Group #21458462 — subgroups

Subgroups — #21458462

Order 7808.

370 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 139 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	#16840491	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic
3	#16840113	C₂	1	3	yes	maximal, complemented
4	#16840048	C₂	1	3	yes	maximal, complemented
5	#16839968	C₂	1	3	yes	maximal
6	C₄×#11758190	C₂	1	3	yes	maximal
7	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₁	—	2	1	no	—
9	C₂×#11758190	C₂²	1	3	yes	—
10	#14519777	C₂²	1	3	yes	—
11	#14519777	C₂²	1	3	yes	—
12	#14519768	—	2	3	no	—
13	#14519757	—	2	3	no	—
14	Q₈×Dic₆₁	—	2	1	no	—
15	C₄×Dic₁₂₂	C₂²	1	3	yes	—
16	C₄×D₂₄₄	C₂²	1	3	yes	—
17	#14519709	C₂²	1	3	yes	—
18	#14519702	C₂²	1	6	yes	—
19	#14519702	C₂²	1	3	yes	—
20	#14519699	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₂²	1	3	yes	—
23	C₄×C₈×C₆₁	C₂²	1	3	yes	—
24	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
25	Q₈×D₆₁	—	4	4	no	—
26	C₂×D₂₄₄	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₆₁	—	2	3	no	—
28	C₂×Dic₁₂₂	—	2	3	no	—
29	C₂×Dic₁₂₂	C₂³	1	3	yes	—
30	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₂³	1	3	yes	—
32	C₄⋊C₄×C₆₁	D₄	1	3	yes	—
33	C₄²×C₆₁	C₂³	1	3	yes	—
34	#11758198	—	2	3	no	—
35	C₄⋊Dic₆₁	—	2	3	no	—
36	C₄⋊Dic₆₁	—	2	3	no	—
37	C₄⋊Dic₆₁	—	2	3	no	—
38	C₄⋊Dic₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊Dic₆₁	C₂³	1	3	yes	—
40	#11758196	—	2	3	no	—
41	C₄×Dic₆₁	—	2	3	no	—
42	#11758190	—	2	6	no	—
43	C₂²×D₆₁	—	2	1	no	—
44	Q₈×C₆₁	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₆₁	—	2	1	no	—
49	C₂×Dic₆₁	—	2	3	no	—
50	C₂×Dic₆₁	—	2	3	no	—
51	D₂₄₄	—	2	1	no	—
52	D₂₄₄	#41	1	2	yes	—
53	C₄×D₆₁	—	4	6	no	—
54	Dic₁₂₂	—	2	3	no	—
55	Dic₁₂₂	—	2	3	no	—
56	Dic₁₂₂	—	4	6	no	—
57	C₈×C₆₁	—	2	3	no	—
58	C₈×C₆₁	—	2	1	no	—
59	C₂²×C₆₁	C₂²×D₄	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	3	no	—
64	C₄×C₆₁	C₂×SD₁₆	1	6	yes	—
65	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
66	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₆₁	—	4	3	no	—
68	Dic₆₁	—	4	3	no	—
69	Dic₆₁	—	4	1	no	—
70	#3359	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	#548	1	1	yes	characteristic
72	C₂×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₆₁	#578	1	1	yes	characteristic
74	D₆₁	—	4	2	no	—
75	#557	—	61	1	no	—
76	D₄×Q₈	—	61	1	no	—
77	#487	—	61	3	no	—
78	#475	—	61	3	no	—
79	#461	—	61	3	no	—
80	C₈×Q₈	D₆₁	1	1	yes	characteristic, complemented
81	C₄×SD₁₆	—	61	3	no	—
82	C₆₁	#3359	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	122	1	no	—
84	C₂×SD₁₆	—	61	3	no	—
85	C₄⋊Q₈	—	61	3	no	—
86	C₄⋊Q₈	—	61	3	no	—
87	#125	—	122	3	no	—
88	C₂²⋊Q₈	—	122	3	no	—
89	C₄×Q₈	—	61	3	no	—
90	C₄×Q₈	—	122	1	no	—
91	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
92	C₄×D₄	—	61	3	no	—
93	#106	—	61	3	no	—
94	C₄⋊C₈	C₂×D₆₁	1	3	yes	—
95	#103	—	61	3	no	—
96	#103	—	61	6	no	—
97	#102	—	61	1	no	—
98	C₄×C₈	C₂×D₆₁	1	3	yes	—
99	C₂×Q₈	—	61	3	no	—
100	C₂×Q₈	—	122	3	no	—
101	C₂×Q₈	—	244	4	no	—
102	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
103	C₂×D₄	—	61	1	no	—
104	C₂²×C₄	—	122	3	no	—
105	SD₁₆	—	122	6	no	—
106	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₆₁	1	3	yes	—
108	C₄⋊C₄	—	61	1	no	—
109	C₄⋊C₄	—	61	3	no	—
110	C₄⋊C₄	—	122	3	no	—
111	C₄⋊C₄	—	122	3	no	—
112	C₄⋊C₄	—	122	3	no	—
113	C₄⋊C₄	—	122	3	no	—
114	C₄⋊C₄	D₂₄₄	1	3	yes	—
115	C₂²⋊C₄	—	122	3	no	—
116	C₄²	—	122	3	no	—
117	C₄²	C₂²×D₆₁	1	3	yes	—
118	C₂³	—	122	1	no	—
119	Q₈	—	122	3	no	—
120	Q₈	—	122	3	no	—
121	Q₈	—	244	6	no	—
122	Q₈	#11758190	1	4	yes	complemented
123	D₄	—	61	2	no	—
124	D₄	—	122	1	no	—
125	C₂×C₄	—	122	1	no	—
126	C₂×C₄	—	122	3	no	—
127	C₂×C₄	—	122	3	no	—
128	C₂×C₄	—	244	6	no	—
129	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
130	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
131	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
132	C₈	—	2	3	no	—
133	C₈	—	2	1	no	—
134	C₂²	—	122	2	no	—
135	C₂²	—	244	1	no	—
136	C₂²	—	244	1	no	—
137	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	244	3	no	—
140	C₄	—	244	1	no	—
141	C₄	—	244	3	no	—
142	C₄	C₂×#11758190	1	6	yes	—
143	C₄	#41×D₆₁	1	1	yes	characteristic
144	C₄	#14519825	1	1	yes	characteristic
145	C₂	—	244	2	no	—
146	C₂	#16840424	1	1	yes	characteristic
147	C₂	C₂²×#11758190	1	1	yes	characteristic
148	C₂	#16840579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840491

