Group #22590562 — subgroups

Subgroups — #22590562

Order 9088.

160 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17986403	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986374	C₂	1	1	yes	maximal, characteristic
3	#17985801	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985800	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985799	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
9	C₂×#11765850	—	2	1	no	—
10	#14556127	—	2	1	no	—
11	#14556124	—	2	1	no	—
12	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14556062	C₂²	1	1	yes	characteristic
14	#14556061	—	2	1	no	—
15	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	#107×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₂	—	4	1	no	—
20	C₂×D₂₈₄	—	2	1	no	—
21	C₂×C₂²⋊D₇₁	—	4	1	no	—
22	C₂×C₂²⋊D₇₁	—	4	1	no	—
23	D₅₆₈	—	2	1	no	—
24	D₅₆₈	—	2	1	no	—
25	#11765850	—	4	2	no	—
26	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
27	C₄⋊Dic₇₁	—	2	1	no	—
28	Dic₇₁⋊C₄	—	4	1	no	—
29	C₄×Dic₇₁	D₄	1	1	yes	characteristic
30	C₇₁⋊C₁₆	—	2	1	no	—
31	C₂×D₄×C₇₁	—	2	1	no	—
32	D₈×C₇₁	—	2	1	no	—
33	D₈×C₇₁	—	2	1	no	—
34	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₁₆×C₇₁	—	2	1	no	—
37	C₂³×C₇₁	—	4	1	no	—
38	C₂²×D₇₁	—	4	1	no	—
39	D₄×C₇₁	—	4	1	no	—
40	D₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	—	4	1	no	—
42	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂²⋊D₇₁	—	8	2	no	—
44	C₂²⋊D₇₁	—	8	2	no	—
45	C₂×Dic₇₁	—	4	1	no	—
46	C₂×Dic₇₁	D₈	1	1	yes	characteristic
47	C₂×Dic₇₁	D₈	1	1	yes	characteristic
48	D₂₈₄	—	4	1	no	—
49	D₂₈₄	—	4	1	no	—
50	Dic₁₄₂	—	8	2	no	—
51	C₈×C₇₁	#41	1	1	yes	characteristic
52	C₈×C₇₁	#41	1	1	yes	characteristic
53	C₇₁⋊C₈	SD₁₆	1	2	yes	—
54	C₂²×C₇₁	—	8	1	no	—
55	C₂²×C₇₁	—	8	1	no	—
56	C₂²×C₇₁	—	8	1	no	—
57	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
58	C₂×D₇₁	—	8	1	no	—
59	C₂×D₇₁	—	8	1	no	—
60	C₂×D₇₁	—	8	1	no	—
61	C₄×C₇₁	—	8	1	no	—
62	C₄×C₇₁	#124	1	1	yes	characteristic
63	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
64	Dic₇₁	—	8	1	no	—
65	Dic₇₁	D₁₆	1	2	yes	—
66	Dic₇₁	SD₃₂	1	2	yes	—
67	C₂×C₇₁	—	16	1	no	—
68	C₂×C₇₁	#486	1	1	yes	characteristic
69	C₂×C₇₁	C₂×D₁₆	1	1	yes	characteristic
70	C₂×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
71	D₇₁	—	16	1	no	—
72	#2240	—	71	1	no	maximal, Sylow
73	C₇₁	#2240	1	1	yes	characteristic, complemented, Sylow
74	#500	—	71	1	no	—
75	#493	—	71	1	no	—
76	#357	—	71	1	no	—
77	#357	—	71	1	no	—
78	#357	—	71	1	no	—
79	#357	D₇₁	1	1	yes	characteristic, complemented
80	C₄×C₁₆	—	71	1	no	—
81	C₂×D₈	—	71	1	no	—
82	C₂×D₈	—	142	1	no	—
83	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
84	C₄⋊Q₈	—	142	1	no	—
85	#127	—	142	1	no	—
86	C₂×C₁₆	—	71	1	no	—
87	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
88	#107	—	71	1	no	—
89	#107	—	142	1	no	—
90	#107	C₂×D₇₁	1	1	yes	characteristic
91	C₄×C₈	—	71	1	no	—
92	C₂×Q₈	—	284	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	284	1	no	—
95	C₂×D₄	—	284	1	no	—
96	C₂×D₄	—	142	1	no	—
97	D₈	—	2	1	no	—
98	D₈	—	2	1	no	—
99	D₈	—	284	2	no	—
100	D₈	—	142	1	no	—
101	D₈	—	142	1	no	—
102	C₂×C₈	—	71	1	no	—
103	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	284	1	no	—
106	C₄⋊C₄	—	142	1	no	—
107	C₄²	—	71	1	no	—
108	C₁₆	—	2	1	no	—
109	C₁₆	—	142	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	284	1	no	—
112	Q₈	—	568	2	no	—
113	D₄	—	4	1	no	—
114	D₄	—	4	1	no	—
115	D₄	—	284	1	no	—
116	D₄	—	284	1	no	—
117	D₄	—	568	2	no	—
118	D₄	—	568	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	284	1	no	—
121	C₂×C₄	—	71	1	no	—
122	C₂×C₄	—	71	1	no	—
123	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
124	C₈	—	71	2	no	—
125	C₈	#11765864	1	1	yes	characteristic
126	C₈	#11765865	1	1	yes	characteristic
127	C₂²	—	8	1	no	—
128	C₂²	—	8	1	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	568	1	no	—
131	C₂²	—	568	1	no	—
132	C₂²	—	568	1	no	—
133	C₂²	D₈×D₇₁	1	1	yes	characteristic
134	C₄	—	8	1	no	—
135	C₄	—	568	1	no	—
136	C₄	—	71	2	no	—
137	C₄	—	71	2	no	—
138	C₄	#14556114	1	1	yes	characteristic
139	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
140	C₂	—	16	1	no	—
141	C₂	—	1136	1	no	—
142	C₂	#17986344	1	1	yes	characteristic
143	C₂	D₁₆×D₇₁	1	1	yes	characteristic
144	C₂	SD₃₂×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986403

