Group #22587967 — subgroups

Subgroups — #22587967

Order 9088.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17986395	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986375	C₂	1	1	yes	maximal, characteristic
3	#17986371	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986347	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986346	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985721	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765853	—	2	2	no	—
9	C₂×#11765850	—	2	2	no	—
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556125	C₂²	1	1	yes	characteristic
12	#14556059	—	2	2	no	—
13	#14556050	—	2	2	no	—
14	#14556048	—	2	2	no	—
15	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₇₁	—	2	2	no	—
18	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
19	#127×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×D₂₈₄	—	2	1	no	—
23	C₂×D₂₈₄	—	2	1	no	—
24	C₂×D₂₈₄	—	2	1	no	—
25	#11765853	—	4	4	no	—
26	#11765850	—	4	4	no	—
27	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
28	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₂×Q₈×C₇₁	—	2	1	no	—
32	C₂×D₄×C₇₁	—	2	1	no	—
33	C₂×D₄×C₇₁	—	2	1	no	—
34	C₂×D₄×C₇₁	—	2	1	no	—
35	SD₁₆×C₇₁	—	4	4	no	—
36	C₂×C₈×C₇₁	D₄	1	2	yes	—
37	C₄⋊C₄×C₇₁	—	2	1	no	—
38	C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	C₂³×C₇₁	—	4	1	no	—
41	C₂²×D₇₁	—	4	1	no	—
42	Q₈×C₇₁	—	4	2	no	—
43	D₄×C₇₁	—	4	2	no	—
44	D₄×C₇₁	—	4	2	no	—
45	D₄×C₇₁	—	4	2	no	—
46	C₂×C₄×C₇₁	—	4	1	no	—
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₇₁	—	4	1	no	—
51	D₂₈₄	—	4	2	no	—
52	D₂₈₄	—	4	2	no	—
53	D₂₈₄	—	4	2	no	—
54	Dic₁₄₂	—	4	2	no	—
55	C₈×C₇₁	—	2	2	no	—
56	C₇₁⋊C₈	—	2	2	no	—
57	C₇₁⋊C₈	SD₁₆	1	4	yes	complemented
58	C₂²×C₇₁	—	8	1	no	—
59	C₂²×C₇₁	—	8	1	no	—
60	C₂²×C₇₁	—	8	1	no	—
61	C₂²×C₇₁	#127	1	1	yes	characteristic
62	C₂×D₇₁	—	8	1	no	—
63	C₂×D₇₁	—	8	1	no	—
64	C₂×D₇₁	—	8	1	no	—
65	C₄×C₇₁	—	8	1	no	—
66	C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
67	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₇₁	—	8	1	no	—
70	C₂×C₇₁	—	16	1	no	—
71	C₂×C₇₁	#492	1	1	yes	characteristic
72	C₂×C₇₁	#496	1	1	yes	characteristic
73	C₂×C₇₁	#496	1	1	yes	characteristic
74	D₇₁	—	16	1	no	—
75	#1715	—	71	1	no	maximal, Sylow
76	C₇₁	#1715	1	1	yes	characteristic, complemented, Sylow
77	#498	—	71	1	no	—
78	#493	—	71	1	no	—
79	#492	—	71	1	no	—
80	#492	D₇₁	1	1	yes	characteristic, complemented
81	#486	—	71	1	no	—
82	#486	—	71	1	no	—
83	#322	—	71	1	no	—
84	C₂×SD₁₆	—	2	2	no	—
85	C₂×SD₁₆	—	142	2	no	—
86	C₂×D₈	—	142	2	no	—
87	C₄⋊Q₈	—	71	1	no	—
88	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
89	#127	—	71	1	no	—
90	#127	C₂×D₇₁	1	1	yes	characteristic
91	#106	—	142	2	no	—
92	#102	—	142	2	no	—
93	#102	—	142	2	no	—
94	C₄×C₈	—	71	1	no	—
95	C₄×C₈	—	71	1	no	—
96	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	142	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	142	1	no	—
103	C₂×D₄	—	142	1	no	—
104	C₂×D₄	—	142	1	no	—
105	SD₁₆	—	4	4	no	—
106	SD₁₆	—	284	4	no	—
107	D₈	—	284	4	no	—
108	C₂×C₈	—	71	2	no	—
109	C₂×C₈	—	71	2	no	—
110	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	142	1	no	—
114	C₄⋊C₄	—	142	1	no	—
115	C₄²	C₂²×D₇₁	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	284	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	284	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	284	2	no	—
124	D₄	—	284	2	no	—
125	D₄	—	284	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	284	1	no	—
128	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	71	4	no	—
133	C₈	—	142	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	568	1	no	—
138	C₂²	—	568	1	no	—
139	C₂²	—	568	1	no	—
140	C₂²	#14556124	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	568	1	no	—
143	C₄	SD₁₆×D₇₁	1	2	yes	—
144	C₄	#14556154	1	2	yes	—
145	C₄	#14556155	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	1136	1	no	—
148	C₂	#17986370	1	1	yes	characteristic
149	C₂	#17986385	1	1	yes	characteristic
150	C₂	#17986386	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986395

