Group #21456478 — subgroups

Subgroups — #21456478

Order 7808.

338 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16840374	C₂	1	2	yes	maximal
2	#16840164	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840142	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840131	C₂	1	2	yes	maximal, complemented
5	#16840020	C₂	1	2	yes	maximal
6	#16840003	C₂	1	2	yes	maximal
7	#16839870	C₂	1	2	yes	maximal, complemented
8	C₄×Dic₂₄₄	C₂	1	2	yes	maximal, complemented
9	C₆₁×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₄₄	C₂²	1	2	yes	—
11	C₂×Dic₂₄₄	C₂²	1	2	yes	—
12	#14519777	C₂²	1	2	yes	—
13	#14519770	—	2	4	no	—
14	#14519769	C₂²	1	2	yes	—
15	#14519767	—	2	4	no	—
16	#14519756	C₂²	1	4	yes	—
17	C₄×Dic₁₂₂	C₂²	1	4	yes	—
18	#14519711	C₂²	1	2	yes	—
19	#14519709	C₂²	1	2	yes	—
20	#14519702	C₂²	1	4	yes	—
21	#14519702	C₂²	1	4	yes	—
22	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	#117×C₆₁	C₂²	1	1	yes	characteristic
25	#97×C₆₁	C₂²	1	1	yes	characteristic
26	#97×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₂₂	C₂³	1	4	yes	—
31	M₁₆×C₆₁	—	2	2	no	—
32	C₂×C₈×C₆₁	—	2	2	no	—
33	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
38	C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
39	C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄²×C₆₁	C₂³	1	1	yes	characteristic
41	C₂²⋊Dic₆₁	—	2	4	no	—
42	C₄⋊Dic₆₁	—	2	4	no	—
43	C₄⋊Dic₆₁	C₂³	1	4	yes	—
44	#11758196	—	2	4	no	—
45	#11758196	—	2	4	no	—
46	#11758196	—	2	4	no	—
47	C₄×Dic₆₁	—	2	4	no	—
48	Dic₂₄₄	—	2	4	no	—
49	Dic₂₄₄	—	2	4	no	—
50	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	—	2	1	no	—
52	C₂×C₄×C₆₁	—	2	1	no	—
53	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
57	C₂×Dic₆₁	—	2	4	no	—
58	C₂×Dic₆₁	—	2	4	no	—
59	Dic₁₂₂	—	2	4	no	—
60	Dic₁₂₂	—	2	4	no	—
61	C₈×C₆₁	—	2	1	no	—
62	C₈×C₆₁	—	2	1	no	—
63	C₈×C₆₁	#41	1	2	yes	—
64	C₈×C₆₁	#41	1	2	yes	—
65	C₂²×C₆₁	—	2	1	no	—
66	C₂²×C₆₁	—	2	2	no	—
67	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₆₁	—	2	1	no	—
69	C₄×C₆₁	—	2	2	no	—
70	C₄×C₆₁	—	2	2	no	—
71	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
72	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
73	Dic₆₁	—	4	4	no	—
74	Dic₆₁	—	4	4	no	—
75	#3177	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	—	4	1	no	—
77	C₂×C₆₁	#532	1	1	yes	characteristic
78	C₂×C₆₁	#578	1	2	yes	—
79	#541	—	61	2	no	—
80	#499	—	61	1	no	—
81	#494	—	61	1	no	—
82	#491	—	61	2	no	—
83	#470	—	61	2	no	—
84	#467	—	61	2	no	—
85	#441	—	61	2	no	—
86	C₄×Q₁₆	—	61	2	no	—
87	#405	D₆₁	1	1	yes	characteristic
88	C₆₁	#3177	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₁₆	—	61	2	no	—
90	C₂×Q₁₆	—	61	2	no	—
91	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
93	C₄⋊Q₈	—	61	2	no	—
94	#126	—	122	4	no	—
95	#125	—	61	2	no	—
96	#125	—	122	4	no	—
97	C₂²⋊Q₈	—	61	4	no	—
98	C₄×Q₈	—	61	4	no	—
99	#117	C₂×D₆₁	1	1	yes	characteristic
100	#107	—	61	2	no	—
101	#106	—	61	2	no	—
102	#103	—	61	4	no	—
103	#103	—	61	4	no	—
104	#97	C₂×D₆₁	1	1	yes	characteristic
105	#97	C₂×D₆₁	1	1	yes	characteristic
106	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
107	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
108	C₂×Q₈	—	61	4	no	—
109	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
110	Q₁₆	—	122	4	no	—
111	Q₁₆	—	122	4	no	—
112	M₁₆	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
115	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
118	C₄⋊C₄	—	61	4	no	—
119	C₄⋊C₄	—	122	4	no	—
120	C₄⋊C₄	—	122	4	no	—
121	C₄⋊C₄	—	122	4	no	—
122	C₄⋊C₄	—	122	4	no	—
123	C₄⋊C₄	D₂₄₄	1	2	yes	—
124	C₂²⋊C₄	—	122	4	no	—
125	C₂²⋊C₄	D₂₄₄	1	2	yes	—
126	C₄²	—	122	4	no	—
127	C₄²	C₂²×D₆₁	1	1	yes	characteristic
128	C₄²	C₂²×D₆₁	1	1	yes	characteristic
129	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
130	Q₈	—	122	4	no	—
131	Q₈	—	122	4	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	122	4	no	—
135	C₂×C₄	—	122	4	no	—
136	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
137	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
138	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
139	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	#11758222	1	2	yes	—
143	C₈	#11758222	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	244	4	no	—
151	C₄	—	244	4	no	—
152	C₄	C₂×#11758222	1	1	yes	characteristic
153	C₄	C₂×#11758222	1	1	yes	characteristic
154	C₂	—	4	1	no	—
155	C₂	#16840312	1	1	yes	characteristic
156	C₂	#16840579	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840374