C₂

yes

maximal, characteristic, complemented

#17986374

C₂

yes

maximal, characteristic

#17985801

C₂

yes

maximal, characteristic, complemented

#17985800

C₂

yes

maximal, characteristic, complemented

#17985799

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₇₁

C₂

yes

maximal, characteristic, complemented

#357×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₂²

yes

—

—

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556061

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×D₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×C₂²⋊D₇₁

—

C₂×C₂²⋊D₇₁

—

D₅₆₈

—

D₅₆₈

—

#11765850

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

—

C₂²×D₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²⋊D₇₁

—

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

SD₁₆

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#124

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

D₁₆

yes

—

Dic₇₁

SD₃₂

yes

—

C₂×C₇₁

—

C₂×C₇₁

#486

yes

characteristic

C₂×C₇₁

C₂×D₁₆

yes

characteristic

C₂×C₇₁

C₂×SD₃₂

yes

characteristic

D₇₁

—

#2240

—

maximal, Sylow

C₇₁

#2240

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₄×C₁₆

—

C₂×D₈

—

C₂×D₈

—

142

—

C₂×D₈

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

142

—

#127

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#107

—

142

—

#107

C₂×D₇₁

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

284

—

C₂×D₄

—

C₂×D₄

—

284

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

D₈

—

D₈

—

D₈

—

284

—

100

D₈

—

142

—

101

D₈

—

142

—

102

C₂×C₈

—

103

C₂×C₈

C₂²×D₇₁

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

284

—

106

C₄⋊C₄

—

142

—

107

C₄²

—

108

C₁₆

—

109

C₁₆

—

142

—

110

C₂³

—

111

C₂³

—

284

—

112

Q₈

—

568

—

113

D₄

—

114

D₄

—

115

D₄

—

284

—

116

D₄

—

284

—

117

D₄

—

568

—

118

D₄

—

568

—

119

C₂×C₄

—

120

C₂×C₄

—

284

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₄×D₇₁

yes

characteristic

124

C₈

—

125

C₈

#11765864

yes

characteristic

126

C₈

#11765865

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

568

—

131

C₂²

—

568

—

132

C₂²

—

568

—

133

C₂²

D₈×D₇₁

yes

characteristic

134

C₄

—

135

C₄

—

568

—

136

C₄

—

137

C₄

—

138

C₄

#14556114

yes

characteristic

139

C₄

SD₁₆×D₇₁

yes

characteristic

140

C₂

—

141

C₂

—

1136

—

142

C₂

#17986344

yes

characteristic

143

C₂

D₁₆×D₇₁

yes

characteristic

144

C₂

SD₃₂×D₇₁

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#357×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#357	D₇₁
hypocenter	C₇₁	#2240
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#357×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #17986403	C₂	semidirect
3. #17985801	C₂	semidirect
4. #17985800	C₂	semidirect
5. #17985799	C₂	semidirect
6. C₁₆×Dic₇₁	C₂	semidirect
7. #357×C₇₁	C₂	semidirect
73. C₇₁	#2240	semidirect
79. #357	D₇₁	semidirect

H ≅ G/N

type

1. #17986403

C₂

semidirect

3. #17985801

C₂

semidirect

4. #17985800

C₂

semidirect

5. #17985799

C₂

semidirect

6. C₁₆×Dic₇₁

C₂

semidirect

7. #357×C₇₁

C₂

semidirect

73. C₇₁

#2240

semidirect

79. #357

D₇₁

semidirect

Subgroups — #22590562

Conjugacy classes of subgroups

Complements