Group #21446780 — subgroups

Subgroups — #21446780

Order 7808.

300 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14519777	C₂	1	1	yes	maximal, characteristic
2	#16839632	C₂	1	1	yes	maximal, characteristic
3	#16839525	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₂₂	—	2	1	no	—
6	#14519777	—	2	2	no	—
7	#14519777	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₆₁	—	2	1	no	—
9	C₂×C₄⋊Dic₆₁	—	2	1	no	—
10	C₂×C₄⋊Dic₆₁	—	2	1	no	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
13	C₂²×C₈×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₁	—	2	2	no	—
16	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₆₁	—	2	2	no	—
20	C₂²×Dic₆₁	—	2	2	no	—
21	C₂×Dic₁₂₂	—	2	4	no	—
22	C₂×Dic₁₂₂	—	4	2	no	—
23	C₂×Dic₁₂₂	—	4	2	no	—
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₂×C₈×C₆₁	—	2	2	no	—
26	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
27	C₄²×C₆₁	C₂×C₄	1	2	yes	—
28	C₄²×C₆₁	D₄	1	2	yes	—
29	C₄⋊Dic₆₁	—	2	4	no	—
30	C₄⋊Dic₆₁	—	2	2	no	—
31	C₄⋊Dic₆₁	—	2	4	no	—
32	C₄⋊Dic₆₁	—	4	2	no	—
33	C₄⋊Dic₆₁	—	4	2	no	—
34	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
35	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₆₁	D₈	1	2	yes	—
39	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
40	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
41	C₂×C₄×C₆₁	Q₁₆	1	2	yes	—
42	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₆₁	#41	1	2	yes	—
46	C₂×Dic₆₁	—	2	4	no	—
47	C₂×Dic₆₁	—	4	2	no	—
48	C₂×Dic₆₁	—	4	2	no	—
49	C₂×Dic₆₁	—	4	2	no	—
50	C₂×Dic₆₁	—	4	2	no	—
51	C₂×Dic₆₁	—	4	2	no	—
52	Dic₁₂₂	—	4	8	no	—
53	C₈×C₆₁	—	2	4	no	—
54	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₆₁	#124	1	1	yes	characteristic
56	C₂²×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
59	C₂²×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₂²×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₆₁	#102	1	4	yes	—
62	C₄×C₆₁	#103	1	4	yes	—
63	C₄×C₆₁	C₄×Q₈	1	2	yes	—
64	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
65	Dic₆₁	—	4	4	no	—
66	Dic₆₁	—	8	2	no	—
67	#1990	—	61	1	no	maximal, Sylow
68	C₂×C₆₁	#391	1	1	yes	characteristic
69	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
70	C₂×C₆₁	C₂×#103	1	1	yes	characteristic
71	C₂×C₆₁	#486	1	1	yes	characteristic
72	C₂×C₆₁	#487	1	1	yes	characteristic
73	C₂×C₆₁	#498	1	1	yes	characteristic
74	C₂×C₆₁	#500	1	1	yes	characteristic
75	C₂×C₄⋊Q₈	—	61	1	no	—
76	C₂×C₄×C₈	D₆₁	1	1	yes	characteristic
77	#384	—	61	1	no	—
78	#340	—	61	4	no	—
79	C₆₁	#1990	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	122	1	no	—
81	C₂²×C₈	C₂×D₆₁	1	2	yes	—
82	C₄⋊Q₈	—	61	4	no	—
83	C₄⋊Q₈	—	122	2	no	—
84	C₂×C₄⋊C₄	—	61	2	no	—
85	C₂×C₄⋊C₄	—	61	2	no	—
86	C₂×C₄⋊C₄	—	122	1	no	—
87	C₂×C₄⋊C₄	—	122	1	no	—
88	C₂×C₄⋊C₄	—	122	1	no	—
89	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
90	C₄×C₈	—	2	2	no	—
91	C₂×Q₈	—	122	4	no	—
92	C₂×Q₈	—	244	2	no	—
93	C₂×Q₈	—	244	2	no	—
94	C₂²×C₄	—	122	2	no	—
95	C₂²×C₄	—	122	2	no	—
96	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
97	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
102	C₄⋊C₄	—	61	4	no	—
103	C₄⋊C₄	—	122	4	no	—
104	C₄⋊C₄	—	122	2	no	—
105	C₄⋊C₄	—	122	4	no	—
106	C₄⋊C₄	—	244	2	no	—
107	C₄⋊C₄	—	244	2	no	—
108	C₄²	C₄×D₆₁	1	2	yes	—
109	C₄²	C₂²⋊D₆₁	1	2	yes	—
110	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
111	Q₈	—	244	8	no	—
112	C₂×C₄	—	122	4	no	—
113	C₂×C₄	—	244	2	no	—
114	C₂×C₄	—	244	2	no	—
115	C₂×C₄	—	244	2	no	—
116	C₂×C₄	—	244	2	no	—
117	C₂×C₄	—	244	2	no	—
118	C₂×C₄	#11758190	1	2	yes	—
119	C₂×C₄	#11758190	1	2	yes	—
120	C₂×C₄	D₄₈₈	1	2	yes	—
121	C₂×C₄	Dic₂₄₄	1	2	yes	—
122	C₂×C₄	#11758198	1	2	yes	—
123	C₂×C₄	#11758198	1	2	yes	—
124	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
125	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
126	C₂×C₄	#11758222	1	2	yes	—
127	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
128	C₈	—	2	4	no	—
129	C₂²	C₂×#11758198	1	1	yes	characteristic
130	C₂²	#14519765	1	1	yes	characteristic
131	C₂²	#14519777	1	1	yes	characteristic
132	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
133	C₂²	C₂×#11758190	1	1	yes	characteristic
134	C₂²	C₂×#11758190	1	1	yes	characteristic
135	C₂²	C₂×Dic₂₄₄	1	1	yes	characteristic
136	C₄	—	488	2	no	—
137	C₄	—	244	4	no	—
138	C₄	#14519699	1	4	yes	—
139	C₄	#14519702	1	4	yes	—
140	C₄	C₄×Dic₁₂₂	1	2	yes	—
141	C₄	#14519756	1	2	yes	—
142	C₂	#16839668	1	1	yes	characteristic
143	C₂	C₂×#14519699	1	1	yes	characteristic
144	C₂	C₂×#14519702	1	1	yes	characteristic
145	C₂	#16840109	1	1	yes	characteristic
146	C₂	#16840113	1	1	yes	characteristic
147	C₂	#16840160	1	1	yes	characteristic
148	C₂	#16840167	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519777

