Finite Group Databasebeta

← #22591739

Subgroups — #22591739

Order 9088.

814 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14556109C211yesmaximal, characteristic, complemented
2C2×#14556096C211yesmaximal, characteristic
3C22×(C2×D71)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D71C211yesmaximal, characteristic, complemented
5#17985928C212yesmaximal, complemented
6#17985914C212yesmaximal, complemented
7#17985875C212yesmaximal, complemented
8#17985872C211yesmaximal, characteristic, complemented
9#17985860C211yesmaximal, characteristic, complemented
10#17985858C212yesmaximal, complemented
11#386×C71C211yesmaximal, characteristic, complemented
12C24×D71C2211yescharacteristic
13C22×C22⋊D71C2211yescharacteristic
14C22×C22⋊D71C2211yescharacteristic
15C23×Dic71C2211yescharacteristic
16C22×C4×D71C2212yes—
17#14556109—24no—
18#14556096—24no—
19C2×Dic71⋊C4C2212yescomplemented
20C2×Dic71⋊C4C2212yescomplemented
21C2×C22⋊Dic71C2211yescharacteristic
22C2×C22⋊Dic71C2211yescharacteristic
23C2×C22⋊Dic71C2212yescomplemented
24C2×(C2×D71)⋊C4—24no—
25C2×(C2×D71)⋊C4C2211yescharacteristic
26C2×(C2×D71)⋊C4C2212yes—
27C2×(C2×D71)⋊C4C2212yes—
28C2×(C2×D71)⋊C4C2212yes—
29C2×(C2×D71)⋊C4C2211yescharacteristic
30C22⋊C4×D71—24no—
31C71⋊#95C2212yescomplemented
32C71⋊#95C2212yescomplemented
33C71⋊#95C2212yescomplemented
34C22×D4×C71C2211yescharacteristic
35C23×C4×C71C2211yescharacteristic
36C2×C22⋊C4×C71C2211yescharacteristic
37C2×C22⋊C4×C71C2212yescomplemented
38#95×C71C2212yescomplemented
39C23×D71—22no—
40C23×D71—22no—
41C23×D71—24no—
42C23×D71—42no—
43C23×D71D414yes—
44C23×D71C2311yescharacteristic
45C23×D71C2311yescharacteristic
46C2×C22⋊D71—22no—
47C2×C22⋊D71—22no—
48C2×C22⋊D71—24no—
49C2×C22⋊D71—42no—
50C22×Dic71—22no—
51C22×Dic71—22no—
52C22×Dic71—22no—
53C22×Dic71C2312yes—
54C22×Dic71C2311yescharacteristic
55C22×Dic71C2311yescharacteristic
56C22×Dic71C2312yes—
57C2×C4×D71—24no—
58C2×C4×D71—24no—
59C2×C4×D71—42no—
60Dic71⋊C4—24no—
61Dic71⋊C4—42no—
62C22⋊Dic71—22no—
63C22⋊Dic71—24no—
64C22⋊Dic71D414yescomplemented
65(C2×D71)⋊C4—24no—
66(C2×D71)⋊C4—24no—
67(C2×D71)⋊C4—22no—
68(C2×D71)⋊C4—22no—
69(C2×D71)⋊C4—24no—
70C24×C71C2311yescharacteristic
71C24×C71C2311yescharacteristic
72C2×D4×C71—22no—
73C2×D4×C71—22no—
74C2×D4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—42no—
77C22×C4×C71C2312yes—
78C22×C4×C71C2311yescharacteristic
79C22×C4×C71C2312yes—
80C22⋊C4×C71—22no—
81C22⋊C4×C71—42no—
82C23×C71—22no—
83C23×C71—22no—
84C23×C71—21no—
85C23×C71—21no—
86C23×C71—21no—
87C23×C71—22no—
88C23×C71—22no—
89C23×C71C2×D412yes—
90C23×C71C2×D412yes—
91C23×C71#4112yes—
92C23×C71C2411yescharacteristic
93C22×D71—21no—
94C22×D71—21no—
