Finite Group Databasebeta

← #22596633

Subgroups — #22596633

Order 9088.

1102 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×C4×D71C211yesmaximal, characteristic, complemented
2C2×#14556112C211yesmaximal, characteristic, complemented
3C2×#14556111C211yesmaximal, characteristic, complemented
4C2×#14556098C211yesmaximal, characteristic
5#17985877C214yesmaximal, complemented
6#17985873C214yesmaximal, complemented
7#17985860C211yesmaximal, characteristic, complemented
8#17985854C211yesmaximal, characteristic, complemented
9#381×C71C211yesmaximal, characteristic, complemented
10C24×D71C2211yescharacteristic
11C22×D284C2212yes—
12C22×C22⋊D71C2211yescharacteristic
13C23×Dic71C2211yescharacteristic
14C22×C4×D71—24no—
15C22×C4×D71—24no—
16C22×C4×D71—24no—
17C22×C4×D71C2214yes—
18#14556112—24no—
19#14556111—24no—
20#14556098—24no—
21C2×C4⋊Dic71C2212yescomplemented
22C2×Dic71⋊C4C2212yescomplemented
23C2×C22⋊Dic71C2211yescharacteristic
24C2×(C2×D71)⋊C4C2214yes—
25C2×(C2×D71)⋊C4C2212yes—
26C2×(C2×D71)⋊C4C2212yescomplemented
27C2×(C2×D71)⋊C4C2212yescomplemented
28C71⋊#95C2214yescomplemented
29C23×C4×C71C2211yescharacteristic
30C2×C22⋊C4×C71C2211yescharacteristic
31C2×C22⋊C4×C71C2211yescharacteristic
32#95×C71C2214yescomplemented
33C23×D71—22no—
34C23×D71—22no—
35C23×D71—28no—
36C23×D71D414yes—
37C23×D71C2311yescharacteristic
38C23×D71C2312yes—
39C2×D284—24no—
40C2×D284—28no—
41C2×C22⋊D71—22no—
42C2×C22⋊D71—22no—
43C2×C22⋊D71—22no—
44C22×Dic71—22no—
45C22×Dic71—22no—
46C22×Dic71D414yescomplemented
47C22×Dic71C2311yescharacteristic
48C22×Dic71C2312yes—
49C22×Dic71C2312yes—
50C2×C4×D71—232no—
51C2×C4×D71—28no—
52C2×C4×D71—28no—
53C2×C4×D71—28no—
54C4⋊Dic71—24no—
55Dic71⋊C4—24no—
56C22⋊Dic71—22no—
57(C2×D71)⋊C4—24no—
58(C2×D71)⋊C4—28no—
59(C2×D71)⋊C4—24no—
60(C2×D71)⋊C4—24no—
61C24×C71C2311yescharacteristic
62C22×C4×C71—22no—
63C22×C4×C71—22no—
64C22×C4×C71D414yescomplemented
65C22×C4×C71C2312yes—
66C22×C4×C71C2312yes—
67C22×C4×C71C2312yes—
68C22⋊C4×C71—22no—
69C22⋊C4×C71—22no—
70C23×C71—24no—
71C23×C71C2×D412yes—
72C23×C71C2×D412yes—
73C23×C71C2×D412yes—
74C23×C71C2411yescharacteristic
75C22×D71—216no—
76C22×D71—216no—
77C22×D71—28no—
78C22×D71—22no—
79C22×D71—22no—
80C22×D71—24no—
81C22×D71—216no—
82C22×D71—21no—
83C22×D71—21no—
84C22×D71—21no—
85C22×D71—24no—
86C22×D71C2×D414yes—
87C22×D71#4118yes—
88C2×C4×C71—28no—
89C2×C4×C71—22no—
90C2×C4×C71—22no—
91C2×C4×C71—22no—
92C2×C4×C71—22no—
93C2×C4×C71—24no—
94C2×C4×C71—24no—
95C2×C4×C71C2×D414yes—
96C22⋊D71—44no—
97C2×Dic71—28no—
98C2×Dic71—22no—
99C2×Dic71—22no—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—21no—
104C2×Dic71—24no—
105C2×Dic71C2×D414yes—
106D284—48no—
107C4×D71—232no—
108C22×C71—24no—
109C22×C71—24no—
110C22×C71—24no—
111C22×C71#12014yes—
112C22×C71C22×D411yescharacteristic
113C22×C71C22×D411yescharacteristic
114C22×C71C22×D411yescharacteristic
115C22×C71C2×#4114yes—
116C2×D71—232no—
117C2×D71—24no—
118C2×D71—24no—
119C2×D71—28no—
120C2×D71—28no—
121C2×D71—41no—
122C2×D71—42no—
123C2×D71—42no—
124C2×D71—42no—
125C2×D71#12318yescomplemented
126C4×C71—24no—
127C4×C71—42no—
128C4×C71—42no—
129Dic71—24no—
130Dic71—41no—
131Dic71—42no—
132C2×C71—24no—
133C2×C71C2×#12011yescharacteristic
134C2×C71C2×#12312yes—
135C2×C71#52512yes—
136C2×C71#52512yes—
137D71—28no—
138D71—81no—
