Finite Group Databasebeta

← #22591950

Subgroups — #22591950

Order 9088.

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

normal subgroups only

This group has 914 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×#11765865C211yesmaximal, characteristic, complemented
2C2×#14556111C211yesmaximal, characteristic, complemented
3C2×#14556102C212yesmaximal, complemented
4#17985920C211yesmaximal, characteristic, complemented
5#17985917C211yesmaximal, characteristic, complemented
6#17985912C211yesmaximal, characteristic, complemented
7#17985909C211yesmaximal, characteristic, complemented
8#17985878C212yesmaximal, complemented
9#17985877C212yesmaximal, complemented
10#17985873C212yesmaximal, complemented
11C2×#95×C71C211yesmaximal, characteristic, complemented
12C2×#11765865—28no—
13C2×#11765865—24no—
14C22×Dic142C2211yescharacteristic
15C22×D284C2211yescharacteristic
16C22×D284C2212yes—
17C22×C22⋊D71C2211yescharacteristic
18C22×C22⋊D71C2212yes—
19C22×C4×D71C2212yes—
20#14556111—24no—
21#14556102—28no—
22C2×C4⋊Dic71C2212yes—
23C2×Dic71⋊C4C2211yescharacteristic
24C2×Dic71⋊C4C2211yescharacteristic
25C2×C22⋊Dic71C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2211yescharacteristic
29C2×(C2×D71)⋊C4C2212yes—
30C2×(C2×D71)⋊C4C2211yescharacteristic
31C2×(C2×D71)⋊C4C2212yes—
32C71⋊#95C2212yes—
33C71⋊#95C2212yes—
34C23×C4×C71C2212yes—
35C23×C4×C71C2211yescharacteristic
36#95×C71C2212yes—
37#95×C71C2211yescharacteristic
38#95×C71C2211yescharacteristic
39C23×D71C2311yescharacteristic
40C23×D71C2312yes—
41#11765865—232no—
42C2×Dic142—22no—
43C2×Dic142—24no—
44C2×D284—24no—
45C2×D284—22no—
46C2×D284—24no—
47C2×D284—24no—
48C2×D284—42no—
49C2×C22⋊D71—24no—
50C2×C22⋊D71—24no—
51C2×C22⋊D71—24no—
52C2×C22⋊D71—22no—
53C2×C22⋊D71D418yescomplemented
54C22×Dic71C2311yescharacteristic
55C22×Dic71C2311yescharacteristic
56C22×Dic71C2311yescharacteristic
57C22×Dic71C2312yes—
58C2×C4×D71—24no—
59C2×C4×D71—24no—
60C2×C4×D71—24no—
61C4⋊Dic71—24no—
62Dic71⋊C4—22no—
63Dic71⋊C4—22no—
64C22⋊Dic71—22no—
65(C2×D71)⋊C4—24no—
66(C2×D71)⋊C4—22no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4—24no—
69(C2×D71)⋊C4—24no—
70(C2×D71)⋊C4—42no—
71C24×C71C2311yescharacteristic
72C22×C4×C71—24no—
73C22×C4×C71—24no—
74C22×C4×C71—24no—
75C22×C4×C71—42no—
76C22×C4×C71D414yescomplemented
77C22×C4×C71C2312yes—
78C22×C4×C71C2311yescharacteristic
79C22×C4×C71C2311yescharacteristic
80C22×C4×C71C2312yes—
81C23×C71—22no—
82C23×C71—21no—
83C23×C71—21no—
84C23×C71C2×D414yes—
85C23×C71C2×D412yes—
86C23×C71C2411yescharacteristic
87C22×D71—22no—
88C22×D71—22no—
89C22×D71—22no—
90C22×D71—22no—
91C22×D71—22no—
92C22×D71—22no—
93C22×D71—21no—
94C22×D71—22no—
95C22×D71—22no—
96C22×D71—21no—
97C22×D71—21no—
98C22×D71C2×D414yes—
99C2×C4×C71—28no—
100C2×C4×C71—22no—
101C2×C4×C71—22no—
102C2×C4×C71—22no—
103C2×C4×C71—22no—
104C2×C4×C71—22no—
105C2×C4×C71—22no—
106C2×C4×C71—22no—
107C2×C4×C71—48no—
108C2×C4×C71C2×D412yes—
109C2×C4×C71C2×D412yes—
110C2×C4×C71#4114yescomplemented
111C22⋊D71—216no—
112C22⋊D71—44no—
113C2×Dic71—22no—
114C2×Dic71—22no—
115C2×Dic71—21no—
116C2×Dic71—21no—
117C2×Dic71—22no—
118C2×Dic71—21no—
119C2×Dic71—22no—
120C2×Dic71—21no—
121C2×Dic71—21no—
122C2×Dic71C2×D414yes—
123D284—28no—
124D284—48no—
125C4×D71—216no—
126Dic142—28no—
127C22×C71—22no—
128C22×C71—22no—
129C22×C71—24no—
130C22×C71—24no—
131C22×C71#12014yes—
132C22×C71C22×D412yes—
133C22×C71C22×D411yescharacteristic
134C22×C71C2×#4112yes—
135C22×C71H(5,𝔽2)11yescharacteristic
136C22×C712−1+411yescharacteristic
137C2×D71—24no—
138C2×D71—24no—
139C2×D71—24no—
140C2×D71—42no—
141C2×D71—42no—
142C2×D71—42no—
143C2×D71—42no—
144C2×D71—41no—
145C2×D71—41no—
146C2×D71—41no—
147C2×D71—42no—
148C2×D71—42no—
149C4×C71—22no—
150C4×C71—22no—
151C4×C71—24no—
152C4×C71—42no—
153Dic71—24no—
154Dic71—41no—
155Dic71—41no—
156Dic71—41no—
157C2×C71—24no—
