Finite Group Databasebeta

← #22586856

Subgroups — #22586856

Order 9088.

346 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17985963C211yesmaximal, characteristic, complemented
2C4×C22⋊Dic71C211yesmaximal, characteristic, complemented
3#17985733C214yesmaximal, complemented
4C2×#121×C71C211yesmaximal, characteristic
5C2×#11765849—21no—
6C2×C22⋊Dic71—21no—
7C2×C4×Dic71—21no—
8C22⋊(C71⋊C8)—21no—
9C22⋊(C71⋊C8)C2212yescomplemented
10C71⋊#95C2212yescomplemented
11C22×D4×C71—21no—
12C22×D4×C71C411yescharacteristic, complemented
13C23×C4×C71C2211yescharacteristic
14#121×C71—21no—
15#121×C71—21no—
16#121×C71C412yescomplemented
17#121×C71C2212yes—
18C2×C4⋊C4×C71C411yescharacteristic, complemented
19C2×C22⋊C4×C71—21no—
20C22×Dic71—22no—
21#11765849—42no—
22C2×C71⋊C8—22no—
23C22⋊Dic71—24no—
24C4×Dic71—42no—
25C24×C71—21no—
26C24×C71D411yescharacteristic
27C2×D4×C71—22no—
28C2×D4×C71—22no—
29C2×D4×C71—21no—
30C2×D4×C71—42no—
31C2×D4×C71—44no—
32C2×D4×C71C2×C412yes—
33C22×C4×C71—21no—
34C22×C4×C71—42no—
35C22×C4×C71C2×C411yescharacteristic
36C22×C4×C71C2×C411yescharacteristic
37C22×C4×C71D411yescharacteristic
38C22×C4×C71D411yescharacteristic
39C22×C4×C71D411yescharacteristic
40C22×C4×C71C2311yescharacteristic
41C4⋊C4×C71—21no—
42C4⋊C4×C71C2×C412yes—
43C22⋊C4×C71—22no—
44C22⋊C4×C71—22no—
45C23×C71—22no—
46C23×C71—22no—
47C23×C71—22no—
48C23×C71—41no—
49C23×C71—42no—
50C23×C71—42no—
51C23×C71—42no—
52C23×C71C22⋊C411yescharacteristic
53C23×C71C22⋊C411yescharacteristic
54C23×C71C2×D411yescharacteristic
55D4×C71—42no—
56D4×C71—44no—
57D4×C71—44no—
58D4×C71—42no—
59C2×C4×C71—22no—
60C2×C4×C71—22no—
61C2×C4×C71—22no—
62C2×C4×C71—22no—
63C2×C4×C71—41no—
64C2×C4×C71—41no—
65C2×C4×C71—41no—
66C2×C4×C71—42no—
67C2×C4×C71C22⋊C411yescharacteristic
68C2×C4×C71C22⋊C411yescharacteristic
69C2×C4×C71C22×C411yescharacteristic
70C2×C4×C71C2×D411yescharacteristic
71C2×Dic71—24no—
72C2×Dic71—42no—
73C2×Dic71—42no—
74C71⋊C8—42no—
75C22×C71—21no—
76C22×C71—21no—
77C22×C71—21no—
78C22×C71—21no—
79C22×C71—21no—
80C22×C71—21no—
81C22×C71—21no—
82C22×C71—44no—
83C22×C71—42no—
84C22×C71—42no—
85C22×C71—42no—
86C22×C71—84no—
87C22×C71#10412yes—
88C22×C71C2×C22⋊C411yescharacteristic
89C22×C71AGL(1,ℤ/8)12yes—
90C4×C71—21no—
91C4×C71—21no—
92C4×C71—21no—
93C4×C71—42no—
94C4×C71#9912yes—
95Dic71—44no—
96C2×C71—22no—
97C2×C71—22no—
98C2×C71—41no—
99C2×C71—82no—
100C2×C71C2×#9911yescharacteristic
101C2×C71#41811yescharacteristic
102C2×C71C2×#10411yescharacteristic
103#1507—711nomaximal, Sylow
104C71#150711yescharacteristic, complemented, Sylow
105C2×#121D7111yescharacteristic
106#407—711no—
107C4×C22⋊C4—711no—
108#327—714no—
109C22×D4—21no—
110C22×D4Dic7111yescharacteristic, complemented
111C23×C4C2×D7111yescharacteristic
112C2×M16—1421no—
113#121—21no—
114#121—21no—
115#121Dic7112yescomplemented
116#121C2×D7112yes—
117C2×C4⋊C4Dic7111yescharacteristic, complemented
118C2×C22⋊C4—21no—
119C2×C22⋊C4—1421no—
120C2×C42—1421no—
121C22⋊C8—712no—
122C22⋊C8—1421no—
123#95—712no—
124C24—21no—
125C24C22⋊D7111yescharacteristic
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—42no—
130C2×D4—44no—
131C2×D4C2×Dic7112yes—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—1422no—
135C22×C4C2×Dic7111yescharacteristic
136C22×C4C2×Dic7111yescharacteristic
137C22×C4C22⋊D7111yescharacteristic
138C22×C4C22⋊D7111yescharacteristic
139C22×C4C22⋊D7111yescharacteristic
140C22×C4C22×D7111yescharacteristic
141M16—2842no—
142C2×C8—1422no—
143C4⋊C4—21no—
144C4⋊C4C2×Dic7112yes—
145C22⋊C4—22no—
146C22⋊C4—22no—
147C22⋊C4—1424no—
148C42—2842no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—41no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23C22⋊Dic7111yescharacteristic
157C23C22⋊Dic7111yescharacteristic
158C23C2×C22⋊D7111yescharacteristic
159D4—42no—
160D4—44no—
161D4—44no—
162D4—42no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—2842no—
172C2×C4—2842no—
173C2×C4—1424no—
174C2×C4C22⋊Dic7111yescharacteristic
175C2×C4C22⋊Dic7111yescharacteristic
176C2×C4C22×Dic7111yescharacteristic
177C2×C4C2×C22⋊D7111yescharacteristic
178C8—2842no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—42no—
188C22—42no—
189C22—42no—
190C22—84no—
191C22#1455605512yes—
192C22C2×C22⋊Dic7111yescharacteristic
193C22#1455615412yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—2844no—
199C4#1455604312yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—82no—
204C2C2×#1455604311yescharacteristic
205C2#1798600411yescharacteristic
206C2C2×#1455605511yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic71
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
FittingC2×#121×C71C2
socleC22×C71C2×C22⋊C4
hypercenterC2×#121D71
hypocenterC71#1507
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17985963C2semidirect
2. C4×C22⋊Dic71C2semidirect
3. #17985733C2semidirect
9. C22⋊(C71⋊C8)C22semidirect
10. C71⋊#95C22semidirect
12. C22×D4×C71C4semidirect
16. #121×C71C4semidirect
18. C2×C4⋊C4×C71C4semidirect
104. C71#1507semidirect
110. C22×D4Dic71semidirect
115. #121Dic71semidirect
117. C2×C4⋊C4Dic71semidirect