yes

—

154

C₂

—

122

—

155

C₂

—

122

—

156

C₂

#16840742

yes

characteristic

157

C₂

C₂×#14519840

yes

characteristic

158

C₂

C₂×#14519842

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#1503
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄⋊Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #14519777	C₄	semidirect
15. Q₈×Dic₆₁	C₄	semidirect
19. C₄⋊Q₈×C₆₁	C₄	semidirect
84. C₆₁	#1503	semidirect
86. C₄⋊Q₈	#6923	semidirect

H ≅ G/N

type

11. #14519777

C₄

semidirect

15. Q₈×Dic₆₁

C₄

semidirect

19. C₄⋊Q₈×C₆₁

C₄

semidirect

84. C₆₁

#1503

semidirect

86. C₄⋊Q₈

#6923

semidirect

Subgroups — #21461203

Conjugacy classes of subgroups

Complements