Group #21445271 — subgroups

Subgroups — #21445271

Order 7808.

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

normal subgroups only

This group has 160 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	#16840113	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840053	C₂	1	1	yes	maximal, characteristic
3	#16840043	C₂	1	1	yes	maximal, characteristic
4	C₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic
5	#16839510	C₂	1	1	yes	maximal, characteristic, complemented
6	#16839507	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₆₁	—	2	1	no	—
9	#14519777	C₂²	1	1	yes	characteristic
10	#14519777	C₂²	1	1	yes	characteristic
11	C₄×D₂₄₄	C₂²	1	1	yes	characteristic
12	#14519711	—	2	1	no	—
13	#14519709	—	2	1	no	—
14	C₄⋊#92418	C₂²	1	1	yes	characteristic
15	#14519702	—	2	1	no	—
16	#14519702	—	2	1	no	—
17	#14519699	—	2	1	no	—
18	#14519699	—	2	1	no	—
19	#14519689	—	2	1	no	—
20	C₄⋊C₈×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×D₂₄₄	D₄	1	1	yes	characteristic
24	C₂×C₄×D₆₁	—	2	1	no	—
25	C₂×Dic₁₂₂	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	1	no	—
27	C₂×C₈×C₆₁	—	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	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	#11758198	—	2	1	no	—
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	—	2	1	no	—
37	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
38	C₂×#92418	—	2	1	no	—
39	C₈×D₆₁	—	4	2	no	—
40	C₂²×D₆₁	—	2	1	no	—
41	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
44	C₂×Dic₆₁	—	2	1	no	—
45	C₂×Dic₆₁	—	4	1	no	—
46	C₂×Dic₆₁	—	4	1	no	—
47	D₂₄₄	—	2	1	no	—
48	D₂₄₄	Q₁₆	1	2	yes	—
49	C₄×D₆₁	—	4	1	no	—
50	C₄×D₆₁	—	4	1	no	—
51	Dic₁₂₂	—	4	1	no	—
52	Dic₁₂₂	—	4	1	no	—
53	Dic₁₂₂	—	8	1	no	—
54	C₈×C₆₁	—	2	1	no	—
55	C₈×C₆₁	—	2	1	no	—
56	C₈×C₆₁	—	4	1	no	—
57	C₈×C₆₁	—	4	1	no	—
58	#92418	—	4	1	no	—
59	C₂²×C₆₁	#121	1	1	yes	characteristic
60	C₂×D₆₁	—	2	2	no	—
61	C₂×D₆₁	—	4	1	no	—
62	C₂×D₆₁	—	4	1	no	—
63	C₄×C₆₁	—	2	1	no	—
64	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₄×C₆₁	#135	1	1	yes	characteristic
66	C₄×C₆₁	#135	1	1	yes	characteristic
67	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	Dic₆₁	—	4	1	no	—
69	Dic₆₁	—	8	1	no	—
70	Dic₆₁	—	8	1	no	—
71	#1683	—	61	1	no	maximal, Sylow
72	C₂×C₆₁	#463	1	1	yes	characteristic
73	C₂×C₆₁	#467	1	1	yes	characteristic
74	C₂×C₆₁	#471	1	1	yes	characteristic
75	D₆₁	—	4	2	no	—
76	#487	—	61	1	no	—
77	#476	—	61	1	no	—
78	#474	—	61	1	no	—
79	C₈×D₄	—	61	1	no	—
80	#335	D₆₁	1	1	yes	characteristic, complemented
81	#333	—	61	1	no	—
82	#332	—	61	1	no	—
83	C₆₁	#1683	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	122	1	no	—
85	C₄⋊Q₈	—	61	1	no	—
86	C₄⋊Q₈	—	61	1	no	—
87	C₄×D₄	—	61	1	no	—
88	#107	—	122	1	no	—
89	#106	—	122	1	no	—
90	C₄⋊C₈	—	61	1	no	—
91	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
92	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
93	#103	—	122	1	no	—
94	#103	—	122	1	no	—
95	#102	—	122	1	no	—
96	#102	—	122	1	no	—
97	C₂²⋊C₈	—	122	1	no	—
98	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
99	C₂×Q₈	—	122	1	no	—
100	C₂×Q₈	—	122	1	no	—
101	C₂×D₄	—	61	1	no	—
102	C₂²×C₄	—	122	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	122	1	no	—
106	C₂×C₈	—	244	2	no	—
107	C₂×C₈	D₂₄₄	1	1	yes	characteristic
108	C₂×C₈	D₂₄₄	1	1	yes	characteristic
109	C₄⋊C₄	—	61	1	no	—
110	C₄⋊C₄	—	122	1	no	—
111	C₄⋊C₄	—	122	1	no	—
112	C₄⋊C₄	—	122	1	no	—
113	C₄⋊C₄	—	122	1	no	—
114	C₂²⋊C₄	—	122	1	no	—
115	C₄²	C₂²×D₆₁	1	1	yes	characteristic
116	C₂³	—	122	1	no	—
117	Q₈	—	488	1	no	—
118	Q₈	—	244	1	no	—
119	Q₈	—	244	1	no	—
120	D₄	—	61	2	no	—
121	D₄	—	122	1	no	—
122	C₂×C₄	—	122	1	no	—
123	C₂×C₄	—	244	1	no	—
124	C₂×C₄	—	244	1	no	—
125	C₂×C₄	—	244	1	no	—
126	C₂×C₄	—	244	1	no	—
127	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
128	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
129	C₂×C₄	#11758226	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	244	1	no	—
135	C₂²	—	122	2	no	—
136	C₂²	—	244	1	no	—
137	C₂²	—	244	1	no	—
138	C₂²	#14519752	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	488	1	no	—
141	C₄	—	488	1	no	—
142	C₄	—	244	1	no	—
143	C₄	Q₁₆×D₆₁	1	1	yes	characteristic
144	C₄	#14519796	1	1	yes	characteristic
145	C₄	#14519799	1	1	yes	characteristic
146	C₄	#14519805	1	1	yes	characteristic
147	C₂	—	244	2	no	—
148	C₂	#16839978	1	1	yes	characteristic
149	C₂	#16840004	1	1	yes	characteristic
150	C₂	#16840026	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840113

