Finite Group Databasebeta

← #22597599

Subgroups — #22597599

Order 9088.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11765866C211yesmaximal, characteristic
2C2×#14556160C212yesmaximal, complemented
3C22⋊Q8×D71C211yesmaximal, characteristic, complemented
4#17986164C212yesmaximal, complemented
5#17986160C212yesmaximal, complemented
6#17986155C211yesmaximal, characteristic, complemented
7#17986152C211yesmaximal, characteristic, complemented
8#17986149C211yesmaximal, characteristic, complemented
9#17985999C212yesmaximal, complemented
10#17985961C211yesmaximal, characteristic, complemented
11#451×C71C211yesmaximal, characteristic, complemented
12C2×#11765866—28no—
13C2×#11765866—21no—
14C2×#11765866—21no—
15C2×#11765866C2212yes—
16C2×#11765866C2212yes—
17C2×Q8×D71C2211yescharacteristic, complemented
18C22×D284—21no—
19C22×D284C2211yescharacteristic
20C22×C4×D71—21no—
21C22×C4×D71C2211yescharacteristic
22#14556160—28no—
23C2×#11765855C2212yescomplemented
24C2×#11765854C2212yescomplemented
25C2×#11765852C2212yescomplemented
26C2×#11765848C2212yescomplemented
27#14556108C2211yescharacteristic, complemented
28#14556107C2211yescharacteristic
29#14556106—21no—
30#14556105—21no—
31#14556104C2211yescharacteristic, complemented
32C4⋊C4×D71—21no—
33C4⋊C4×D71C2211yescharacteristic, complemented
34C22⋊C4×D71—21no—
35#14556052C2212yescomplemented
36#14556050C2212yescomplemented
37#14556048C2212yescomplemented
38C22⋊(C71⋊C8)C2211yescharacteristic
39(C2×D71)⋊C8C2212yescomplemented
40C22×Q8×C71C2211yescharacteristic
41C2×Q16×C71C2212yescomplemented
42C22⋊Q8×C71C2211yescharacteristic
43#103×C71C2212yescomplemented
44C22⋊C8×C71C2211yescharacteristic
45C23×D71—21no—
46C23×D71D411yescharacteristic
47#11765866—28no—
48#11765866—28no—
49#11765866—24no—
50#11765866—24no—
51#11765866—24no—
52#11765866D418yes—
53C2×Dic142—21no—
54C2×Dic142C2311yescharacteristic
55Q8×D71—24no—
56C2×D284—28no—
57C2×D284—22no—
58C2×D284—22no—
59C2×D284—21no—
60C2×D284—24no—
61C2×D284C2312yes—
62C22×Dic71D411yescharacteristic
63C2×C4×D71—28no—
64C2×C4×D71—22no—
65C2×C4×D71—22no—
66C2×C4×D71—24no—
67C2×C4×D71—21no—
68C2×C4×D71—21no—
69C2×C4×D71D411yescharacteristic
70C2×C4×D71D411yescharacteristic
71C2×C4×D71C2311yescharacteristic
72C2×C4×D71C2311yescharacteristic
73#11765855—24no—
74#11765854—24no—
75#11765852—24no—
76#11765848—24no—
77C2×C71⋊C8C2312yes—
78C4⋊Dic71—21no—
79C4⋊Dic71C2311yescharacteristic
80Dic71⋊C4—21no—
81Dic71⋊C4—21no—
82Dic71⋊C4—21no—
83C22⋊Dic71—21no—
84(C2×D71)⋊C4—21no—
85(C2×D71)⋊C4—21no—
86C2×Q8×C71—24no—
87C2×Q8×C71—21no—
88C2×Q8×C71C2311yescharacteristic
89C2×Q8×C71C2312yes—
90C22×C4×C71—21no—
91C22×C4×C71C2311yescharacteristic
92Q16×C71—24no—
93C2×C8×C71C2312yes—
94C4⋊C4×C71—21no—
95C4⋊C4×C71C2311yescharacteristic
