Group #22590179 — subgroups

Subgroups — #22590179

Order 9088.

144 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 63 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	#17986417	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986416	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986114	C₂	1	1	yes	maximal, characteristic
4	#17985819	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985782	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985774	C₂	1	1	yes	maximal, characteristic, complemented
7	#350×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556148	C₄	1	1	yes	characteristic
9	#14556134	C₂²	1	1	yes	characteristic
10	#14556068	—	2	1	no	—
11	#14556068	C₂²	1	1	yes	characteristic
12	#14556067	—	2	1	no	—
13	C₂×C₇₁⋊C₁₆	—	2	1	no	—
14	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₁₆×D₇₁	—	2	1	no	—
16	#14556063	C₄	1	1	yes	characteristic
17	#14556055	C₄	1	1	yes	characteristic
18	#14556054	C₄	1	1	yes	characteristic
19	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
20	#131×C₇₁	C₂²	1	1	yes	characteristic
21	M₃₂×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
23	#11765865	C₂×C₄	1	1	yes	characteristic
24	#11765855	C₈	1	1	yes	characteristic
25	#11765853	C₈	1	1	yes	characteristic
26	#11765852	C₈	1	1	yes	characteristic
27	#11765850	C₈	1	1	yes	characteristic
28	#11765849	C₂×C₄	1	1	yes	characteristic
29	#11765848	—	2	1	no	—
30	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₈×D₇₁	—	2	1	no	—
32	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
33	C₇₁⋊C₁₆	—	2	1	no	—
34	C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
35	C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
36	#41×C₇₁	C₂×C₄	1	1	yes	characteristic
37	M₁₆×C₇₁	—	2	1	no	—
38	M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	C₁₆×C₇₁	—	2	1	no	—
42	C₁₆×C₇₁	—	2	1	no	—
43	Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
44	D₄×C₇₁	—	2	1	no	—
45	D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₇₁	—	2	1	no	—
47	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²⋊D₇₁	—	2	1	no	—
49	C₂×Dic₇₁	—	2	1	no	—
50	D₂₈₄	C₂×C₈	1	1	yes	characteristic
51	C₄×D₇₁	—	2	1	no	—
52	Dic₁₄₂	C₂×C₈	1	1	yes	characteristic
53	C₈×C₇₁	—	2	1	no	—
54	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₈×C₇₁	#41	1	1	yes	characteristic
56	C₇₁⋊C₈	—	2	1	no	—
57	C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
58	C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₇₁	—	2	1	no	—
60	C₂²×C₇₁	#131	1	1	yes	characteristic
61	C₂×D₇₁	—	2	1	no	—
62	C₄×C₇₁	—	2	1	no	—
63	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
64	C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
65	Dic₇₁	—	2	1	no	—
66	Dic₇₁	—	4	1	no	—
67	C₂×C₇₁	—	2	1	no	—
68	C₂×C₇₁	—	4	1	no	—
69	C₂×C₇₁	C₈×D₄	1	1	yes	characteristic
70	D₇₁	—	4	1	no	—
71	#2171	—	71	1	no	maximal, Sylow
72	C₇₁	#2171	1	1	yes	characteristic, complemented, Sylow
73	#504	—	71	1	no	—
74	#504	—	71	1	no	—
75	#443	—	71	1	no	—
76	#364	—	71	1	no	—
77	#350	—	71	1	no	—
78	#350	D₇₁	1	1	yes	characteristic, complemented
79	#346	—	71	1	no	—
80	#135	—	71	1	no	—
81	#131	—	71	1	no	—
82	#131	C₂×D₇₁	1	1	yes	characteristic
83	M₃₂	—	71	1	no	—
84	M₃₂	—	142	1	no	—
85	M₃₂	—	142	1	no	—
86	M₃₂	C₂×D₇₁	1	1	yes	characteristic
87	C₂×C₁₆	—	71	1	no	—
88	C₂×C₁₆	—	142	1	no	—
89	C₂×C₁₆	—	142	1	no	—
90	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
91	#108	—	71	1	no	—
92	#104	—	71	1	no	—
93	#104	—	71	1	no	—
94	C₄×C₈	—	71	1	no	—
95	#41	—	71	1	no	—
96	#41	C₄×D₇₁	1	1	yes	characteristic
97	Q₁₆	—	71	1	no	—
98	SD₁₆	—	71	1	no	—
99	SD₁₆	—	71	1	no	—
100	D₈	—	71	1	no	—
101	M₁₆	—	2	1	no	—
102	M₁₆	—	71	1	no	—
103	M₁₆	—	142	1	no	—
104	M₁₆	C₄×D₇₁	1	1	yes	characteristic
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	71	1	no	—
107	C₂×C₈	—	142	1	no	—
108	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
109	C₄²	—	71	1	no	—
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	—	71	1	no	—
113	C₁₆	—	71	1	no	—
114	C₁₆	—	142	1	no	—
115	Q₈	—	71	1	no	—
116	Q₈	C₈×D₇₁	1	1	yes	characteristic
117	D₄	—	2	1	no	—
118	D₄	—	71	1	no	—
119	D₄	—	142	1	no	—
120	D₄	C₈×D₇₁	1	1	yes	characteristic
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	142	1	no	—
123	C₂×C₄	—	142	1	no	—
124	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	71	1	no	—
127	C₈	—	71	1	no	—
128	C₈	—	142	1	no	—
129	C₈	D₄×D₇₁	1	1	yes	characteristic
130	C₈	#11765864	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	142	1	no	—
133	C₂²	#14556136	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	284	1	no	—
136	C₄	—	142	1	no	—
137	C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
138	C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	4	1	no	—
141	C₂	—	284	1	no	—
142	C₂	(C₂²⋊D₇₁)⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986417