C₂

yes

maximal, characteristic, complemented

Q₈×D₂₄₄

C₂

yes

maximal, characteristic

#16840113

C₂

yes

maximal, complemented

#16840048

C₂

yes

maximal, complemented

#16839968

C₂

yes

maximal

C₄×#11758190

C₂

yes

maximal

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₁

—

C₂×#11758190

C₂²

yes

—

#14519777

C₂²

yes

—

#14519777

C₂²

yes

—

#14519768

—

#14519757

—

Q₈×Dic₆₁

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

—

#14519709

C₂²

yes

—

#14519702

C₂²

yes

—

#14519702

C₂²

yes

—

#14519699

C₂²

yes

characteristic

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

Q₈×D₆₁

—

C₂×D₂₄₄

C₂³

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

#11758190

—

C₂²×D₆₁

—

Q₈×C₆₁

SD₁₆

yes

complemented

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

#41

yes

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂²×SD₁₆

yes

characteristic

C₂×C₆₁

#578

yes

characteristic

D₆₁

—

#557

—

D₄×Q₈

—

#487

—

#475

—

#461

—

C₈×Q₈

D₆₁

yes

characteristic, complemented

C₄×SD₁₆

—

C₆₁

#3359

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

122

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

C₂×D₆₁

yes

—

#103

—

#103

—

#102

—

C₄×C₈

C₂×D₆₁

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

122

—

101

C₂×Q₈

—

244

—

102

C₂×Q₈

D₂₄₄

yes

characteristic

103

C₂×D₄

—

104

C₂²×C₄

—

122

—

105

SD₁₆

—

122

—

106

C₂×C₈

C₂²×D₆₁

yes

characteristic

107

C₂×C₈

C₂²×D₆₁

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

122

—

111

C₄⋊C₄

—

122

—

112

C₄⋊C₄

—

122

—

113

C₄⋊C₄

—

122

—

114

C₄⋊C₄

D₂₄₄

yes

—

115

C₂²⋊C₄

—

122

—

116

C₄²

—

122

—

117

C₄²

C₂²×D₆₁

yes

—

118

C₂³

—

122

—

119

Q₈

—

122

—

120

Q₈

—

122

—

121

Q₈

—

244

—

122

Q₈

#11758190

yes

complemented

123

D₄

—

124

D₄

—

122

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

244

—

129

C₂×C₄

C₂×D₂₄₄

yes

—

130

C₂×C₄

C₂×D₂₄₄

yes

—

131

C₂×C₄

C₂³×D₆₁

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

122

—

135

C₂²

—

244

—

136

C₂²

—

244

—

137

C₂²

C₂²×D₂₄₄

yes

characteristic

138

C₄

—

139

C₄

—

244

—

140

C₄

—

244

—

141

C₄

—

244

—

142

C₄

C₂×#11758190

yes

—

143

C₄

#41×D₆₁

yes

characteristic

144

C₄

#14519825

yes

characteristic

145

C₂

—

244

—

146

C₂

#16840424

yes

characteristic

147

C₂

C₂²×#11758190

yes

characteristic

148

C₂

#16840579

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#3359
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

#3359

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #16840491	C₂	semidirect
3. #16840113	C₂	semidirect
4. #16840048	C₂	semidirect
7. C₈×Q₈×C₆₁	C₂	semidirect
44. Q₈×C₆₁	SD₁₆	semidirect
80. C₈×Q₈	D₆₁	semidirect
82. C₆₁	#3359	semidirect
122. Q₈	#11758190	semidirect

H ≅ G/N

type

1. #16840491

C₂

semidirect

3. #16840113

C₂

semidirect

4. #16840048

C₂

semidirect

7. C₈×Q₈×C₆₁

C₂

semidirect

44. Q₈×C₆₁

SD₁₆

semidirect

80. C₈×Q₈

D₆₁

semidirect

82. C₆₁

#3359

semidirect

122. Q₈

#11758190

semidirect

Subgroups — #21458462

Conjugacy classes of subgroups

Complements