Finite Group Databasebeta

← #22593272

Subgroups — #22593272

Order 9088.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C2×#14556109C212yesmaximal, complemented
2C2×#14556096C212yesmaximal, complemented
3C22×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
4#17985934C212yesmaximal, complemented
5#17985907C212yesmaximal, complemented
6#17985899C214yesmaximal, complemented
7C4×C22⋊Dic71C211yesmaximal, characteristic, complemented
8C2×#121×C71C211yesmaximal, characteristic, complemented
9C24×D71C2211yescharacteristic
10C22×C22⋊D71C2214yes—
11C23×Dic71C2211yescharacteristic
12#14556109—28no—
13#14556096—28no—
14C2×Dic71⋊C4C2214yescomplemented
15C2×C22⋊Dic71C2212yes—
16C2×C22⋊Dic71C2214yescomplemented
17C2×(C2×D71)⋊C4—24no—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2214yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×C4×Dic71C2212yescomplemented
22C71⋊#95C2212yescomplemented
23C22×D4×C71C2212yes—
24C22×D4×C71C2211yescharacteristic, complemented
25C23×C4×C71C2211yescharacteristic
26#121×C71—24no—
27C2×C4⋊C4×C71C2211yescharacteristic, complemented
28C2×C22⋊C4×C71C2212yescomplemented
29C23×D71—22no—
30C23×D71—24no—
31C23×D71—24no—
32C23×D71—42no—
33C23×D71C2312yes—
34C2×C22⋊D71—28no—
35C2×C22⋊D71—28no—
36C2×C22⋊D71—44no—
37C22×Dic71—22no—
38C22×Dic71—24no—
39C22×Dic71C2312yes—
40C22×Dic71C2314yes—
41Dic71⋊C4—28no—
42C22⋊Dic71—28no—
43C22⋊Dic71D418yescomplemented
44(C2×D71)⋊C4—24no—
45(C2×D71)⋊C4—24no—
46(C2×D71)⋊C4—28no—
47C4×Dic71—42no—
48C24×C71C2312yes—
49C24×C71C2311yescharacteristic
50C2×D4×C71—24no—
51C2×D4×C71—24no—
52C2×D4×C71—24no—
53C2×D4×C71—22no—
54C2×D4×C71—42no—
55C22×C4×C71—42no—
56C22×C4×C71D414yescomplemented
57C22×C4×C71C2312yes—
58C22×C4×C71C2311yescharacteristic
59C22×C4×C71C2311yescharacteristic
60C4⋊C4×C71—22no—
61C22⋊C4×C71—24no—
62C23×C71—22no—
63C23×C71—22no—
64C23×C71—22no—
65C23×C71—22no—
66C23×C71—22no—
67C23×C71—24no—
68C23×C71—24no—
69C23×C71C2×D414yes—
70C23×C71C2×D412yes—
71C23×C71C2411yescharacteristic
72C22×D71—28no—
73C22×D71—22no—
74C22×D71—24no—
75C22×D71—22no—
76C22×D71—24no—
77C22×D71—416no—
78C22×D71—44no—
79C22×D71—44no—
80C22×D71#4114yes—
81D4×C71—44no—
82D4×C71—48no—
83C2×C4×C71—22no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×C4×C71—44no—
89C2×C4×C71C2×D412yes—
90C2×C4×C71C2×D412yes—
91C22⋊D71—416no—
92C2×Dic71—24no—
93C2×Dic71—24no—
94C2×Dic71—22no—
95C2×Dic71—24no—
96C2×Dic71—44no—
97C2×Dic71C2×D418yes—
98C22×C71—22no—
99C22×C71—22no—
100C22×C71—28no—
101C22×C71—42no—
102C22×C71—42no—
103C22×C71—42no—
104C22×C71—44no—
105C22×C71—44no—
106C22×C71#12714yes—
107C22×C71C22×D412yes—
108C22×C71C22×D411yescharacteristic
109C22×C71C2×#4112yes—
110C22×C71H(5,𝔽2)12yes—
111C2×D71—24no—
112C2×D71—28no—
113C2×D71—416no—
114C2×D71—42no—
115C2×D71—42no—
116C2×D71—44no—
117C4×C71—22no—
118C4×C71—22no—
119C4×C71—42no—
120Dic71—28no—
121Dic71—42no—
122C2×C71—24no—
123C2×C71—82no—
124C2×C71#52511yescharacteristic
125C2×C71C2×#12711yescharacteristic
126C2×C71#53411yescharacteristic
127C2×C71#54614yes—
128D71—44no—
129#2539—711nomaximal, Sylow
130C71#253911yescharacteristic, complemented, Sylow
131C2×#123—712no—