C₂

yes

maximal, characteristic, complemented

#17986416

C₂

yes

maximal, characteristic, complemented

#17986114

C₂

yes

maximal, characteristic

#17985819

C₂

yes

maximal, characteristic, complemented

#17985782

C₂

yes

maximal, characteristic, complemented

#17985774

C₂

yes

maximal, characteristic, complemented

#350×C₇₁

C₂

yes

maximal, characteristic, complemented

#14556148

C₄

yes

characteristic

#14556134

C₂²

yes

characteristic

#14556068

—

#14556068

C₂²

yes

characteristic

#14556067

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₇₁

—

#14556063

C₄

yes

characteristic

#14556055

C₄

yes

characteristic

#14556054

C₄

yes

characteristic

C₈×Dic₇₁

C₂²

yes

characteristic

#131×C₇₁

C₂²

yes

characteristic

M₃₂×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#11765865

C₂×C₄

yes

characteristic

#11765855

C₈

yes

characteristic

#11765853

C₈

yes

characteristic

#11765852

C₈

yes

characteristic

#11765850

C₈

yes

characteristic

#11765849

C₂×C₄

yes

characteristic

#11765848

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₇₁

—

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

D₄

yes

characteristic

C₇₁⋊C₁₆

D₄

yes

characteristic

#41×C₇₁

C₂×C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

Q₈×C₇₁

C₂×C₈

yes

characteristic

D₄×C₇₁

—

D₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

C₂×C₈

yes

characteristic

C₄×D₇₁

—

Dic₁₄₂

C₂×C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#131

yes

characteristic

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₂²×C₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₈×D₄

yes

characteristic

D₇₁

—

#2171

—

maximal, Sylow

C₇₁

#2171

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic, complemented

—

—

—

C₂×D₇₁

yes

characteristic

M₃₂

—

M₃₂

—

142

—

M₃₂

—

142

—

M₃₂

C₂×D₇₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂×D₇₁

yes

—

—

—

C₄×C₈

—

#41

—

#41

C₄×D₇₁

yes

characteristic

Q₁₆

—

SD₁₆

—

SD₁₆

—

100

D₈

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

142

—

104

M₁₆

C₄×D₇₁

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

142

—

108

C₂×C₈

C₂²×D₇₁

yes

characteristic

109

C₄²

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

142

—

115

Q₈

—

116

Q₈

C₈×D₇₁

yes

characteristic

117

D₄

—

118

D₄

—

119

D₄

—

142

—

120

D₄

C₈×D₇₁

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

142

—

123

C₂×C₄

—

142

—

124

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

142

—

129

C₈

D₄×D₇₁

yes

characteristic

130

C₈

#11765864

yes

characteristic

131

C₂²

—

132

C₂²

—

142

—

133

C₂²

#14556136

yes

characteristic

134

C₄

—

135

C₄

—

284

—

136

C₄

—

142

—

137

C₄

(C₂²⋊D₇₁)⋊C₄

yes

characteristic

138

C₄

C₂×C₈×D₇₁

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

284

—

142

C₂

(C₂²⋊D₇₁)⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₇₁
commutator	C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#350×C₇₁	C₂
socle	C₂×C₇₁	C₈×D₄
hypercenter	#350	D₇₁
hypocenter	C₇₁	#2171
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

subgroup

G/H

center

C₈

D₄×D₇₁

commutator

C₄×C₇₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#350×C₇₁

C₂

socle

C₂×C₇₁

C₈×D₄

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #350

N	H ≅ G/N	type
1. #17986417	C₂	semidirect
2. #17986416	C₂	semidirect
4. #17985819	C₂	semidirect
5. #17985782	C₂	semidirect
6. #17985774	C₂	semidirect
7. #350×C₇₁	C₂	semidirect
72. C₇₁	#2171	semidirect
78. #350	D₇₁	semidirect

H ≅ G/N

type

1. #17986417

C₂

semidirect

2. #17986416

C₂

semidirect

4. #17985819

C₂

semidirect

5. #17985782

C₂

semidirect

6. #17985774

C₂

semidirect

7. #350×C₇₁

C₂

semidirect

72. C₇₁

#2171

semidirect

78. #350

D₇₁

semidirect

Subgroups — #22590179

Conjugacy classes of subgroups

Complements