95C22×D71—22no—
96C22×D71—28no—
97C22×D71—21no—
98C22×D71—22no—
99C22×D71—21no—
100C22×D71—22no—
101C22×D71—28no—
102C22×D71—44no—
103C22×D71—48no—
104C22×D71—48no—
105C22×D71C2×D412yes—
106C22×D71C2×D412yes—
107C22×D71#4114yes—
108D4×C71—44no—
109C2×C4×C71—22no—
110C2×C4×C71—22no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—22no—
117C2×C4×C71—22no—
118C2×C4×C71—44no—
119C22⋊D71—44no—
120C22⋊D71—44no—
121C2×Dic71—22no—
122C2×Dic71—22no—
123C2×Dic71—22no—
124C2×Dic71—21no—
125C2×Dic71—21no—
126C2×Dic71—22no—
127C2×Dic71—22no—
128C2×Dic71—22no—
129C2×Dic71—22no—
130C2×Dic71—44no—
131C2×Dic71C2×D414yes—
132C4×D71—48no—
133C22×C71—22no—
134C22×C71—22no—
135C22×C71—24no—
136C22×C71—24no—
137C22×C71—41no—
138C22×C71—41no—
139C22×C71—41no—
140C22×C71—42no—
141C22×C71—42no—
142C22×C71#12114yes—
143C22×C71C22×D411yescharacteristic
144C22×C71C22×D411yescharacteristic
145C22×C71C2×#4112yes—
146C22×C71C2×#4111yescharacteristic
147C22×C71H(5,𝔽2)12yes—
148C2×D71—22no—
149C2×D71—22no—
150C2×D71—24no—
151C2×D71—24no—
152C2×D71—22no—
153C2×D71—22no—
154C2×D71—22no—
155C2×D71—416no—
156C2×D71—42no—
157C2×D71—42no—
158C2×D71#12314yescomplemented
159C4×C71—41no—
160C4×C71—42no—
161C4×C71—42no—
162Dic71—24no—
163Dic71—42no—
164Dic71—41no—
165Dic71—41no—
166C2×C71—24no—
167C2×C71—81no—
168C2×C71C2×#12111yescharacteristic
169C2×C71C2×#12311yescharacteristic
170C2×C71#53411yescharacteristic
171C2×C71#54612yes—
172C2×C71#55112yes—
173D71—24no—
174D71—42no—
175#2412—711nomaximal, Sylow
176C71#241211yescharacteristic, complemented, Sylow
177C2×#123—711no—
178C2×#120—711no—
179C22×C22⋊C4—711no—
180C22×C22⋊C4—711no—
181#397—712no—
182#394—712no—
183#386—712no—
184#386D7111yescharacteristic, complemented
185#385—711no—
186#381—711no—
187#380—712no—
188C25—711no—
189C22×D4—711no—
190C22×D4—711no—
191C22×D4C2×D7111yescharacteristic
192C23×C4—712no—
193C23×C4—711no—
194C23×C4C2×D7111yescharacteristic
195#123—1424no—
196#120—1424no—
197C2×C4⋊C4—712no—
198C2×C4⋊C4—712no—
199C2×C22⋊C4—711no—
200C2×C22⋊C4—711no—
201C2×C22⋊C4—712no—
202C2×C22⋊C4—712no—
203C2×C22⋊C4—712no—
204C2×C22⋊C4—712no—
205C2×C22⋊C4—711no—
206C2×C22⋊C4—711no—
207C2×C22⋊C4—1424no—
208C2×C22⋊C4—1424no—
209C2×C22⋊C4C2×D7111yescharacteristic
210C2×C22⋊C4C2×D7112yescomplemented
211#95—712no—
212#95—712no—
213#95—712no—
214#95C2×D7112yescomplemented
215C24—2842no—
216C24—714no—
217C24—711no—
218C24—711no—
219C24—1424no—
220C24—1422no—
221C24—1422no—
222C24C22×D7111yescharacteristic
223C24C22×D7111yescharacteristic
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—2842no—
228C2×D4—1422no—
229C2×D4—1422no—
230C2×D4—1424no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—2842no—
