Group #22590420 — subgroups

Subgroups — #22590420

Order 9088.

150 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 43 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	#17986442	C₂	1	1	yes	maximal, characteristic
2	#17986440	C₂	1	1	yes	maximal, characteristic
3	#17986377	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986115	C₂	1	1	yes	maximal, characteristic
5	#17985819	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985814	C₂	1	1	yes	maximal, characteristic, complemented
7	#362×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556150	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
10	C₂×#11765855	—	2	1	no	—
11	#14556136	—	2	1	no	—
12	#14556126	—	2	1	no	—
13	Dic₅₆₈	—	2	1	no	—
14	#14556074	—	2	1	no	—
15	#14556073	—	2	1	no	—
16	#14556071	—	2	1	no	—
17	#14556068	C₂²	1	1	yes	characteristic
18	#14556054	—	2	1	no	—
19	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	M₃₂×C₇₁	C₂²	1	1	yes	characteristic
22	#108×C₇₁	C₂²	1	1	yes	characteristic
23	#11765865	—	2	1	no	—
24	C₂×Dic₁₄₂	—	2	1	no	—
25	Dic₂₈₄	—	2	1	no	—
26	Dic₂₈₄	—	2	1	no	—
27	Dic₂₈₄	D₄	1	1	yes	characteristic
28	#11765855	—	4	1	no	—
29	#11765855	—	4	1	no	—
30	#11765854	—	2	1	no	—
31	D₅₆₈	D₄	1	1	yes	characteristic
32	#11765848	—	4	1	no	—
33	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
34	C₈×D₇₁	—	4	1	no	—
35	Dic₇₁⋊C₄	—	4	2	no	—
36	C₄×Dic₇₁	D₄	1	1	yes	characteristic
37	C₇₁⋊C₁₆	—	2	1	no	—
38	C₂×Q₈×C₇₁	—	2	1	no	—
39	Q₁₆×C₇₁	—	2	1	no	—
40	Q₁₆×C₇₁	—	2	1	no	—
41	M₁₆×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₁₆×C₇₁	—	2	1	no	—
44	Q₈×C₇₁	—	4	1	no	—
45	Q₈×C₇₁	—	4	1	no	—
46	C₂×C₄×C₇₁	—	4	1	no	—
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₇₁	—	4	1	no	—
49	C₂×Dic₇₁	—	2	1	no	—
50	C₂×Dic₇₁	—	4	1	no	—
51	D₂₈₄	—	2	1	no	—
52	C₄×D₇₁	—	4	1	no	—
53	Dic₁₄₂	—	2	1	no	—
54	Dic₁₄₂	—	4	1	no	—
55	Dic₁₄₂	—	4	1	no	—
56	C₈×C₇₁	—	4	1	no	—
57	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₈×C₇₁	#41	1	1	yes	characteristic
59	C₇₁⋊C₈	D₈	1	1	yes	characteristic
60	C₇₁⋊C₈	D₈	1	1	yes	characteristic
61	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₂×D₇₁	—	4	1	no	—
63	C₄×C₇₁	—	8	1	no	—
64	C₄×C₇₁	#121	1	1	yes	characteristic
65	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
66	Dic₇₁	—	2	2	no	—
67	Dic₇₁	—	4	1	no	—
68	Dic₇₁	—	8	1	no	—
69	C₂×C₇₁	—	2	1	no	—
70	C₂×C₇₁	#459	1	1	yes	characteristic
71	D₇₁	—	8	1	no	—
72	#2214	—	71	1	no	maximal, Sylow
73	C₇₁	#2214	1	1	yes	characteristic, complemented, Sylow
74	#510	—	71	1	no	—
75	#510	—	71	1	no	—
76	#494	—	71	1	no	—
77	#443	—	71	1	no	—
78	#364	—	71	1	no	—
79	#362	—	71	1	no	—
80	#362	D₇₁	1	1	yes	characteristic, complemented
81	#135	—	71	1	no	—
82	C₂×Q₁₆	—	71	1	no	—
83	C₂×Q₁₆	—	142	1	no	—
84	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
85	#131	—	142	1	no	—
86	C₄⋊Q₈	—	142	1	no	—
87	Q₃₂	—	142	1	no	—
88	Q₃₂	—	142	1	no	—
89	SD₃₂	—	142	1	no	—
90	SD₃₂	—	142	1	no	—
91	M₃₂	—	71	1	no	—
92	M₃₂	C₂×D₇₁	1	1	yes	characteristic
93	#108	C₂×D₇₁	1	1	yes	characteristic
94	#104	—	142	1	no	—
95	C₄×C₈	—	71	1	no	—
96	#41	—	142	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	142	1	no	—
99	Q₁₆	—	2	1	no	—
100	Q₁₆	—	2	1	no	—
101	Q₁₆	—	284	1	no	—
102	Q₁₆	—	284	1	no	—
103	Q₁₆	—	71	1	no	—
104	Q₁₆	—	142	1	no	—
105	Q₁₆	—	142	1	no	—
106	SD₁₆	—	142	1	no	—
107	D₈	—	71	1	no	—
108	M₁₆	—	2	1	no	—
109	M₁₆	—	284	1	no	—
110	C₂×C₈	—	284	1	no	—
111	C₂×C₈	—	71	1	no	—
112	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
113	C₄⋊C₄	—	284	2	no	—
114	C₄²	—	71	1	no	—
115	C₁₆	—	2	1	no	—
116	C₁₆	—	142	1	no	—
117	Q₈	—	4	1	no	—
118	Q₈	—	4	1	no	—
119	Q₈	—	284	1	no	—
120	Q₈	—	284	1	no	—
121	Q₈	—	142	1	no	—
122	D₄	—	284	1	no	—
123	D₄	—	142	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	284	1	no	—
126	C₂×C₄	—	284	1	no	—
127	C₂×C₄	—	142	1	no	—
128	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
129	C₈	—	4	1	no	—
130	C₈	—	71	1	no	—
131	C₈	—	71	1	no	—
132	C₈	D₄×D₇₁	1	1	yes	characteristic
133	C₈	#11765865	1	1	yes	characteristic
134	C₂²	—	284	1	no	—
135	C₂²	#14556155	1	1	yes	characteristic
136	C₄	—	8	1	no	—
137	C₄	—	284	1	no	—
138	C₄	—	568	1	no	—
139	C₄	—	142	2	no	—
140	C₄	#14556101	1	1	yes	characteristic
141	C₄	D₈×D₇₁	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	568	1	no	—
144	C₂	#17986193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986442

