Group #22590553 — subgroups

Subgroups — #22590553

Order 9088.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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	#17986249	C₂	1	1	yes	maximal, characteristic
2	#17986046	C₂	1	1	yes	maximal, characteristic, complemented
3	#17985823	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985822	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985812	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985780	C₂	1	1	yes	maximal, characteristic, complemented
7	#361×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765848	C₂²	1	1	yes	characteristic
9	M₁₆×D₇₁	—	2	1	no	—
10	#14556099	—	2	1	no	—
11	#14556068	C₂²	1	1	yes	characteristic
12	#14556064	C₂²	1	1	yes	characteristic
13	#14556063	—	2	1	no	—
14	#14556060	C₂²	1	1	yes	characteristic
15	#14556049	—	2	1	no	—
16	C₂×D₈×C₇₁	C₂²	1	1	yes	characteristic
17	M₃₂×C₇₁	C₂²	1	1	yes	characteristic
18	#108×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊D₇₁	—	4	1	no	—
20	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
21	#11765849	—	2	1	no	—
22	#11765848	—	2	1	no	—
23	#11765848	—	2	1	no	—
24	#11765848	—	4	1	no	—
25	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₇₁	—	4	1	no	—
27	C₄⋊Dic₇₁	—	2	1	no	—
28	C₂²⋊Dic₇₁	—	4	1	no	—
29	C₇₁⋊C₁₆	—	2	1	no	—
30	C₂×D₄×C₇₁	—	2	1	no	—
31	D₈×C₇₁	—	2	1	no	—
32	D₈×C₇₁	—	2	1	no	—
33	M₁₆×C₇₁	—	2	1	no	—
34	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₁₆×C₇₁	—	2	1	no	—
36	C₂³×C₇₁	—	4	1	no	—
37	C₂²×D₇₁	SD₁₆	1	1	yes	characteristic
38	D₄×C₇₁	—	4	1	no	—
39	D₄×C₇₁	—	4	1	no	—
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂²⋊D₇₁	—	8	2	no	—
42	C₂×Dic₇₁	—	4	1	no	—
43	C₂×Dic₇₁	SD₁₆	1	1	yes	characteristic
44	C₄×D₇₁	—	2	1	no	—
45	C₄×D₇₁	—	2	1	no	—
46	C₈×C₇₁	—	4	1	no	—
47	C₈×C₇₁	#41	1	1	yes	characteristic
48	C₈×C₇₁	#41	1	1	yes	characteristic
49	C₇₁⋊C₈	—	2	1	no	—
50	C₇₁⋊C₈	—	4	1	no	—
51	C₂²×C₇₁	—	8	1	no	—
52	C₂²×C₇₁	—	8	1	no	—
53	C₂²×C₇₁	—	8	1	no	—
54	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×D₇₁	—	2	1	no	—
56	C₂×D₇₁	—	4	1	no	—
57	C₄×C₇₁	#123	1	1	yes	characteristic
58	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₇₁	—	2	1	no	—
60	Dic₇₁	—	8	1	no	—
61	C₂×C₇₁	—	2	1	no	—
62	C₂×C₇₁	—	16	1	no	—
63	C₂×C₇₁	#481	1	1	yes	characteristic
64	D₇₁	—	4	1	no	—
65	#2238	—	71	1	no	maximal, Sylow
66	C₇₁	#2238	1	1	yes	characteristic, complemented, Sylow
67	#469	—	71	1	no	—
68	#430	—	71	1	no	—
69	#365	—	71	1	no	—
70	#365	—	71	1	no	—
71	#361	—	71	1	no	—
72	#361	D₇₁	1	1	yes	characteristic, complemented
73	#349	—	71	1	no	—
74	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
75	C₂×M₁₆	—	71	1	no	—
76	C₂×M₁₆	—	142	1	no	—
77	#121	—	142	1	no	—
78	M₃₂	—	71	1	no	—
79	M₃₂	C₂×D₇₁	1	1	yes	characteristic
80	#108	—	71	1	no	—
81	#108	—	142	1	no	—
82	#108	C₂×D₇₁	1	1	yes	characteristic
83	#106	—	71	1	no	—
84	#102	—	142	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	284	1	no	—
87	C₂²×C₄	—	71	1	no	—
88	D₈	—	2	1	no	—
89	D₈	—	2	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	284	1	no	—
92	M₁₆	—	142	1	no	—
93	M₁₆	—	142	1	no	—
94	M₁₆	—	142	1	no	—
95	C₂×C₈	—	284	1	no	—
96	C₂×C₈	—	71	1	no	—
97	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
98	C₄⋊C₄	—	142	1	no	—
99	C₂²⋊C₄	—	284	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	142	1	no	—
102	C₂³	—	4	1	no	—
103	C₂³	—	71	1	no	—
104	D₄	—	4	1	no	—
105	D₄	—	4	1	no	—
106	D₄	—	568	2	no	—
107	C₂×C₄	—	284	1	no	—
108	C₂×C₄	—	71	1	no	—
109	C₂×C₄	—	142	1	no	—
110	C₂×C₄	—	142	1	no	—
111	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	284	1	no	—
114	C₈	—	142	1	no	—
115	C₈	#11765864	1	1	yes	characteristic
116	C₈	#11765865	1	1	yes	characteristic
117	C₂²	—	8	1	no	—
118	C₂²	—	8	1	no	—
119	C₂²	—	8	1	no	—
120	C₂²	—	284	1	no	—
121	C₂²	—	142	1	no	—
122	C₂²	SD₁₆×D₇₁	1	1	yes	characteristic
123	C₄	—	568	1	no	—
124	C₄	—	142	1	no	—
125	C₄	#14556110	1	1	yes	characteristic
126	C₄	#14556152	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	16	1	no	—
129	C₂	—	284	1	no	—
130	C₂	#17986316	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986249

