Finite Group Databasebeta

← #22591988

Subgroups — #22591988

Order 9088.

1088 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×D4×D71C211yesmaximal, characteristic
2C2×#14556117C211yesmaximal, characteristic, complemented
3C2×#14556104C211yesmaximal, characteristic, complemented
4#17985914C214yesmaximal, complemented
5#17985902C211yesmaximal, characteristic, complemented
6#17985901C211yesmaximal, characteristic, complemented
7#17985854C214yesmaximal, complemented
8C4×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
9C2×#124×C71C211yesmaximal, characteristic, complemented
10C24×D71C2212yes—
11C22×D284C2211yescharacteristic
12C22×C22⋊D71C2212yes—
13C2×D4×D71—24no—
14C2×D4×D71—24no—
15C2×D4×D71—24no—
16C22×C4×D71C2211yescharacteristic
17#14556117—24no—
18#14556104—24no—
19C2×C4⋊Dic71C2211yescharacteristic, complemented
20C2×Dic71⋊C4C2212yescomplemented
21C2×C22⋊Dic71C2214yescomplemented
22C2×(C2×D71)⋊C4C2212yes—
23C2×(C2×D71)⋊C4C2212yes—
24C2×(C2×D71)⋊C4C2214yescomplemented
25C2×(C2×D71)⋊C4C2214yescomplemented
26C2×C4×Dic71C2211yescharacteristic, complemented
27C71⋊#95C2212yescomplemented
28C22×Q8×C71C2211yescharacteristic, complemented
29C22×D4×C71C2211yescharacteristic
30#124×C71—24no—
31C2×C22⋊C4×C71C2214yescomplemented
32C2×C42×C71C2211yescharacteristic, complemented
33C23×D71—24no—
34C23×D71—24no—
35C23×D71—24no—
36C23×D71—28no—
37C23×D71—44no—
38C23×D71C2312yes—
39C23×D71C2311yescharacteristic
40C2×D284—22no—
41C2×D284—22no—
42C2×D284D414yes—
43C2×C22⋊D71—24no—
44C2×C22⋊D71—24no—
45C2×C22⋊D71—24no—
46D4×D71—232no—
47C22×Dic71C2312yes—
48C22×Dic71C2312yes—
49C22×Dic71C2311yescharacteristic
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71D414yes—
53C4⋊Dic71—22no—
54Dic71⋊C4—24no—
55C22⋊Dic71—44no—
56(C2×D71)⋊C4—24no—
57(C2×D71)⋊C4—28no—
58(C2×D71)⋊C4—24no—
59(C2×D71)⋊C4—44no—
60C4×Dic71—22no—
61C24×C71C2312yes—
62C2×Q8×C71—22no—
63C2×Q8×C71—22no—
64C2×Q8×C71—22no—
65C2×D4×C71—22no—
66C2×D4×C71—22no—
67C2×D4×C71D414yes—
68C22×C4×C71C2312yes—
69C22×C4×C71C2312yes—
70C22×C4×C71C2311yescharacteristic
71C22⋊C4×C71—28no—
72C42×C71—22no—
73C23×C71—22no—
74C23×C71—22no—
75C23×C71—24no—
76C23×C71—24no—
77C23×C71C2×D414yes—
78C23×C71C2411yescharacteristic
79C22×D71—21no—
80C22×D71—21no—
81C22×D71—22no—
82C22×D71—216no—
83C22×D71—22no—
84C22×D71—22no—
85C22×D71—24no—
86C22×D71—24no—
87C22×D71—416no—
88C22×D71—416no—
89C22×D71—48no—
90C22×D71—48no—
91C22×D71C2×D414yes—
92C22×D71C2×D412yes—
93C22×D71#4114yes—
94Q8×C71—44no—
95D4×C71—28no—
96C2×C4×C71—22no—
97C2×C4×C71—22no—
98C2×C4×C71—22no—
99C2×C4×C71—22no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71C2×D412yes—
103C2×C4×C71C2×D412yes—
104C2×C4×C71C2×D412yes—
105C22⋊D71—216no—
106C2×Dic71—21no—
107C2×Dic71—21no—
108C2×Dic71—22no—
109C2×Dic71—22no—
110C2×Dic71—22no—
111C2×Dic71—24no—
112C2×Dic71—22no—
113C2×Dic71C2×D412yes—
114D284—28no—
115C4×D71—28no—
116C22×C71—24no—
117C22×C71—28no—
118C22×C71—42no—
119C22×C71—42no—
120C22×C71—44no—
121C22×C71C22×D411yescharacteristic
122C22×C71C22×D411yescharacteristic
123C22×C71C22×D411yescharacteristic
124C22×C71C2×#4112yes—
125C22×C71H(5,𝔽2)12yes—
126C2×D71—22no—
127C2×D71—22no—
128C2×D71—24no—
129C2×D71—24no—
130C2×D71—24no—
131C2×D71—28no—
132C2×D71—432no—
133C2×D71—42no—
134C2×D71—42no—
135C2×D71—44no—
136C2×D71#12414yescomplemented
137C4×C71—42no—
138C4×C71—42no—
139C4×C71#12014yes—
140Dic71—22no—
141Dic71—42no—
142Dic71—42no—
143C2×C71—44no—
144C2×C71C2×#12011yescharacteristic
145C2×C71C2×#12411yescharacteristic
146C2×C71#53511yescharacteristic
147C2×C71#54612yes—
148C2×C71#54612yes—
