Group #21445263 — subgroups

Subgroups — #21445263

Order 7808.

166 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840109	C₂	1	1	yes	maximal, characteristic
2	#16839972	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839962	C₂	1	1	yes	maximal, characteristic
4	#16839515	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839507	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839505	C₂	1	1	yes	maximal, characteristic
7	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758202	—	2	1	no	—
9	C₂×#11758201	—	2	1	no	—
10	#14519777	C₂²	1	1	yes	characteristic
11	#14519774	C₂²	1	1	yes	characteristic
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	C₄⋊#92418	C₂²	1	1	yes	characteristic
14	#14519700	—	2	1	no	—
15	#14519699	—	2	1	no	—
16	#14519699	—	2	1	no	—
17	#14519697	—	2	1	no	—
18	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₆₁	—	2	1	no	—
20	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₆₁	—	2	1	no	—
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
24	C₂×D₂₄₄	—	2	1	no	—
25	C₂×D₂₄₄	—	2	1	no	—
26	C₂×D₂₄₄	—	2	1	no	—
27	C₂×Dic₁₂₂	—	2	1	no	—
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
33	C₄²×C₆₁	—	2	1	no	—
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	#11758202	—	4	2	no	—
36	#11758201	—	4	2	no	—
37	C₄⋊Dic₆₁	—	2	1	no	—
38	C₄⋊Dic₆₁	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	C₂×#92418	—	2	1	no	—
41	C₂²×D₆₁	—	4	1	no	—
42	Q₈×C₆₁	—	2	1	no	—
43	Q₈×C₆₁	D₈	1	2	yes	—
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
49	C₂×Dic₆₁	—	4	1	no	—
50	D₂₄₄	—	4	1	no	—
51	D₂₄₄	—	4	1	no	—
52	D₂₄₄	—	4	1	no	—
53	D₂₄₄	—	4	1	no	—
54	D₂₄₄	—	8	1	no	—
55	Dic₁₂₂	—	4	1	no	—
56	Dic₁₂₂	—	4	1	no	—
57	C₈×C₆₁	—	2	1	no	—
58	C₈×C₆₁	—	2	1	no	—
59	C₈×C₆₁	—	4	1	no	—
60	#92418	—	4	1	no	—
61	#92418	—	4	1	no	—
62	C₂²×C₆₁	#121	1	1	yes	characteristic
63	C₂×D₆₁	—	8	1	no	—
64	C₂×D₆₁	—	8	1	no	—
65	C₂×D₆₁	—	8	1	no	—
66	C₄×C₆₁	—	2	2	no	—
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	4	1	no	—
69	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
70	C₄×C₆₁	#135	1	1	yes	characteristic
71	C₄×C₆₁	#135	1	1	yes	characteristic
72	C₄×C₆₁	#137	1	1	yes	characteristic
73	Dic₆₁	—	8	1	no	—
74	#1682	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	#464	1	1	yes	characteristic
76	C₂×C₆₁	#466	1	1	yes	characteristic
77	C₂×C₆₁	#471	1	1	yes	characteristic
78	D₆₁	—	16	1	no	—
79	#486	—	61	1	no	—
80	#462	—	61	1	no	—
81	#460	—	61	1	no	—
82	C₈×Q₈	D₆₁	1	1	yes	characteristic, complemented
83	#335	—	61	1	no	—
84	#332	—	61	1	no	—
85	#331	—	61	1	no	—
86	C₆₁	#1682	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	122	1	no	—
88	C₂×SD₁₆	—	122	1	no	—
89	C₄⋊Q₈	—	61	1	no	—
90	#127	—	61	1	no	—
91	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	—	61	1	no	—
94	C₄⋊C₈	—	61	1	no	—
95	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
96	#103	—	122	1	no	—
97	#102	—	122	1	no	—
98	#102	—	122	1	no	—
99	#102	—	122	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
102	C₂×Q₈	—	122	1	no	—
103	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
104	C₂×D₄	—	122	1	no	—
105	C₂×D₄	—	122	1	no	—
106	C₂×D₄	—	122	1	no	—
107	Q₁₆	—	244	2	no	—
108	SD₁₆	—	244	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	122	1	no	—
111	C₂×C₈	—	122	1	no	—
112	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
113	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	122	1	no	—
116	C₄⋊C₄	—	122	1	no	—
117	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
118	C₄²	—	2	1	no	—
119	C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	C₂³	—	244	1	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	244	1	no	—
123	Q₈	—	244	1	no	—
124	Q₈	D₄₈₈	1	2	yes	—
125	D₄	—	488	1	no	—
126	D₄	—	244	1	no	—
127	D₄	—	244	1	no	—
128	D₄	—	244	1	no	—
129	D₄	—	244	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	244	1	no	—
133	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
134	C₂×C₄	#11758222	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	4	1	no	—
139	C₈	—	244	1	no	—
140	C₈	—	244	1	no	—
141	C₂²	—	488	1	no	—
142	C₂²	—	488	1	no	—
143	C₂²	—	488	1	no	—
144	C₂²	#14519751	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	488	1	no	—
149	C₄	C₂×D₄₈₈	1	1	yes	characteristic
150	C₄	#14519797	1	1	yes	characteristic
151	C₄	#14519799	1	1	yes	characteristic
152	C₄	#14519808	1	1	yes	characteristic
153	C₂	—	976	1	no	—
154	C₂	#16839984	1	1	yes	characteristic
155	C₂	#16839998	1	1	yes	characteristic
156	C₂	#16840025	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840109