158C2×C71C2×#12011yescharacteristic
159C2×C71#53211yescharacteristic
160C2×C71#53711yescharacteristic
161C2×C71#54612yes—
162C2×C71#54712yes—
163D71—44no—
164D71—81no—
165#2430—711nomaximal, Sylow
166C71#243011yescharacteristic, complemented, Sylow
167C22×#41—711no—
168C2×#123—711no—
169C2×#121—712no—
170#395—711no—
171#394—711no—
172#393—711no—
173#392—711no—
174#386—712no—
175#386—712no—
176#385—712no—
177C2×#95D7111yescharacteristic, complemented
178C2×#41—1428no—
179C2×#41—1424no—
180C22×Q8—711no—
181C22×D4—712no—
182C22×D4—711no—
183C22×D4—712no—
184C22×D4—711no—
185C23×C4—712no—
186C23×C4C2×D7112yes—
187C23×C4C2×D7111yescharacteristic
188#123—1424no—
189#121—1428no—
190C2×C4⋊C4—712no—
191C2×C4⋊C4—711no—
192C2×C4⋊C4—711no—
193C2×C22⋊C4—712no—
194C2×C22⋊C4—711no—
195C2×C22⋊C4—712no—
196C2×C22⋊C4—712no—
197C2×C22⋊C4—711no—
198C2×C22⋊C4—711no—
199C2×C22⋊C4—712no—
200#95—712no—
201#95—712no—
202#95C2×D7112yes—
203#95C2×D7111yescharacteristic
204#95C2×D7111yescharacteristic
205C24—712no—
206C24—711no—
207C24C22×D7111yescharacteristic
208#41—14232no—
209C2×Q8—1424no—
210C2×Q8—1422no—
211C2×D4—2842no—
212C2×D4—718no—
213C2×D4—1424no—
214C2×D4—1424no—
215C2×D4—1424no—
216C2×D4—1422no—
217C2×D4—1424no—
218C2×D4—1424no—
219C2×D4—1422no—
220C2×D4—1424no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—712no—
226C22×C4—711no—
227C22×C4—711no—
228C22×C4—711no—
229C22×C4—1424no—
230C22×C4—1424no—
231C22×C4—1424no—
232C22×C4D28414yescomplemented
233C22×C4C22×D7112yes—
234C22×C4C22×D7111yescharacteristic
235C22×C4C22×D7111yescharacteristic
236C22×C4C22×D7112yes—
237C4⋊C4—1424no—
238C4⋊C4—1422no—
239C4⋊C4—1422no—
240C22⋊C4—2842no—
241C22⋊C4—1424no—
242C22⋊C4—1424no—
243C22⋊C4—1422no—
244C22⋊C4—1422no—
245C22⋊C4—1422no—
246C22⋊C4—1424no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—714no—
251C23—1422no—
252C23—1421no—
253C23—1422no—
254C23—1421no—
255C23—1421no—
256C23—1422no—
257C23—1422no—
258C23—1422no—
259C23—1422no—
260C23—1422no—
261C23—1422no—
262C23D4×D7114yes—
263C23C2×D28412yes—
264C23C23×D7111yescharacteristic
265Q8—1428no—
266D4—2848no—
267D4—2844no—
268D4—1428no—
269D4—14216no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—28no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—48no—
279C2×C4—714no—
280C2×C4—14216no—
281C2×C4—1422no—
282C2×C4—1421no—
283C2×C4—1421no—
284C2×C4—1421no—
285C2×C4—1422no—
286C2×C4—1421no—
287C2×C4—1421no—
288C2×C4—1422no—
289C2×C4—1422no—
290C2×C4C2×D28412yes—
291C2×C4C2×D28412yes—
292C2×C4#1176586514yescomplemented
293C22—22no—
294C22—24no—
295C22—24no—
296C22—22no—
297C22—2842no—
298C22—2842no—
299C22—2842no—
300C22—2842no—
301C22—2841no—
302C22—2842no—
303C22—2842no—
304C22—2841no—
305C22—2841no—
306C22—1424no—
307C22—1424no—
308C22—1424no—
309C22#1455609814yes—
310C22C2×D4×D7112yes—
311C22C22×D28411yescharacteristic
312C22C2×#1176586512yes—
313C22#1455617311yescharacteristic
314C22#1455617611yescharacteristic
315C4—24no—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—2841no—
320C4—2841no—
321C4—2841no—
322C4—1424no—
323C2—24no—
324C2—2844no—
325C2—5681no—
326C2C2×#1455609811yescharacteristic
327C2#1798654911yescharacteristic
328C2#1798658011yescharacteristic
329C2#1798663912yes—
330C2#1798665012yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
FittingC2×#95×C71C2
hypercenterC2×#95D71
hypocenterC71#2430
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
1. C22×#11765865C2semidirect
2. C2×#14556111C2semidirect
3. C2×#14556102C2semidirect
4. #17985920C2semidirect
5. #17985917C2semidirect
6. #17985912C2semidirect
7. #17985909C2semidirect
8. #17985878C2semidirect
9. #17985877C2semidirect
10. #17985873C2semidirect
11. C2×#95×C71C2semidirect
53. C2×C22⋊D71D4semidirect
76. C22×C4×C71D4semidirect
110. C2×C4×C71#41semidirect
166. C71#2430semidirect
177. C2×#95D71semidirect
232. C22×C4D284semidirect
292. C2×C4#11765865semidirect