Group #21454688 — subgroups

Subgroups — #21454688

Order 7808.

456 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 189 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	C₂²×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#16840239	C₂	1	2	yes	maximal
3	#16839870	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₂₄₄	C₂	1	4	yes	maximal, complemented
5	#16839800	C₂	1	1	yes	maximal, characteristic
6	#16839772	C₂	1	2	yes	maximal
7	C₆₁×#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
9	C₂×Dic₂₄₄	C₄	1	8	yes	complemented
10	C₂×Dic₂₄₄	C₂²	1	2	yes	—
11	C₂×Dic₂₄₄	C₂²	1	2	yes	—
12	C₄×Dic₁₂₂	C₂²	1	8	yes	—
13	#14519742	—	2	4	no	—
14	#14519736	C₂²	1	2	yes	—
15	#14519733	—	2	4	no	—
16	#14519711	C₂²	1	2	yes	—
17	#14519709	C₂²	1	2	yes	—
18	#14519702	C₂²	1	8	yes	—
19	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	#117×C₆₁	C₂²	1	1	yes	characteristic
22	#97×C₆₁	C₂²	1	2	yes	—
23	C₄×C₈×C₆₁	C₂²	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
25	C₂²×Dic₆₁	—	2	2	no	—
26	C₂×Dic₁₂₂	—	2	2	no	—
27	C₂×Dic₁₂₂	—	2	4	no	—
28	C₂×Dic₁₂₂	C₂×C₄	1	8	yes	—
29	C₂×Dic₁₂₂	C₂³	1	4	yes	—
30	M₁₆×C₆₁	—	2	2	no	—
31	C₂×C₈×C₆₁	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₆₁	C₂³	1	2	yes	—
33	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
36	C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
37	C₄²×C₆₁	C₂³	1	2	yes	—
38	C₂²⋊Dic₆₁	—	2	2	no	—
39	C₄⋊Dic₆₁	C₂³	1	4	yes	—
40	#11758196	—	2	8	no	—
41	C₄×Dic₆₁	—	2	8	no	—
42	C₄×Dic₆₁	—	2	2	no	—
43	Dic₂₄₄	C₂×C₄	1	8	yes	—
44	Dic₂₄₄	C₂×C₄	1	8	yes	—
45	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	#41	1	2	yes	—
50	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₆₁	—	2	8	no	—
52	C₂×Dic₆₁	—	2	4	no	—
53	C₂×Dic₆₁	—	2	4	no	—
54	Dic₁₂₂	—	2	8	no	—
55	Dic₁₂₂	—	2	4	no	—
56	Dic₁₂₂	C₂²×C₄	1	8	yes	—
57	C₈×C₆₁	—	2	2	no	—
58	C₈×C₆₁	C₂²×C₄	1	2	yes	—
59	C₈×C₆₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₆₁	—	2	2	no	—
61	C₂²×C₆₁	C₄×D₄	1	2	yes	—
62	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₆₁	—	2	4	no	—
64	C₄×C₆₁	C₄×D₄	1	2	yes	—
65	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
67	Dic₆₁	—	2	8	no	—
68	Dic₆₁	—	4	4	no	—
69	#2950	—	61	1	no	maximal, Sylow
70	C₂×C₆₁	—	2	2	no	—
71	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
72	C₂×C₆₁	#578	1	2	yes	—
73	C₂²×Q₁₆	—	61	1	no	—
74	#519	—	61	2	no	—
75	#441	—	61	4	no	—
76	C₄×Q₁₆	—	61	4	no	—
77	#427	—	61	1	no	—
78	#419	—	61	2	no	—
79	#405	D₆₁	1	1	yes	characteristic
80	C₆₁	#2950	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	—	61	2	no	—
82	C₂×Q₁₆	—	61	8	no	—
83	C₂×Q₁₆	—	61	2	no	—
84	C₂×Q₁₆	—	61	2	no	—
85	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
87	C₄×Q₈	—	61	8	no	—
88	C₄×Q₈	—	122	4	no	—
89	#117	—	61	2	no	—
90	#117	—	122	4	no	—
91	#117	C₂×D₆₁	1	1	yes	characteristic
92	#107	—	61	2	no	—
93	#106	—	61	2	no	—
94	#103	—	61	8	no	—
95	#97	C₂×D₆₁	1	2	yes	—
96	C₄×C₈	C₂×D₆₁	1	2	yes	—
97	C₂×Q₈	—	61	8	no	—
98	C₂×Q₈	—	61	4	no	—
99	C₂×Q₈	—	122	2	no	—
100	C₂×Q₈	—	122	4	no	—
101	C₂²×C₄	—	122	2	no	—
102	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
103	Q₁₆	—	61	8	no	—
104	Q₁₆	—	61	8	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	C₄×D₆₁	1	4	yes	—
107	C₂×C₈	C₂²×D₆₁	1	2	yes	—
108	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
110	C₄⋊C₄	—	61	4	no	—
111	C₄⋊C₄	—	122	8	no	—
112	C₄⋊C₄	D₂₄₄	1	2	yes	—
113	C₂²⋊C₄	—	122	2	no	—
114	C₂²⋊C₄	D₂₄₄	1	2	yes	—
115	C₄²	—	122	2	no	—
116	C₄²	—	122	8	no	—
117	C₄²	C₂²×D₆₁	1	2	yes	—
118	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
119	Q₈	—	61	8	no	—
120	Q₈	—	122	4	no	—
121	Q₈	—	122	8	no	—
122	C₂×C₄	—	122	4	no	—
123	C₂×C₄	—	122	8	no	—
124	C₂×C₄	—	122	4	no	—
125	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
126	C₂×C₄	C₂×D₂₄₄	1	4	yes	—
127	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
128	C₂×C₄	#11758222	1	2	yes	—
129	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	C₂×C₄×D₆₁	1	2	yes	—
132	C₈	C₂×C₄×D₆₁	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	C₄×D₂₄₄	1	2	yes	—
135	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	122	8	no	—
138	C₄	—	244	4	no	—
139	C₄	C₄×D₂₄₄	1	2	yes	—
140	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
141	C₄	C₂×#11758222	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×C₄×D₂₄₄	1	1	yes	characteristic
144	C₂	#16840579	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₄₄