132C2×#121D7111yescharacteristic, complemented
133C2×#120—712no—
134C22×C22⋊C4—711no—
135#399—712no—
136#392—712no—
137#390—714no—
138C4×C22⋊C4—711no—
139C25—711no—
140C22×D4—714no—
141C22×D4C2×D7112yes—
142C22×D4C2×D7111yescharacteristic, complemented
143C23×C4—711no—
144C23×C4C2×D7111yescharacteristic
145#123—1428no—
146#121—24no—
147#120—1428no—
148C2×C4⋊C4—714no—
149C2×C4⋊C4C2×D7111yescharacteristic, complemented
150C2×C22⋊C4—712no—
151C2×C22⋊C4—714no—
152C2×C22⋊C4—712no—
153C2×C22⋊C4—712no—
154C2×C22⋊C4—714no—
155C2×C22⋊C4—1424no—
156C2×C22⋊C4C2×D7112yescomplemented
157C2×C42—712no—
158#95—712no—
159C24—2842no—
160C24—712no—
161C24—1424no—
162C24—1424no—
163C24—1422no—
164C24C22×D7112yes—
165C24C22×D7111yescharacteristic
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—22no—
170C2×D4—42no—
171C2×D4—2844no—
172C2×D4—1428no—
173C2×D4—1428no—
174C22×C4—42no—
175C22×C4—714no—
176C22×C4—712no—
177C22×C4—1424no—
178C22×C4—1422no—
179C22×C4C22⋊D7114yescomplemented
180C22×C4C22×D7112yes—
181C22×C4C22×D7111yescharacteristic
182C22×C4C22×D7111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—1428no—
185C22⋊C4—24no—
186C22⋊C4—718no—
187C22⋊C4—1428no—
188C22⋊C4—1428no—
189C22⋊C4—1424no—
190C22⋊C4—1424no—
191C42—2842no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—24no—
198C23—24no—
199C23—2844no—
200C23—28416no—
201C23—2844no—
202C23—714no—
203C23—1428no—
204C23—1424no—
205C23—1422no—
206C23—1422no—
207C23—1424no—
208C23D4×D7114yes—
209C23C2×C22⋊D7112yes—
210C23C23×D7111yescharacteristic
211D4—44no—
212D4—48no—
213D4—28416no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—44no—
220C2×C4—2844no—
221C2×C4—718no—
222C2×C4—1424no—
223C2×C4—1422no—
224C2×C4—1424no—
225C2×C4—1424no—
226C2×C4C2×C22⋊D7112yes—
227C2×C4C2×C22⋊D7112yes—
228C22—28no—
229C22—22no—
230C22—22no—
231C22—44no—
232C22—44no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—28416no—
237C22—2844no—
238C22—2842no—
239C22—2842no—
240C22—1428no—
241C22—1424no—
242C22#1455612414yes—
243C22C2×D4×D7112yes—
244C22C22×C22⋊D7111yescharacteristic
245C22#41×D7112yes—
246C22#1455617312yes—
247C4—22no—
248C4—22no—
249C4—42no—
250C4—2842no—
251C4—1428no—
252C2—24no—
253C2—82no—
254C2—2844no—
255C2#1798650611yescharacteristic
256C2C2×#1455612411yescharacteristic
257C2#1798655811yescharacteristic
258C2#1798663814yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#121×C71C2
hypercenterC2×#121D71
hypocenterC71#2539
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#14556109C2semidirect
2. C2×#14556096C2semidirect
3. C22×(C2×D71)⋊C4C2semidirect
4. #17985934C2semidirect
5. #17985907C2semidirect
6. #17985899C2semidirect
7. C4×C22⋊Dic71C2semidirect
8. C2×#121×C71C2semidirect
14. C2×Dic71⋊C4C22semidirect
16. C2×C22⋊Dic71C22semidirect
21. C2×C4×Dic71C22semidirect
22. C71⋊#95C22semidirect
24. C22×D4×C71C22semidirect
27. C2×C4⋊C4×C71C22semidirect
28. C2×C22⋊C4×C71C22semidirect
43. C22⋊Dic71D4semidirect
56. C22×C4×C71D4semidirect
130. C71#2539semidirect
132. C2×#121D71semidirect
142. C22×D4C2×D71semidirect
149. C2×C4⋊C4C2×D71semidirect
156. C2×C22⋊C4C2×D71semidirect
179. C22×C4C22⋊D71semidirect