Finite Group Databasebeta

← #22596628

Subgroups — #22596628

Order 9088.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
2C2×#14556108C212yesmaximal, complemented
3C2×#14556107C211yesmaximal, characteristic, complemented
4#17985874C212yesmaximal, complemented
5#17985873C214yesmaximal, complemented
6#17985856C212yesmaximal, complemented
7#17985854C212yesmaximal
8#380×C71C211yesmaximal, characteristic, complemented
9C24×D71C2211yescharacteristic
10C22×Dic142C2211yescharacteristic, complemented
11C23×Dic71C2211yescharacteristic
12C22×C4×D71—28no—
13C22×C4×D71—24no—
14C22×C4×D71C2212yes—
15C22×C4×D71C2212yes—
16#14556108—28no—
17#14556107—24no—
18C2×C4⋊Dic71C2211yescharacteristic, complemented
19C2×C4⋊Dic71C2214yescomplemented
20C2×Dic71⋊C4C2212yescomplemented
21C2×C22⋊Dic71C2212yes—
22C2×(C2×D71)⋊C4C2214yes—
23C2×(C2×D71)⋊C4C2214yes—
24C71⋊#95C2214yescomplemented
25C23×C4×C71C2211yescharacteristic
26C2×C4⋊C4×C71C2212yescomplemented
27C2×C22⋊C4×C71C2212yes—
28#95×C71C2212yescomplemented
29C23×D71—22no—
30C23×D71—22no—
31C23×D71—24no—
32C23×D71—24no—
33C23×D71Q814yescomplemented
34C23×D71C2312yes—
35C2×Dic142—22no—
36C2×Dic142—24no—
37C22×Dic71—22no—
38C22×Dic71—24no—
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C2×C4×D71—232no—
43C2×C4×D71—24no—
44C2×C4×D71—24no—
45C2×C4×D71—24no—
46C2×C4×D71—24no—
47C2×C4×D71—24no—
48C2×C4×D71D418yes—
49C4⋊Dic71—22no—
50C4⋊Dic71—28no—
51Dic71⋊C4—24no—
52C22⋊Dic71—24no—
53(C2×D71)⋊C4—28no—
54(C2×D71)⋊C4—28no—
55C24×C71C2311yescharacteristic
56C22×C4×C71—24no—
57C22×C4×C71D414yes—
58C22×C4×C71C2314yes—
59C22×C4×C71C2311yescharacteristic
60C22×C4×C71C2311yescharacteristic
61C4⋊C4×C71—24no—
62C22⋊C4×C71—24no—
63C23×C71—22no—
64C23×C71—22no—
65C23×C71C2×D412yes—
66C23×C71C2×Q812yes—
67C23×C71#4112yes—
68C23×C71C2411yescharacteristic
69C22×D71—28no—
70C22×D71—216no—
71C22×D71—28no—
72C22×D71—28no—
73C22×D71—22no—
74C22×D71—22no—
75C22×D71—22no—
76C22×D71—22no—
77C22×D71—216no—
78C22×D71C2×D414yes—
79C22×D71C2×Q814yes—
80C22×D71#4114yes—
81C2×C4×C71—22no—
82C2×C4×C71—28no—
83C2×C4×C71—24no—
84C2×C4×C71—24no—
85C2×C4×C71—24no—
86C2×C4×C71C2×D414yes—
87C2×C4×C71C2×D412yes—
88C2×C4×C71C2×D412yes—
89C2×Dic71—28no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—24no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—24no—
96C2×Dic71C2×D414yes—
97C4×D71—216no—
98C4×D71—216no—
99Dic142—44no—
100C22×C71—22no—
101C22×C71—24no—
102C22×C71—22no—
103C22×C71—24no—
104C22×C71C22⋊Q814yescomplemented
105C22×C71C22×D412yes—
106C22×C71C22×D411yescharacteristic
107C22×C71C22×Q811yescharacteristic
108C22×C71C2×#4112yes—
109C22×C71C2×#4111yescharacteristic
110C2×D71—232no—
111C2×D71—24no—
112C2×D71—24no—
113C2×D71—24no—
114C2×D71—24no—
115C2×D71—24no—
116C2×D71—24no—
117C2×D71C22⋊Q818yescomplemented
118C4×C71—22no—
119C4×C71—44no—
120C4×C71#12014yes—
121Dic71—24no—
122Dic71—42no—
123Dic71—42no—
124C2×C71—24no—
125C2×C71C2×#12011yescharacteristic
126C2×C71C2×C22⋊Q812yes—
127C2×C71C2×C22⋊Q811yescharacteristic
128C2×C71#52512yes—
129C2×C71#52511yescharacteristic
130D71—28no—
131#2848—711nomaximal, Sylow
132C71#284811yescharacteristic, complemented, Sylow
