Group #21445193 — subgroups

Subgroups — #21445193

Order 7808.

166 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16840158	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840053	C₂	1	1	yes	maximal, characteristic
3	#16839964	C₂	1	1	yes	maximal, characteristic
4	#16839827	C₂	1	1	yes	maximal, characteristic
5	#16839512	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839510	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758201	—	2	1	no	—
9	C₂×C₈×D₆₁	—	2	1	no	—
10	#14519777	C₂²	1	1	yes	characteristic
11	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
12	#14519709	—	2	1	no	—
13	#14519708	—	2	1	no	—
14	#14519708	—	2	1	no	—
15	C₄⋊#92418	C₂²	1	1	yes	characteristic
16	#14519701	—	2	1	no	—
17	#14519699	—	2	1	no	—
18	#14519689	—	2	1	no	—
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	—	2	1	no	—
24	C₂×D₂₄₄	D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	#11758201	—	4	2	no	—
33	#11758198	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
37	C₂×#92418	—	2	1	no	—
38	C₂×#92418	D₄	1	1	yes	characteristic
39	C₂×#92418	D₄	1	1	yes	characteristic
40	C₈×D₆₁	—	4	2	no	—
41	C₂²×D₆₁	—	2	1	no	—
42	Q₈×C₆₁	—	4	1	no	—
43	Q₈×C₆₁	—	4	1	no	—
44	C₂×C₄×C₆₁	—	4	1	no	—
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
48	C₂×Dic₆₁	—	2	1	no	—
49	C₂×Dic₆₁	—	4	1	no	—
50	D₂₄₄	—	2	1	no	—
51	D₂₄₄	SD₁₆	1	2	yes	—
52	C₄×D₆₁	—	4	1	no	—
53	C₄×D₆₁	—	4	1	no	—
54	Dic₁₂₂	—	8	1	no	—
55	C₈×C₆₁	—	4	1	no	—
56	C₈×C₆₁	—	4	1	no	—
57	#92418	—	2	1	no	—
58	#92418	—	4	1	no	—
59	#92418	SD₁₆	1	2	yes	complemented
60	C₂²×C₆₁	#121	1	1	yes	characteristic
61	C₂×D₆₁	—	2	2	no	—
62	C₂×D₆₁	—	4	1	no	—
63	C₂×D₆₁	—	4	1	no	—
64	C₄×C₆₁	—	2	1	no	—
65	C₄×C₆₁	—	8	1	no	—
66	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₆₁	#135	1	1	yes	characteristic
69	C₄×C₆₁	#137	1	1	yes	characteristic
70	Dic₆₁	—	4	1	no	—
71	Dic₆₁	—	8	1	no	—
72	#1671	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	#461	1	1	yes	characteristic
74	C₂×C₆₁	#465	1	1	yes	characteristic
75	C₂×C₆₁	#471	1	1	yes	characteristic
76	D₆₁	—	4	2	no	—
77	#498	—	61	1	no	—
78	#476	—	61	1	no	—
79	#461	—	61	1	no	—
80	C₈×D₄	—	61	1	no	—
81	#334	—	61	1	no	—
82	#333	—	61	1	no	—
83	#332	D₆₁	1	1	yes	characteristic, complemented
84	C₆₁	#1671	1	1	yes	characteristic, complemented, Sylow
85	C₂×SD₁₆	—	122	1	no	—
86	C₂²×C₈	—	122	1	no	—
87	C₄⋊Q₈	—	61	1	no	—
88	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
89	C₄×D₄	—	61	1	no	—
90	#106	—	122	1	no	—
91	#106	—	122	1	no	—
92	#106	—	122	1	no	—
93	C₄⋊C₈	—	61	1	no	—
94	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
95	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
96	#103	—	122	1	no	—
97	#102	—	122	1	no	—
98	C₂²⋊C₈	—	122	1	no	—
99	C₄×C₈	—	61	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	122	1	no	—
102	C₂×D₄	—	61	1	no	—
103	C₂²×C₄	—	122	1	no	—
104	SD₁₆	—	244	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	122	1	no	—
110	C₂×C₈	—	244	2	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	61	1	no	—
114	C₄⋊C₄	—	122	1	no	—
115	C₄⋊C₄	—	122	1	no	—
116	C₂²⋊C₄	—	122	1	no	—
117	C₄²	C₂²×D₆₁	1	1	yes	characteristic
118	C₂³	—	122	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	488	1	no	—
122	D₄	—	61	2	no	—
123	D₄	—	122	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	122	1	no	—
126	C₂×C₄	—	244	1	no	—
127	C₂×C₄	—	244	1	no	—
128	C₂×C₄	—	244	1	no	—
129	C₂×C₄	#11758222	1	1	yes	characteristic
130	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
131	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	61	2	no	—
135	C₈	—	122	1	no	—
136	C₈	—	244	1	no	—
137	C₂²	—	122	2	no	—
138	C₂²	—	244	1	no	—
139	C₂²	—	244	1	no	—
140	C₂²	#14519750	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	8	1	no	—
143	C₄	—	488	1	no	—
144	C₄	—	244	1	no	—
145	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
146	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
147	C₄	#14519796	1	1	yes	characteristic
148	C₄	#14519809	1	1	yes	characteristic
149	C₂	—	244	2	no	—
150	C₂	#16839966	1	1	yes	characteristic
151	C₂	#16839990	1	1	yes	characteristic
152	C₂	#16840028	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840158