C₂

yes

maximal, characteristic

#16840239

C₂

yes

maximal

#16839870

C₂

yes

maximal, complemented

C₄×Dic₂₄₄

C₂

yes

maximal, complemented

#16839800

C₂

yes

maximal, characteristic

#16839772

C₂

yes

maximal

C₆₁×#405

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

C₂×Dic₂₄₄

C₄

yes

complemented

C₂×Dic₂₄₄

C₂²

yes

—

C₂×Dic₂₄₄

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519742

—

#14519736

C₂²

yes

—

#14519733

—

#14519711

C₂²

yes

—

#14519709

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

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×Dic₁₂₂

C₂³

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

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

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

Dic₂₄₄

C₂×C₄

yes

—

Dic₂₄₄

C₂×C₄

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

C₂²×C₄

yes

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₄

yes

—

C₈×C₆₁

C₂²×C₄

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

C₂×#41

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2950

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×D₄

yes

characteristic

C₂×C₆₁

#578

yes

—

C₂²×Q₁₆

—

#519

—

#441

—

C₄×Q₁₆

—

#427

—

#419

—

#405

D₆₁

yes

characteristic

C₆₁

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

#117

—

122

—

#117

C₂×D₆₁

yes

—

—

—

C₂×D₆₁

yes

—

C₄×C₈

C₂×D₆₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

100

C₂×Q₈

—

122

—

101

C₂²×C₄

—

122

—

102

C₂²×C₄

C₂²×D₆₁

yes

characteristic

103

Q₁₆

—

104

Q₁₆

—

105

M₁₆

—

106

C₂×C₈

C₄×D₆₁

yes

—

107

C₂×C₈

C₂²×D₆₁

yes

—

108

C₂×C₈

C₂²×D₆₁

yes

characteristic

109

C₂×C₈

C₂²×D₆₁

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

D₂₄₄

yes

—

113

C₂²⋊C₄

—

122

—

114

C₂²⋊C₄

D₂₄₄

yes

—

115

C₄²

—

122

—

116

C₄²

—

122

—

117

C₄²

C₂²×D₆₁

yes

—

118

C₂³

C₂×D₂₄₄

yes

characteristic

119

Q₈

—

120

Q₈

—

122

—

121

Q₈

—

122

—

122

C₂×C₄

—

122

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

122

—

125

C₂×C₄

C₂×C₄×D₆₁

yes

—

126

C₂×C₄

C₂×D₂₄₄

yes

—

127

C₂×C₄

C₂×D₂₄₄

yes

characteristic

128

C₂×C₄

#11758222

yes

—

129

C₂×C₄

C₂³×D₆₁

yes

characteristic

130

C₈

—

131

C₈

C₂×C₄×D₆₁

yes

—

132

C₈

C₂×C₄×D₆₁

yes

—

133

C₂²

—

134

C₂²

C₄×D₂₄₄

yes

—

135

C₂²

C₂²×D₂₄₄

yes

characteristic

136

C₄

—

137

C₄

—

122

—

138

C₄

—

244

—

139

C₄

C₄×D₂₄₄

yes

—

140

C₄

C₂²×C₄×D₆₁

yes

characteristic

141

C₄

C₂×#11758222

yes

characteristic

142

C₂

—

143

C₂

C₂×C₄×D₂₄₄

yes

characteristic

144

C₂

#16840579

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#405	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#405	D₆₁
hypocenter	C₆₁	#2950
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₄×C₆₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#405

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
3. #16839870	C₂	semidirect
4. C₄×Dic₂₄₄	C₂	semidirect
9. C₂×Dic₂₄₄	C₄	semidirect
80. C₆₁	#2950	semidirect

H ≅ G/N

type

3. #16839870

C₂

semidirect

4. C₄×Dic₂₄₄

C₂

semidirect

9. C₂×Dic₂₄₄

C₄

semidirect

80. C₆₁

#2950

semidirect

Subgroups — #21454688

Conjugacy classes of subgroups

Complements