C₂

yes

maximal, characteristic, complemented

#16840053

C₂

yes

maximal, characteristic

#16840043

C₂

yes

maximal, characteristic

C₈×D₂₄₄

C₂

yes

maximal, characteristic

#16839510

C₂

yes

maximal, characteristic, complemented

#16839507

C₂

yes

maximal, characteristic, complemented

C₆₁×#335

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₆₁

—

#14519777

C₂²

yes

characteristic

#14519777

C₂²

yes

characteristic

C₄×D₂₄₄

C₂²

yes

—

—

C₂²

yes

—

—

—

—

—

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₂₄₄

D₄

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

characteristic

C₂×#92418

—

C₈×D₆₁

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

Q₁₆

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#487

—

#476

—

#474

—

C₈×D₄

—

#335

D₆₁

yes

characteristic, complemented

#333

—

#332

—

C₆₁

#1683

yes

characteristic, complemented, Sylow

C₂²×C₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

122

—

#106

—

122

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#103

—

122

—

#103

—

122

—

#102

—

122

—

#102

—

122

—

C₂²⋊C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

100

C₂×Q₈

—

122

—

101

C₂×D₄

—

102

C₂²×C₄

—

122

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

122

—

106

C₂×C₈

—

244

—

107

C₂×C₈

D₂₄₄

yes

characteristic

108

C₂×C₈

D₂₄₄

yes

characteristic

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

C₂²×D₆₁

yes

characteristic

116

C₂³

—

122

—

117

Q₈

—

488

—

118

Q₈

—

244

—

119

Q₈

—

244

—

120

D₄

—

121

D₄

—

122

—

122

C₂×C₄

—

122

—

123

C₂×C₄

—

244

—

124

C₂×C₄

—

244

—

125

C₂×C₄

—

244

—

126

C₂×C₄

—

244

—

127

C₂×C₄

C₂×D₂₄₄

yes

characteristic

128

C₂×C₄

D₄×D₆₁

yes

characteristic

129

C₂×C₄

#11758226

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

244

—

135

C₂²

—

122

—

136

C₂²

—

244

—

137

C₂²

—

244

—

138

C₂²

#14519752

yes

characteristic

139

C₄

—

140

C₄

—

488

—

141

C₄

—

488

—

142

C₄

—

244

—

143

C₄

Q₁₆×D₆₁

yes

characteristic

144

C₄

#14519796

yes

characteristic

145

C₄

#14519799

yes

characteristic

146

C₄

#14519805

yes

characteristic

147

C₂

—

244

—

148

C₂

#16839978

yes

characteristic

149

C₂

#16840004

yes

characteristic

150

C₂

#16840026

yes

characteristic

	subgroup	G/H
center	C₂²	#14519752
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#335	C₂
socle	C₂²×C₆₁	#121
hypercenter	#335	D₆₁
hypocenter	C₆₁	#1683
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14519752

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#335

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #16840113	C₂	semidirect
5. #16839510	C₂	semidirect
6. #16839507	C₂	semidirect
7. C₆₁×#335	C₂	semidirect
80. #335	D₆₁	semidirect
83. C₆₁	#1683	semidirect

H ≅ G/N

type

1. #16840113

C₂

semidirect

5. #16839510

C₂

semidirect

6. #16839507

C₂

semidirect

7. C₆₁×#335

C₂

semidirect

80. #335

D₆₁

semidirect

83. C₆₁

#1683

semidirect

Subgroups — #21445271

Conjugacy classes of subgroups

Complements