C₂

yes

maximal

#16840164

C₂

yes

maximal, characteristic, complemented

#16840142

C₂

yes

maximal, characteristic, complemented

#16840131

C₂

yes

maximal, complemented

#16840020

C₂

yes

maximal

#16840003

C₂

yes

maximal

#16839870

C₂

yes

maximal, complemented

C₄×Dic₂₄₄

C₂

yes

maximal, complemented

C₆₁×#405

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

C₂²

yes

—

C₂×Dic₂₄₄

C₂²

yes

—

#14519777

C₂²

yes

—

#14519770

—

#14519769

C₂²

yes

—

#14519767

—

#14519756

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519711

C₂²

yes

—

#14519709

C₂²

yes

—

#14519702

C₂²

yes

—

#14519702

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

#117×C₆₁

C₂²

yes

characteristic

#97×C₆₁

C₂²

yes

characteristic

#97×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

C₂³

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

—

C₂²⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

#41

yes

—

C₈×C₆₁

#41

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#3177

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#532

yes

characteristic

C₂×C₆₁

#578

yes

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

#405

D₆₁

yes

characteristic

C₆₁

#3177

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

#126

—

122

—

#125

—

#125

—

122

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₆₁

yes

characteristic

100

#107

—

101

#106

—

102

#103

—

103

#103

—

104

#97

C₂×D₆₁

yes

characteristic

105

#97

C₂×D₆₁

yes

characteristic

106

C₄×C₈

C₂×D₆₁

yes

characteristic

107

C₄×C₈

C₂×D₆₁

yes

characteristic

108

C₂×Q₈

—

109

C₂²×C₄

C₂²×D₆₁

yes

characteristic

110

Q₁₆

—

122

—

111

Q₁₆

—

122

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×D₆₁

yes

characteristic

115

C₂×C₈

C₂²×D₆₁

yes

characteristic

116

C₂×C₈

C₂²×D₆₁

yes

characteristic

117

C₂×C₈

C₂²×D₆₁

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

122

—

120

C₄⋊C₄

—

122

—

121

C₄⋊C₄

—

122

—

122

C₄⋊C₄

—

122

—

123

C₄⋊C₄

D₂₄₄

yes

—

124

C₂²⋊C₄

—

122

—

125

C₂²⋊C₄

D₂₄₄

yes

—

126

C₄²

—

122

—

127

C₄²

C₂²×D₆₁

yes

characteristic

128

C₄²

C₂²×D₆₁

yes

characteristic

129

C₂³

C₂×D₂₄₄

yes

characteristic

130

Q₈

—

122

—

131

Q₈

—

122

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

122

—

135

C₂×C₄

—

122

—

136

C₂×C₄

C₂×D₂₄₄

yes

—

137

C₂×C₄

C₂×D₂₄₄

yes

—

138

C₂×C₄

C₂×D₂₄₄

yes

characteristic

139

C₂×C₄

C₂³×D₆₁

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

#11758222

yes

—

143

C₈

#11758222

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂²×D₂₄₄

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

244

—

151

C₄

—

244

—

152

C₄

C₂×#11758222

yes

characteristic

153

C₄

C₂×#11758222

yes

characteristic

154

C₂

—

155

C₂

#16840312

yes

characteristic

156

C₂

#16840579

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#405	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#405	D₆₁
hypocenter	C₆₁	#3177
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#405

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #16840164	C₂	semidirect
3. #16840142	C₂	semidirect
4. #16840131	C₂	semidirect
7. #16839870	C₂	semidirect
8. C₄×Dic₂₄₄	C₂	semidirect
88. C₆₁	#3177	semidirect

H ≅ G/N

type

2. #16840164

C₂

semidirect

3. #16840142

C₂

semidirect

4. #16840131

C₂

semidirect

7. #16839870

C₂

semidirect

8. C₄×Dic₂₄₄

C₂

semidirect

88. C₆₁

#3177

semidirect

Subgroups — #21456478

Conjugacy classes of subgroups

Complements