149D71—24no—
150D71—44no—
151#2435—711nomaximal, Sylow
152C71#243511yescharacteristic, complemented, Sylow
153C23×D4—711no—
154C2×#124—711no—
155C2×#124D7111yescharacteristic, complemented
156C2×C22⋊Q8—711no—
157#394—714no—
158#390—711no—
159#390—711no—
160#379—714no—
161C4×C22⋊C4—711no—
162C25—712no—
163C22×Q8C2×D7111yescharacteristic, complemented
164C22×D4—712no—
165C22×D4—711no—
166C22×D4—1424no—
167C22×D4—1424no—
168C22×D4—1424no—
169C22×D4C2×D7111yescharacteristic
170C23×C4—711no—
171#124—24no—
172#124—1424no—
173C22⋊Q8—1424no—
174C2×C4⋊C4—711no—
175C2×C4⋊C4—712no—
176C2×C22⋊C4—714no—
177C2×C22⋊C4—714no—
178C2×C22⋊C4—712no—
179C2×C22⋊C4—714no—
180C2×C22⋊C4—712no—
181C2×C22⋊C4C2×D7114yescomplemented
182C2×C42—711no—
183C2×C42C2×D7111yescharacteristic, complemented
184#95—712no—
185C24—2844no—
186C24—711no—
187C24—712no—
188C24—1428no—
189C24—1424no—
190C24—1424no—
191C24—1424no—
192C24C22×D7112yes—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—714no—
199C2×D4—1424no—
200C2×D4—14232no—
201C2×D4—1424no—
202C2×D4—1424no—
203C2×D4—1422no—
204C2×D4—1422no—
205C2×D4C22⋊D7114yes—
206C22×C4—711no—
207C22×C4—714no—
208C22×C4—712no—
209C22×C4—712no—
210C22×C4—1422no—
211C22×C4—1422no—
212C22×C4C22×D7112yes—
213C22×C4C22×D7112yes—
214C22×C4C22×D7111yescharacteristic
215C4⋊C4—1424no—
216C4⋊C4—1422no—
217C22⋊C4—28no—
218C22⋊C4—2844no—
219C22⋊C4—2844no—
220C22⋊C4—1424no—
221C22⋊C4—1424no—
222C22⋊C4—1428no—
223C42—22no—
224C42—1422no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—28416no—
230C23—2848no—
231C23—2848no—
232C23—28416no—
233C23—712no—
234C23—714no—
235C23—714no—
236C23—14216no—
237C23—1422no—
238C23—1422no—
239C23—1424no—
240C23—1424no—
241C23—1421no—
242C23—1421no—
243C23—1422no—
244C23C2×C22⋊D7114yes—
245C23C23×D7111yescharacteristic
246Q8—44no—
247D4—28no—
248D4—14216no—
249D4—1428no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—712no—
257C2×C4—1422no—
258C2×C4—1422no—
259C2×C4—1422no—
260C2×C4—1424no—
261C2×C4—1422no—
262C2×C4—1421no—
263C2×C4—1421no—
264C2×C4—1428no—
265C2×C4D4×D7112yes—
266C2×C4D4×D7112yes—
267C2×C4C2×C22⋊D7112yes—
268C22—28no—
269C22—24no—
270C22—42no—
271C22—44no—
272C22—42no—
273C22—28432no—
274C22—2842no—
275C22—2844no—
276C22—2842no—
277C22—714no—
278C22—1428no—
279C22—1424no—
280C22—1422no—
281C22—1424no—
282C22—1424no—
283C22—1422no—
284C22C2×D4×D7111yescharacteristic
285C22C2×D4×D7111yescharacteristic
286C22C22×C22⋊D7111yescharacteristic
287C22#41×D7112yes—
288C22#1455617412yes—
289C4—42no—
290C4—42no—
291C4—2842no—
292C4—2842no—
293C4—1422no—
294C4#1455609614yes—
295C2—44no—
296C2—2844no—
297C2—1424no—
298C2C2×#1455609611yescharacteristic
299C2#124×D7111yescharacteristic
300C2#1798656711yescharacteristic
301C2#1798663712yes—
302C2#1798664212yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#124×C71C2
hypercenterC2×#124D71
hypocenterC71#2435
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#14556117C2semidirect
3. C2×#14556104C2semidirect
4. #17985914C2semidirect
5. #17985902C2semidirect
6. #17985901C2semidirect
7. #17985854C2semidirect
8. C4×(C2×D71)⋊C4C2semidirect
9. C2×#124×C71C2semidirect
19. C2×C4⋊Dic71C22semidirect
20. C2×Dic71⋊C4C22semidirect
21. C2×C22⋊Dic71C22semidirect
24. C2×(C2×D71)⋊C4C22semidirect
25. C2×(C2×D71)⋊C4C22semidirect
26. C2×C4×Dic71C22semidirect
27. C71⋊#95C22semidirect
28. C22×Q8×C71C22semidirect
31. C2×C22⋊C4×C71C22semidirect
32. C2×C42×C71C22semidirect
136. C2×D71#124semidirect
152. C71#2435semidirect
155. C2×#124D71semidirect
163. C22×Q8C2×D71semidirect
181. C2×C22⋊C4C2×D71semidirect
183. C2×C42C2×D71semidirect