C₂

yes

maximal, characteristic

#17986046

C₂

yes

maximal, characteristic, complemented

#17985823

C₂

yes

maximal, characteristic, complemented

#17985822

C₂

yes

maximal, characteristic, complemented

#17985812

C₂

yes

maximal, characteristic, complemented

#17985780

C₂

yes

maximal, characteristic, complemented

#361×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765848

C₂²

yes

characteristic

M₁₆×D₇₁

—

#14556099

—

#14556068

C₂²

yes

characteristic

#14556064

C₂²

yes

characteristic

#14556063

—

#14556060

C₂²

yes

characteristic

#14556049

—

C₂×D₈×C₇₁

C₂²

yes

characteristic

M₃₂×C₇₁

C₂²

yes

characteristic

#108×C₇₁

C₂²

yes

characteristic

C₂×C₂²⋊D₇₁

—

C₂×C₄×D₇₁

D₄

yes

—

—

—

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₈×D₇₁

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

—

C₂²×D₇₁

SD₁₆

yes

characteristic

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

SD₁₆

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

yes

C₈×C₇₁

yes

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#481

yes

characteristic

D₇₁

—

#2238

—

maximal, Sylow

C₇₁

#2238

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic, complemented

#349

—

C₂×D₈

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

142

—

#121

—

142

—

M₃₂

—

M₃₂

C₂×D₇₁

yes

characteristic

#108

—

#108

—

142

—

#108

C₂×D₇₁

yes

characteristic

#106

—

#102

—

142

—

C₂×D₄

—

C₂×D₄

—

284

—

C₂²×C₄

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

284

—

M₁₆

—

142

—

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

284

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

100

C₁₆

—

101

C₁₆

—

142

—

102

C₂³

—

103

C₂³

—

104

D₄

—

105

D₄

—

106

D₄

—

568

—

107

C₂×C₄

—

284

—

108

C₂×C₄

—

109

C₂×C₄

—

142

—

110

C₂×C₄

—

142

—

111

C₂×C₄

D₄×D₇₁

yes

characteristic

112

C₈

—

113

C₈

—

284

—

114

C₈

—

142

—

115

C₈

#11765864

yes

characteristic

116

C₈

#11765865

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

284

—

121

C₂²

—

142

—

122

C₂²

SD₁₆×D₇₁

yes

characteristic

123

C₄

—

568

—

124

C₄

—

142

—

125

C₄

#14556110

yes

characteristic

126

C₄

#14556152

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

284

—

130

C₂

#17986316

yes

characteristic

	subgroup	G/H
center	C₂	#17986316
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#361×C₇₁	C₂
socle	C₂×C₇₁	#481
hypercenter	#361	D₇₁
hypocenter	C₇₁	#2238
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

subgroup

G/H

center

C₂

#17986316

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#361×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

N	H ≅ G/N	type
2. #17986046	C₂	semidirect
3. #17985823	C₂	semidirect
4. #17985822	C₂	semidirect
5. #17985812	C₂	semidirect
6. #17985780	C₂	semidirect
7. #361×C₇₁	C₂	semidirect
66. C₇₁	#2238	semidirect
72. #361	D₇₁	semidirect

H ≅ G/N

type

2. #17986046

C₂

semidirect

3. #17985823

C₂

semidirect

4. #17985822

C₂

semidirect

5. #17985812

C₂

semidirect

6. #17985780

C₂

semidirect

7. #361×C₇₁

C₂

semidirect

66. C₇₁

#2238

semidirect

72. #361

D₇₁

semidirect

Subgroups — #22590553

Conjugacy classes of subgroups

Complements