C₂

yes

maximal, characteristic

#16839972

C₂

yes

maximal, characteristic, complemented

#16839962

C₂

yes

maximal, characteristic

#16839515

C₂

yes

maximal, characteristic, complemented

#16839507

C₂

yes

maximal, characteristic, complemented

#16839505

C₂

yes

maximal, characteristic

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#11758202

—

C₂×#11758201

—

#14519777

C₂²

yes

characteristic

#14519774

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

—

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

—

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758202

—

#11758201

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

D₈

yes

—

C₂×C₄×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

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#486

—

#462

—

#460

—

C₈×Q₈

D₆₁

yes

characteristic, complemented

#335

—

#332

—

#331

—

C₆₁

#1682

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×SD₁₆

—

122

—

C₄⋊Q₈

—

#127

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

122

—

#102

—

122

—

#102

—

122

—

#102

—

122

—

100

C₄×C₈

—

101

C₄×C₈

C₂×D₆₁

yes

characteristic

102

C₂×Q₈

—

122

—

103

C₂×Q₈

D₂₄₄

yes

characteristic

104

C₂×D₄

—

122

—

105

C₂×D₄

—

122

—

106

C₂×D₄

—

122

—

107

Q₁₆

—

244

—

108

SD₁₆

—

244

—

109

C₂×C₈

—

110

C₂×C₈

—

122

—

111

C₂×C₈

—

122

—

112

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

113

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

122

—

116

C₄⋊C₄

—

122

—

117

C₄⋊C₄

D₂₄₄

yes

characteristic

118

C₄²

—

119

C₄²

C₂²×D₆₁

yes

characteristic

120

C₂³

—

244

—

121

Q₈

—

122

Q₈

—

244

—

123

Q₈

—

244

—

124

Q₈

D₄₈₈

yes

—

125

D₄

—

488

—

126

D₄

—

244

—

127

D₄

—

244

—

128

D₄

—

244

—

129

D₄

—

244

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

244

—

133

C₂×C₄

C₂×D₂₄₄

yes

characteristic

134

C₂×C₄

#11758222

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

244

—

140

C₈

—

244

—

141

C₂²

—

488

—

142

C₂²

—

488

—

143

C₂²

—

488

—

144

C₂²

#14519751

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

488

—

149

C₄

C₂×D₄₈₈

yes

characteristic

150

C₄

#14519797

yes

characteristic

151

C₄

#14519799

yes

characteristic

152

C₄

#14519808

yes

characteristic

153

C₂

—

976

—

154

C₂

#16839984

yes

characteristic

155

C₂

#16839998

yes

characteristic

156

C₂

#16840025

yes

characteristic

	subgroup	G/H
center	C₂²	#14519751
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	#121
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#1682
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14519751

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

#121

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

#1682

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
2. #16839972	C₂	semidirect
4. #16839515	C₂	semidirect
5. #16839507	C₂	semidirect
7. C₈×Q₈×C₆₁	C₂	semidirect
82. C₈×Q₈	D₆₁	semidirect
86. C₆₁	#1682	semidirect

H ≅ G/N

type

2. #16839972

C₂

semidirect

4. #16839515

C₂

semidirect

5. #16839507

C₂

semidirect

7. C₈×Q₈×C₆₁

C₂

semidirect

82. C₈×Q₈

D₆₁

semidirect

86. C₆₁

#1682

semidirect

Subgroups — #21445263

Conjugacy classes of subgroups

Complements