C₂

yes

maximal, characteristic, complemented

#16840053

C₂

yes

maximal, characteristic

#16839964

C₂

yes

maximal, characteristic

#16839827

C₂

yes

maximal, characteristic

#16839512

C₂

yes

maximal, characteristic, complemented

#16839510

C₂

yes

maximal, characteristic, complemented

C₆₁×#332

C₂

yes

maximal, characteristic, complemented

C₂×#11758201

—

C₂×C₈×D₆₁

—

#14519777

C₂²

yes

characteristic

C₄×D₂₄₄

C₂²

yes

—

—

—

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

D₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758201

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₂×#92418

—

C₂×#92418

D₄

yes

characteristic

C₂×#92418

D₄

yes

characteristic

C₈×D₆₁

—

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

SD₁₆

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

—

#92418

SD₁₆

yes

complemented

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#498

—

#476

—

#461

—

C₈×D₄

—

#334

—

#333

—

#332

D₆₁

yes

characteristic, complemented

C₆₁

#1671

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂²×C₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

#106

—

122

—

#106

—

122

—

#106

—

122

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

122

—

#102

—

122

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

122

—

102

C₂×D₄

—

103

C₂²×C₄

—

122

—

104

SD₁₆

—

244

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

122

—

110

C₂×C₈

—

244

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

122

—

115

C₄⋊C₄

—

122

—

116

C₂²⋊C₄

—

122

—

117

C₄²

C₂²×D₆₁

yes

characteristic

118

C₂³

—

122

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

488

—

122

D₄

—

123

D₄

—

122

—

124

C₂×C₄

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

244

—

127

C₂×C₄

—

244

—

128

C₂×C₄

—

244

—

129

C₂×C₄

#11758222

yes

characteristic

130

C₂×C₄

D₄×D₆₁

yes

characteristic

131

C₂×C₄

D₄×D₆₁

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

122

—

136

C₈

—

244

—

137

C₂²

—

122

—

138

C₂²

—

244

—

139

C₂²

—

244

—

140

C₂²

#14519750

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

488

—

144

C₄

—

244

—

145

C₄

SD₁₆×D₆₁

yes

characteristic

146

C₄

SD₁₆×D₆₁

yes

characteristic

147

C₄

#14519796

yes

characteristic

148

C₄

#14519809

yes

characteristic

149

C₂

—

244

—

150

C₂

#16839966

yes

characteristic

151

C₂

#16839990

yes

characteristic

152

C₂

#16840028

yes

characteristic

	subgroup	G/H
center	C₂²	#14519750
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#332	C₂
socle	C₂²×C₆₁	#121
hypercenter	#332	D₆₁
hypocenter	C₆₁	#1671
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14519750

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#332

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #16840158	C₂	semidirect
5. #16839512	C₂	semidirect
6. #16839510	C₂	semidirect
7. C₆₁×#332	C₂	semidirect
59. #92418	SD₁₆	semidirect
83. #332	D₆₁	semidirect
84. C₆₁	#1671	semidirect

H ≅ G/N

type

1. #16840158

C₂

semidirect

5. #16839512

C₂

semidirect

6. #16839510

C₂

semidirect

7. C₆₁×#332

C₂

semidirect

59. #92418

SD₁₆

semidirect

83. #332

D₆₁

semidirect

84. C₆₁

#1671

semidirect

Subgroups — #21445193

Conjugacy classes of subgroups

Complements