C₂

yes

maximal, characteristic

#17986440

C₂

yes

maximal, characteristic

#17986377

C₂

yes

maximal, characteristic, complemented

#17986115

C₂

yes

maximal, characteristic

#17985819

C₂

yes

maximal, characteristic, complemented

#17985814

C₂

yes

maximal, characteristic, complemented

#362×C₇₁

C₂

yes

maximal, characteristic, complemented

#14556150

C₂²

yes

characteristic

C₂×Dic₂₈₄

C₂²

yes

—

—

—

Dic₅₆₈

—

—

—

—

C₂²

yes

characteristic

#14556054

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

M₃₂×C₇₁

C₂²

yes

characteristic

#108×C₇₁

C₂²

yes

characteristic

#11765865

—

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

D₄

yes

—

—

—

D₅₆₈

D₄

yes

characteristic

#11765848

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₈×D₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

D₈

yes

characteristic

C₇₁⋊C₈

D₈

yes

characteristic

C₂²×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#121

yes

characteristic

C₄×C₇₁

C₂×D₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#459

yes

characteristic

D₇₁

—

#2214

—

maximal, Sylow

C₇₁

#2214

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

#135

—

C₂×Q₁₆

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

#131

—

142

—

C₄⋊Q₈

—

142

—

Q₃₂

—

142

—

Q₃₂

—

142

—

SD₃₂

—

142

—

SD₃₂

—

142

—

M₃₂

—

M₃₂

C₂×D₇₁

yes

characteristic

#108

C₂×D₇₁

yes

characteristic

#104

—

142

—

C₄×C₈

—

#41

—

142

—

C₂×Q₈

—

C₂×Q₈

—

142

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

284

—

102

Q₁₆

—

284

—

103

Q₁₆

—

104

Q₁₆

—

142

—

105

Q₁₆

—

142

—

106

SD₁₆

—

142

—

107

D₈

—

108

M₁₆

—

109

M₁₆

—

284

—

110

C₂×C₈

—

284

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₇₁

yes

characteristic

113

C₄⋊C₄

—

284

—

114

C₄²

—

115

C₁₆

—

116

C₁₆

—

142

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

284

—

120

Q₈

—

284

—

121

Q₈

—

142

—

122

D₄

—

284

—

123

D₄

—

142

—

124

C₂×C₄

—

125

C₂×C₄

—

284

—

126

C₂×C₄

—

284

—

127

C₂×C₄

—

142

—

128

C₂×C₄

D₄×D₇₁

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

D₄×D₇₁

yes

characteristic

133

C₈

#11765865

yes

characteristic

134

C₂²

—

284

—

135

C₂²

#14556155

yes

characteristic

136

C₄

—

137

C₄

—

284

—

138

C₄

—

568

—

139

C₄

—

142

—

140

C₄

#14556101

yes

characteristic

141

C₄

D₈×D₇₁

yes

characteristic

142

C₂

—

143

C₂

—

568

—

144

C₂

#17986193

yes

characteristic

	subgroup	G/H
center	C₂	#17986193
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#362×C₇₁	C₂
socle	C₂×C₇₁	#459
hypercenter	#362	D₇₁
hypocenter	C₇₁	#2214
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

subgroup

G/H

center

C₂

#17986193

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#362×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

N	H ≅ G/N	type
3. #17986377	C₂	semidirect
5. #17985819	C₂	semidirect
6. #17985814	C₂	semidirect
7. #362×C₇₁	C₂	semidirect
73. C₇₁	#2214	semidirect
80. #362	D₇₁	semidirect

H ≅ G/N

type

3. #17986377

C₂

semidirect

5. #17985819

C₂

semidirect

6. #17985814

C₂

semidirect

7. #362×C₇₁

C₂

semidirect

73. C₇₁

#2214

semidirect

80. #362

D₇₁

semidirect

Subgroups — #22590420

Conjugacy classes of subgroups

Complements