96C22⋊C4×C71—21no—
97C23×C71C2×D411yescharacteristic
98C22×D71—24no—
99C22×D71—22no—
100C22×D71—22no—
101C22×D71—22no—
102C22×D71—41no—
103C22×D71—42no—
104C22×D71—42no—
105C22×D71C2×D411yescharacteristic
106C22×D71C2×D411yescharacteristic
107Q8×C71—24no—
108Q8×C71—22no—
109Q8×C71—22no—
110Q8×C71C2×D414yes—
111C2×C4×C71—24no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71C2×D411yescharacteristic
117C2×C4×C71C2411yescharacteristic
118C2×Dic71—22no—
119C2×Dic71—21no—
120C2×Dic71—21no—
121C2×Dic71C2×D411yescharacteristic
122C2×Dic71C2×D411yescharacteristic
123D284—28no—
124D284—24no—
125D284—28no—
126D284—22no—
127D284C2×D414yes—
128C4×D71—28no—
129C4×D71—22no—
130C4×D71—22no—
131C4×D71—28no—
132C4×D71—22no—
133C4×D71—42no—
134C4×D71—42no—
135C4×D71C2×D414yes—
136Dic142—22no—
137Dic142—42no—
138C8×C71—22no—
139C71⋊C8—22no—
140C22×C71—21no—
141C22×C71—21no—
142C22×C71#13712yescomplemented
143C22×C71C22×D411yescharacteristic
144C2×D71—21no—
145C2×D71—21no—
146C2×D71—21no—
147C2×D71—24no—
148C2×D71—21no—
149C2×D71—21no—
150C2×D71—44no—
151C2×D71—48no—
152C2×D71—42no—
153C2×D71—42no—
154C2×D71#12012yes—
155C4×C71—24no—
156C4×C71—21no—
157C4×C71—41no—
158C4×C71—41no—
159C4×C71C22×D412yes—
160Dic71—21no—
161Dic71—41no—
162Dic71—41no—
163Dic71#12012yes—
164C2×C71—22no—
165C2×C71C2×#12011yescharacteristic
166C2×C71C2×#13711yescharacteristic
167C2×C71#57511yescharacteristic
168D71—22no—
169D71—41no—
170D71—44no—
171#3007—711nomaximal, Sylow
172C71#300711yescharacteristic, complemented, Sylow
173C22×#41—711no—
174C2×#137—712no—
175C2×C22⋊Q8—711no—
176#452—712no—
177#452—712no—
178#451—711no—
179#451D7111yescharacteristic, complemented
180#450—711no—
181#450—711no—
182#417—712no—
183#407—711no—
184C2×#41—712no—
185C2×#41—712no—
186C2×#41—1421no—
187C2×#41—1421no—
188C2×#41—1428no—
189C22×Q8—711no—
190C22×Q8C2×D7111yescharacteristic
191C22×D4—711no—
192C22×D4—1421no—
193C23×C4—711no—
194C23×C4—1421no—
195#137—1428no—
196C2×Q16—712no—
197C2×Q16C2×D7112yescomplemented
198C2×SD16—712no—
199C2×SD16—712no—
200C2×M16—712no—
201C22⋊Q8—711no—
202C22⋊Q8—711no—
203C22⋊Q8—711no—
204C22⋊Q8—1421no—
205C22⋊Q8—1421no—
206C22⋊Q8C2×D7111yescharacteristic
207C2×C4⋊C4—711no—
208C2×C4⋊C4—1421no—
209C2×C22⋊C4—1421no—
210#103—712no—
211#103C2×D7112yescomplemented
212#102—712no—
213#102—712no—
214C22⋊C8—711no—
215C22⋊C8—712no—
216C22⋊C8C2×D7111yescharacteristic
217C24—711no—
218C24—1421no—
219#41—718no—
220#41—1424no—
221#41—1424no—
222#41—1428no—
223#41—1428no—
224#41—1424no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—711no—
228C2×Q8—1421no—
229C2×Q8—1424no—
230C2×Q8C22×D7111yescharacteristic
231C2×Q8C22×D7112yes—
232C2×D4—712no—
233C2×D4—1424no—
234C2×D4—1422no—
235C2×D4—1422no—
236C2×D4—1428no—