139#2849—711nomaximal, Sylow
140C71#284911yescharacteristic, complemented, Sylow
141C24×C4—711no—
142C2×#123—711no—
143C2×#123—711no—
144C2×#120—711no—
145#386—714no—
146#385—714no—
147#381—711no—
148#381D7111yescharacteristic, complemented
149#379—711no—
150C25—711no—
151C22×D4—711no—
152C22×D4—712no—
153C23×C4—711no—
154C23×C4—714no—
155C23×C4—1424no—
156C23×C4—1424no—
157C23×C4—1424no—
158C23×C4C2×D7111yescharacteristic
159#123—1424no—
160#123—1424no—
161#120—1424no—
162C2×C4⋊C4—712no—
163C2×C4⋊C4—712no—
164C2×C22⋊C4—711no—
165C2×C22⋊C4—712no—
166C2×C22⋊C4—712no—
167C2×C22⋊C4—712no—
168C2×C22⋊C4—714no—
169C2×C22⋊C4C2×D7111yescharacteristic
170C2×C22⋊C4C2×D7111yescharacteristic
171#95—714no—
172#95C2×D7114yescomplemented
173C24—712no—
174C24—714no—
175C24—711no—
176C24—1428no—
177C24—1422no—
178C24—1422no—
179C24C22×D7111yescharacteristic
180C2×D4—1424no—
181C2×D4—1428no—
182C2×D4—1422no—
183C2×D4—1422no—
184C2×D4—1422no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—714no—
188C22×C4—712no—
189C22×C4—712no—
190C22×C4—711no—
191C22×C4—1428no—
192C22×C4—1428no—
193C22×C4—14232no—
194C22×C4—1422no—
195C22×C4—1422no—
196C22×C4—1428no—
197C22×C4D28414yescomplemented
198C22×C4C22×D7112yes—
199C22×C4C22×D7112yes—
200C22×C4C22×D7112yes—
201C4⋊C4—1424no—
202C4⋊C4—1424no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—1424no—
206C22⋊C4—1428no—
207C22⋊C4—1424no—
208C22⋊C4—1422no—
209C22⋊C4—1424no—
210C23—24no—
211C23—714no—
212C23—718no—
213C23—14216no—
214C23—14216no—
215C23—14216no—
216C23—1421no—
217C23—1421no—
218C23—1421no—
219C23—1424no—
220C23—1422no—
221C23—1422no—
222C23—1424no—
223C23—1428no—
224C23D4×D7112yes—
225C23D4×D7112yes—
226C23C2×D28412yes—
227C23C23×D7111yescharacteristic
228D4—2848no—
229D4—2844no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—28no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—714no—
238C2×C4—14232no—
239C2×C4—1421no—
240C2×C4—1421no—
241C2×C4—1422no—
242C2×C4—1421no—
243C2×C4—1424no—
244C2×C4—1422no—
245C2×C4—1422no—
246C2×C4—1428no—
247C2×C4C2×D28414yes—
248C22—24no—
249C22—24no—
250C22—24no—
251C22—2841no—
252C22—2842no—
253C22—2842no—
254C22—2842no—
255C22—718no—
256C22—14232no—
257C22—1424no—
258C22—1424no—
259C22—1428no—
260C22—1428no—
261C22#1455609814yes—
262C22C2×D4×D7111yescharacteristic
263C22C2×D4×D7111yescharacteristic
264C22C22×D28411yescharacteristic
265C22#41×D7114yes—
266C4—24no—
267C4—42no—
268C4—42no—
269C4—2841no—
270C4—2842no—
271C4—1424no—
272C2—24no—
273C2—5681no—
274C2—1428no—
275C2C2×#1455609811yescharacteristic
276C2#123×D7112yes—
277C2#1798650712yes—
278C2#1798651112yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#381×C71C2
hypercenter#381D71
hypocenterC71#2849
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C23×C4×D71C2semidirect
2. C2×#14556112C2semidirect
3. C2×#14556111C2semidirect
5. #17985877C2semidirect
6. #17985873C2semidirect
7. #17985860C2semidirect
8. #17985854C2semidirect
9. #381×C71C2semidirect
21. C2×C4⋊Dic71C22semidirect
22. C2×Dic71⋊C4C22semidirect
26. C2×(C2×D71)⋊C4C22semidirect
27. C2×(C2×D71)⋊C4C22semidirect
28. C71⋊#95C22semidirect
32. #95×C71C22semidirect
46. C22×Dic71D4semidirect
64. C22×C4×C71D4semidirect
125. C2×D71#123semidirect
140. C71#2849semidirect
148. #381D71semidirect
172. #95C2×D71semidirect
197. C22×C4D284semidirect