C₂

yes

maximal, characteristic

#16839632

C₂

yes

maximal, characteristic

#16839525

C₂

yes

maximal

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

—

#14519777

—

#14519777

C₄

yes

complemented

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

#124

yes

characteristic

C₂²×C₆₁

C₄⋊Q₈

yes

characteristic

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₂²×C₆₁

C₂×SD₁₆

yes

characteristic

C₂²×C₆₁

C₂×SD₁₆

yes

characteristic

C₂²×C₆₁

C₂×Q₁₆

yes

characteristic

C₄×C₆₁

#102

yes

—

C₄×C₆₁

#103

yes

—

C₄×C₆₁

C₄×Q₈

yes

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₄⋊Q₈

—

C₂×C₄×C₈

D₆₁

yes

characteristic

#384

—

#340

—

C₆₁

#1990

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂²×C₈

C₂×D₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

D₂₄₄

yes

characteristic

C₂²×C₄

D₂₄₄

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

Dic₁₂₂

yes

complemented

102

C₄⋊C₄

—

103

C₄⋊C₄

—

122

—

104

C₄⋊C₄

—

122

—

105

C₄⋊C₄

—

122

—

106

C₄⋊C₄

—

244

—

107

C₄⋊C₄

—

244

—

108

C₄²

C₄×D₆₁

yes

—

109

C₄²

C₂²⋊D₆₁

yes

—

110

C₂³

C₂×D₂₄₄

yes

characteristic

111

Q₈

—

244

—

112

C₂×C₄

—

122

—

113

C₂×C₄

—

244

—

114

C₂×C₄

—

244

—

115

C₂×C₄

—

244

—

116

C₂×C₄

—

244

—

117

C₂×C₄

—

244

—

118

C₂×C₄

#11758190

yes

—

119

C₂×C₄

#11758190

yes

—

120

C₂×C₄

D₄₈₈

yes

—

121

C₂×C₄

Dic₂₄₄

yes

—

122

C₂×C₄

#11758198

yes

—

123

C₂×C₄

#11758198

yes

—

124

C₂×C₄

C₂×Dic₁₂₂

yes

—

125

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

126

C₂×C₄

#11758222

yes

—

127

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

128

C₈

—

129

C₂²

C₂×#11758198

yes

characteristic

130

C₂²

#14519765

yes

characteristic

131

C₂²

#14519777

yes

characteristic

132

C₂²

C₂×D₄₈₈

yes

characteristic

133

C₂²

C₂×#11758190

yes

characteristic

134

C₂²

C₂×#11758190

yes

characteristic

135

C₂²

C₂×Dic₂₄₄

yes

characteristic

136

C₄

—

488

—

137

C₄

—

244

—

138

C₄

#14519699

yes

—

139

C₄

#14519702

yes

—

140

C₄

C₄×Dic₁₂₂

yes

—

141

C₄

#14519756

yes

—

142

C₂

#16839668

yes

characteristic

143

C₂

C₂×#14519699

yes

characteristic

144

C₂

C₂×#14519702

yes

characteristic

145

C₂

#16840109

yes

characteristic

146

C₂

#16840113

yes

characteristic

147

C₂

#16840160

yes

characteristic

148

C₂

#16840167

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×C₄×C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₆₁
hypocenter	C₆₁	#1990
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×C₄×C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₆₁

hypocenter

C₆₁

#1990

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #14519777	C₄	semidirect
26. C₂×C₈×C₆₁	Q₈	semidirect
79. C₆₁	#1990	semidirect
101. C₂×C₈	Dic₁₂₂	semidirect

H ≅ G/N

type

7. #14519777

C₄

semidirect

26. C₂×C₈×C₆₁

Q₈

semidirect

79. C₆₁

#1990

semidirect

101. C₂×C₈

Dic₁₂₂

semidirect

Subgroups — #21446780

Conjugacy classes of subgroups

Complements