C₂

yes

maximal, characteristic, complemented

#17986375

C₂

yes

maximal, characteristic

#17986371

C₂

yes

maximal, characteristic, complemented

#17986347

C₂

yes

maximal, characteristic, complemented

#17986346

C₂

yes

maximal, characteristic, complemented

#17985721

C₂

yes

maximal, characteristic, complemented

#492×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765853

—

C₂×#11765850

—

#14556128

C₂²

yes

characteristic

#14556125

C₂²

yes

—

—

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

#127×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

#11765853

—

#11765850

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

SD₁₆×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

C₂²×D₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

SD₁₆

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#127

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1715

—

maximal, Sylow

C₇₁

#1715

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

characteristic, complemented

#486

—

#486

—

#322

—

C₂×SD₁₆

—

C₂×SD₁₆

—

142

—

C₂×D₈

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

#127

—

#127

C₂×D₇₁

yes

characteristic

#106

—

142

—

#102

—

142

—

#102

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

142

—

103

C₂×D₄

—

142

—

104

C₂×D₄

—

142

—

105

SD₁₆

—

106

SD₁₆

—

284

—

107

D₈

—

284

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂²⋊D₇₁

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

142

—

114

C₄⋊C₄

—

142

—

115

C₄²

C₂²×D₇₁

yes

characteristic

116

C₂³

—

117

C₂³

—

284

—

118

Q₈

—

119

Q₈

—

284

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

284

—

124

D₄

—

284

—

125

D₄

—

284

—

126

C₂×C₄

—

127

C₂×C₄

—

284

—

128

C₂×C₄

D₄×D₇₁

yes

characteristic

129

C₂×C₄

D₄×D₇₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

142

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

568

—

138

C₂²

—

568

—

139

C₂²

—

568

—

140

C₂²

#14556124

yes

characteristic

141

C₄

—

142

C₄

—

568

—

143

C₄

SD₁₆×D₇₁

yes

—

144

C₄

#14556154

yes

—

145

C₄

#14556155

yes

—

146

C₂

—

147

C₂

—

1136

—

148

C₂

#17986370

yes

characteristic

149

C₂

#17986385

yes

characteristic

150

C₂

#17986386

yes

characteristic

	subgroup	G/H
center	C₂²	#14556124
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#492×C₇₁	C₂
socle	C₂²×C₇₁	#127
hypercenter	#492	D₇₁
hypocenter	C₇₁	#1715
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#14556124

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#492×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #17986395	C₂	semidirect
3. #17986371	C₂	semidirect
4. #17986347	C₂	semidirect
5. #17986346	C₂	semidirect
6. #17985721	C₂	semidirect
7. #492×C₇₁	C₂	semidirect
57. C₇₁⋊C₈	SD₁₆	semidirect
76. C₇₁	#1715	semidirect
80. #492	D₇₁	semidirect

H ≅ G/N

type

1. #17986395

C₂

semidirect

3. #17986371

C₂

semidirect

4. #17986347

C₂

semidirect

5. #17986346

C₂

semidirect

6. #17985721

C₂

semidirect

7. #492×C₇₁

C₂

semidirect

57. C₇₁⋊C₈

SD₁₆

semidirect

76. C₇₁

#1715

semidirect

80. #492

D₇₁

semidirect

Subgroups — #22587967

Conjugacy classes of subgroups

Complements