237C2×D4—1421no—
238C22×C4—21no—
239C22×C4—711no—
240C22×C4—711no—
241C22×C4—711no—
242C22×C4—711no—
243C22×C4—711no—
244C22×C4—1421no—
245C22×C4—1421no—
246C22×C4—1424no—
247C22×C4—1422no—
248C22×C4—1422no—
249C22×C4—1428no—
250C22×C4C22×D7111yescharacteristic
251Q16—24no—
252Q16—1424no—
253SD16—1424no—
254SD16—1424no—
255M16—1424no—
256C2×C8—712no—
257C2×C8C22×D7112yes—
258C4⋊C4—21no—
259C4⋊C4—711no—
260C4⋊C4—1421no—
261C4⋊C4—1421no—
262C4⋊C4—1421no—
263C4⋊C4—1421no—
264C4⋊C4C22×D7111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—1421no—
267C22⋊C4—1421no—
268C22⋊C4—1421no—
269C23—2842no—
270C23—2841no—
271C23—2842no—
272C23—711no—
273C23—711no—
274C23—1422no—
275C23—1424no—
276C23—1422no—
277C23—1422no—
278C23D4×D7111yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—2842no—
283Q8—1422no—
284Q8D4×D7114yes—
285D4—714no—
286D4—1428no—
287D4—1428no—
288D4—1424no—
289D4—1422no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—2842no—
296C2×C4—2842no—
297C2×C4—711no—
298C2×C4—714no—
299C2×C4—711no—
300C2×C4—1422no—
301C2×C4—1428no—
302C2×C4—1422no—
303C2×C4—1428no—
304C2×C4—1421no—
305C2×C4—1421no—
306C2×C4—1422no—
307C2×C4—1422no—
308C2×C4D4×D7111yescharacteristic
309C2×C4C23×D7111yescharacteristic
310C8—22no—
311C8—1422no—
312C22—21no—
313C22—21no—
314C22—2848no—
315C22—2844no—
316C22—2842no—
317C22—2842no—
318C22—712no—
319C22—1424no—
320C22—1421no—
321C22—1421no—
322C22—1421no—
323C22—1421no—
324C22—1421no—
325C22#1455616012yescomplemented
326C22C2×D4×D7111yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—2841no—
332C4—2841no—
333C4—712no—
334C4—1421no—
335C4C2×D4×D7112yes—
336C2—22no—
337C2—2844no—
338C2—2841no—
339C2—1422no—
340C2#120×D7111yescharacteristic
341C2C2×#1455616011yescharacteristic
342C2#1798679811yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#451×C71C2
socleC22×C71C22×D4
hypercenter#451D71
hypocenterC71#3007
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×#14556160C2semidirect
3. C22⋊Q8×D71C2semidirect
4. #17986164C2semidirect
5. #17986160C2semidirect
6. #17986155C2semidirect
7. #17986152C2semidirect
8. #17986149C2semidirect
9. #17985999C2semidirect
10. #17985961C2semidirect
11. #451×C71C2semidirect
17. C2×Q8×D71C22semidirect
23. C2×#11765855C22semidirect
24. C2×#11765854C22semidirect
25. C2×#11765852C22semidirect
26. C2×#11765848C22semidirect
27. #14556108C22semidirect
31. #14556104C22semidirect
33. C4⋊C4×D71C22semidirect
35. #14556052C22semidirect
36. #14556050C22semidirect
37. #14556048C22semidirect
39. (C2×D71)⋊C8C22semidirect
41. C2×Q16×C71C22semidirect
43. #103×C71C22semidirect
142. C22×C71#137semidirect
172. C71#3007semidirect
179. #451D71semidirect
197. C2×Q16C2×D71semidirect
211. #103C2×D71semidirect
325. C22#14556160semidirect