234C22×C4—712no—
235C22×C4—711no—
236C22×C4—712no—
237C22×C4—711no—
238C22×C4—1424no—
239C22×C4—1424no—
240C22×C4—1422no—
241C22×C4—1422no—
242C22×C4—1422no—
243C22×C4C22×D7112yes—
244C22×C4C22×D7112yes—
245C22×C4C22×D7111yescharacteristic
246C4⋊C4—2842no—
247C4⋊C4—1424no—
248C22⋊C4—22no—
249C22⋊C4—42no—
250C22⋊C4—714no—
251C22⋊C4—1424no—
252C22⋊C4—1424no—
253C22⋊C4—1424no—
254C22⋊C4—1422no—
255C22⋊C4—1422no—
256C22⋊C4—1424no—
257C22⋊C4—1422no—
258C23—22no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—22no—
265C23—2848no—
266C23—2848no—
267C23—2844no—
268C23—712no—
269C23—714no—
270C23—712no—
271C23—1428no—
272C23—1421no—
273C23—1422no—
274C23—1421no—
275C23—1422no—
276C23—1428no—
277C23—1421no—
278C23—1421no—
279C23—1422no—
280C23D4×D7112yes—
281C23D4×D7112yes—
282C23#1176586512yes—
283C23C23×D7111yescharacteristic
284D4—44no—
285D4—2844no—
286D4—2844no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—44no—
297C2×C4—2848no—
298C2×C4—2844no—
299C2×C4—714no—
300C2×C4—1422no—
301C2×C4—1421no—
302C2×C4—1421no—
303C2×C4—1422no—
304C2×C4—1422no—
305C2×C4—1422no—
306C2×C4—1422no—
307C2×C4—1422no—
308C2×C4—1422no—
309C22—24no—
310C22—24no—
311C22—22no—
312C22—22no—
313C22—42no—
314C22—42no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—28416no—
319C22—2842no—
320C22—2842no—
321C22—714no—
322C22—1422no—
323C22—1422no—
324C22—1422no—
325C22—1424no—
326C22—1424no—
327C22—1422no—
328C22—1422no—
329C22#1455610114yes—
330C22C2×D4×D7111yescharacteristic
331C22C2×D4×D7111yescharacteristic
332C22#41×D7112yes—
333C22C2×#1176586511yescharacteristic
334C22#1455617312yes—
335C4—42no—
336C4—41no—
337C4—42no—
338C4—2842no—
339C4—2841no—
340C4—2841no—
341C4—1424no—
342C2—24no—
343C2—81no—
344C2—2842no—
345C2—1424no—
346C2C2×#1455610111yescharacteristic
347C2#123×D7111yescharacteristic
348C2#1798655911yescharacteristic
349C2#1798663812yes—
350C2#1798667812yes—
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#386×C71C2
hypercenter#386D71
hypocenterC71#2412
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#14556109C2semidirect
3. C22×(C2×D71)⋊C4C2semidirect
4. C2×C22⋊C4×D71C2semidirect
5. #17985928C2semidirect
6. #17985914C2semidirect
7. #17985875C2semidirect
8. #17985872C2semidirect
9. #17985860C2semidirect
10. #17985858C2semidirect
11. #386×C71C2semidirect
19. C2×Dic71⋊C4C22semidirect
20. C2×Dic71⋊C4C22semidirect
23. C2×C22⋊Dic71C22semidirect
31. C71⋊#95C22semidirect
32. C71⋊#95C22semidirect
33. C71⋊#95C22semidirect
37. C2×C22⋊C4×C71C22semidirect
38. #95×C71C22semidirect
64. C22⋊Dic71D4semidirect
158. C2×D71#123semidirect
176. C71#2412semidirect
184. #386D71semidirect
210. C2×C22⋊C4C2×D71semidirect
214. #95C2×D71semidirect