Group #22587713 — subgroups

Subgroups — #22587713

Order 9088.

166 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17986289	C₂	1	1	yes	maximal, characteristic
3	D₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#17985749	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985747	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985745	C₂	1	1	yes	maximal, characteristic, complemented
7	#461×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₁⋊C₈	—	2	1	no	—
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556059	—	2	1	no	—
11	#14556059	—	2	1	no	—
12	#14556059	—	2	1	no	—
13	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	#14556049	—	2	1	no	—
16	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
17	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₇₁	—	2	1	no	—
19	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	#103×C₇₁	—	2	1	no	—
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×C₇₁⋊C₈	—	2	1	no	—
25	C₂×C₇₁⋊C₈	—	2	1	no	—
26	C₂×C₇₁⋊C₈	—	4	2	no	—
27	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₂×Q₈×C₇₁	—	2	1	no	—
32	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	—	2	1	no	—
34	SD₁₆×C₇₁	—	4	2	no	—
35	C₂×C₈×C₇₁	—	2	1	no	—
36	C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₄⋊C₄×C₇₁	—	2	1	no	—
38	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₇₁	—	2	1	no	—
40	C₄²×C₇₁	C₂³	1	1	yes	characteristic
41	C₂³×C₇₁	—	2	1	no	—
42	Q₈×C₇₁	—	4	1	no	—
43	Q₈×C₇₁	—	4	1	no	—
44	D₄×C₇₁	—	2	1	no	—
45	D₄×C₇₁	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₁	—	2	1	no	—
47	C₂×C₄×C₇₁	—	4	1	no	—
48	C₂×C₄×C₇₁	—	4	1	no	—
49	C₂×C₄×C₇₁	—	4	1	no	—
50	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
53	C₂×Dic₇₁	—	4	1	no	—
54	Dic₁₄₂	—	8	1	no	—
55	C₈×C₇₁	—	4	1	no	—
56	C₇₁⋊C₈	—	2	1	no	—
57	C₇₁⋊C₈	—	4	1	no	—
58	C₇₁⋊C₈	—	4	1	no	—
59	C₇₁⋊C₈	SD₁₆	1	2	yes	complemented
60	C₂²×C₇₁	—	2	2	no	—
61	C₂²×C₇₁	—	4	1	no	—
62	C₂²×C₇₁	—	4	1	no	—
63	C₂²×C₇₁	#121	1	1	yes	characteristic
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	—	4	1	no	—
66	C₄×C₇₁	—	8	1	no	—
67	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₁	#135	1	1	yes	characteristic
70	C₄×C₇₁	#137	1	1	yes	characteristic
71	Dic₇₁	—	8	1	no	—
72	C₂×C₇₁	—	4	2	no	—
73	C₂×C₇₁	#461	1	1	yes	characteristic
74	C₂×C₇₁	#465	1	1	yes	characteristic
75	C₂×C₇₁	#471	1	1	yes	characteristic
76	#1671	—	71	1	no	maximal, Sylow
77	C₇₁	#1671	1	1	yes	characteristic, complemented, Sylow
78	#498	—	71	1	no	—
79	#476	—	71	1	no	—
80	#461	D₇₁	1	1	yes	characteristic
81	C₈×D₄	—	71	1	no	—
82	#334	—	71	1	no	—
83	#333	—	71	1	no	—
84	#332	—	71	1	no	—
85	C₂×SD₁₆	—	2	1	no	—
86	C₂²×C₈	—	142	1	no	—
87	C₄⋊Q₈	—	71	1	no	—
88	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
89	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
90	#106	—	142	1	no	—
91	#106	—	142	1	no	—
92	#106	—	142	1	no	—
93	C₄⋊C₈	—	71	1	no	—
94	C₄⋊C₈	—	71	1	no	—
95	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
96	#103	—	2	1	no	—
97	#102	—	142	1	no	—
98	C₂²⋊C₈	—	142	1	no	—
99	C₄×C₈	—	71	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	142	1	no	—
102	C₂×D₄	C₂²⋊D₇₁	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	4	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	284	2	no	—
107	C₂×C₈	—	71	1	no	—
108	C₂×C₈	—	71	1	no	—
109	C₂×C₈	—	142	1	no	—
110	C₂×C₈	—	142	1	no	—
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	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	C₂²×D₇₁	1	1	yes	characteristic
118	C₂³	—	2	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	568	1	no	—
122	D₄	—	2	1	no	—
123	D₄	#11765853	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	284	1	no	—
129	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
131	C₂×C₄	#11765864	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	284	1	no	—
134	C₈	—	284	1	no	—
135	C₈	—	71	2	no	—
136	C₈	—	142	1	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	#14556100	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	568	1	no	—
145	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
146	C₄	C₂×#11765853	1	1	yes	characteristic
147	C₄	#14556148	1	1	yes	characteristic
148	C₄	#14556157	1	1	yes	characteristic
149	C₂	—	4	2	no	—
150	C₂	#17986202	1	1	yes	characteristic
151	C₂	#17986226	1	1	yes	characteristic
152	C₂	#17986264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986395

C₂

yes

maximal, characteristic, complemented

#17986289

C₂

yes

maximal, characteristic

D₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985749

C₂

yes

maximal, characteristic, complemented

#17985747

C₂

yes

maximal, characteristic, complemented

#17985745

C₂

yes

maximal, characteristic, complemented

#461×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

#14556128

C₂²

yes

—

—

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556049

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₂×Dic₁₄₂

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

SD₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

SD₁₆

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#498

—

#476

—

#461

D₇₁

yes

characteristic

C₈×D₄

—

#334

—

#333

—

#332

—

C₂×SD₁₆

—

C₂²×C₈

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

C₂×D₇₁

yes

characteristic

#106

—

142

—

#106

—

142

—

#106

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

#102

—

142

—

C₂²⋊C₈

—

142

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

142

—

102

C₂×D₄

C₂²⋊D₇₁

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

284

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

142

—

110

C₂×C₈

—

142

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

142

—

114

C₄⋊C₄

—

142

—

115

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

C₂²×D₇₁

yes

characteristic

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

568

—

122

D₄

—

123

D₄

#11765853

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

284

—

129

C₂×C₄

D₄×D₇₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

131

C₂×C₄

#11765864

yes

characteristic

132

C₈

—

133

C₈

—

284

—

134

C₈

—

284

—

135

C₈

—

136

C₈

—

142

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#14556100

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

568

—

145

C₄

SD₁₆×D₇₁

yes

characteristic

146

C₄

C₂×#11765853

yes

characteristic

147

C₄

#14556148

yes

characteristic

148

C₄

#14556157

yes

characteristic

149

C₂

—

150

C₂

#17986202

yes

characteristic

151

C₂

#17986226

yes

characteristic

152

C₂

#17986264

yes

characteristic

	subgroup	G/H
center	C₂²	#14556100
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#461×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#461	D₇₁
hypocenter	C₇₁	#1671
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#14556100

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#461×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #17986395	C₂	semidirect
4. #17985749	C₂	semidirect
5. #17985747	C₂	semidirect
6. #17985745	C₂	semidirect
59. C₇₁⋊C₈	SD₁₆	semidirect
77. C₇₁	#1671	semidirect

H ≅ G/N

type

1. #17986395

C₂

semidirect

4. #17985749

C₂

semidirect

5. #17985747

C₂

semidirect

6. #17985745

C₂

semidirect

59. C₇₁⋊C₈

SD₁₆

semidirect

77. C₇₁

#1671

semidirect

Subgroups — #22587713

Conjugacy classes of subgroups

Complements