133C24×C4—711no—
134C2×C22⋊Q8—711no—
135C2×C22⋊Q8—712no—
136#385—712no—
137#385—714no—
138#380—712no—
139#380D7111yescharacteristic, complemented
140#379—712no—
141C25—711no—
142C22×Q8—711no—
143C23×C4—712no—
144C23×C4—711no—
145C23×C4—712no—
146C23×C4—1428no—
147C23×C4—1424no—
148C23×C4C2×D7111yescharacteristic
149C22⋊Q8—1428no—
150C22⋊Q8—1424no—
151C2×C4⋊C4—711no—
152C2×C4⋊C4—712no—
153C2×C4⋊C4—714no—
154C2×C4⋊C4C2×D7112yescomplemented
155C2×C22⋊C4—712no—
156C2×C22⋊C4—714no—
157C2×C22⋊C4—714no—
158C2×C22⋊C4C2×D7112yes—
159#95—714no—
160#95C2×D7112yescomplemented
161C24—712no—
162C24—714no—
163C24—1424no—
164C24—1424no—
165C24—1422no—
166C24—1422no—
167C24C22×D7111yescharacteristic
168C2×Q8—1424no—
169C2×Q8—1422no—
170C22×C4—24no—
171C22×C4—712no—
172C22×C4—718no—
173C22×C4—712no—
174C22×C4—712no—
175C22×C4—1424no—
176C22×C4—1424no—
177C22×C4—14232no—
178C22×C4—1424no—
179C22×C4—1422no—
180C22×C4—1424no—
181C22×C4—1424no—
182C22×C4—1424no—
183C22×C4D28414yes—
184C22×C4C22×D7114yes—
185C22×C4C22×D7111yescharacteristic
186C22×C4C22×D7111yescharacteristic
187C4⋊C4—24no—
188C4⋊C4—1428no—
189C4⋊C4—1424no—
190C4⋊C4—1422no—
191C22⋊C4—24no—
192C22⋊C4—1428no—
193C22⋊C4—1428no—
194C22⋊C4—1424no—
195C23—22no—
196C23—22no—
197C23—714no—
198C23—714no—
199C23—714no—
200C23—14216no—
201C23—1428no—
202C23—14216no—
203C23—1428no—
204C23—1422no—
205C23—1422no—
206C23—1422no—
207C23—1422no—
208C23—1428no—
209C23C2×D28412yes—
210C23Q8×D7112yes—
211C23#1176586412yes—
212C23C23×D7111yescharacteristic
213Q8—2844no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—714no—
220C2×C4—14216no—
221C2×C4—1424no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4—1424no—
225C2×C4—1428no—
226C2×C4—1422no—
227C2×C4—1422no—
228C2×C4—14216no—
229C2×C4D4×D7114yes—
230C2×C4C2×D28412yes—
231C2×C4C2×D28412yes—
232C22—22no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—718no—
237C22—14232no—
238C22—1424no—
239C22—1424no—
240C22—1424no—
241C22—1424no—
242C22—1424no—
243C22—1424no—
244C22#1455610814yescomplemented
245C22C2×D4×D7112yes—
246C22C22×D28411yescharacteristic
247C22C2×Q8×D7111yescharacteristic
248C22#41×D7112yes—
249C22C2×#1176586411yescharacteristic
250C4—22no—
251C4—44no—
252C4—2842no—
253C4—2842no—
254C4—1424no—
255C4#1455609814yes—
256C2—24no—
257C2—1428no—
258C2C2×#1455609811yescharacteristic
259C2C22⋊Q8×D7112yes—
260C2C2×#1455610811yescharacteristic
261C2#1798650912yes—
262C2#1798651111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#380×C71C2
hypercenter#380D71
hypocenterC71#2848
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#14556108C2semidirect
3. C2×#14556107C2semidirect
4. #17985874C2semidirect
5. #17985873C2semidirect
6. #17985856C2semidirect
8. #380×C71C2semidirect
10. C22×Dic142C22semidirect
18. C2×C4⋊Dic71C22semidirect
19. C2×C4⋊Dic71C22semidirect
20. C2×Dic71⋊C4C22semidirect
24. C71⋊#95C22semidirect
26. C2×C4⋊C4×C71C22semidirect
28. #95×C71C22semidirect
33. C23×D71Q8semidirect
104. C22×C71C22⋊Q8semidirect
117. C2×D71C22⋊Q8semidirect
132. C71#2848semidirect
139. #380D71semidirect
154. C2×C4⋊C4C2×D71semidirect
160. #95C2×D71semidirect
244. C22#14556108semidirect