Finite Group Databasebeta

← #22600535

Subgroups — #22600535

Order 9088.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986645C211yesmaximal, characteristic, complemented
2#17986356C211yesmaximal, characteristic, complemented
3#17986340C211yesmaximal, characteristic, complemented
4#17986312C211yesmaximal, characteristic, complemented
5#17986296C211yesmaximal, characteristic, complemented
6#17986250C211yesmaximal, characteristic, complemented
7#17986243C211yesmaximal, characteristic, complemented
8#17986214C211yesmaximal, characteristic, complemented
9#17986142C211yesmaximal, characteristic, complemented
10#17986132C211yesmaximal, characteristic, complemented
11C4×#11765850C211yesmaximal, characteristic, complemented
12#17986077C211yesmaximal, characteristic, complemented
13#17986003C211yesmaximal, characteristic, complemented
14#17985998C211yesmaximal, characteristic, complemented
15#553×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×D284C2211yescharacteristic
18C2×#11765853C2211yescharacteristic, complemented
19C2×#11765850C2211yescharacteristic
20C2×#11765850C2211yescharacteristic
21C2×#11765849C2211yescharacteristic
22C2×#11765849C2211yescharacteristic, complemented
23#14556116C2211yescharacteristic
24#14556113—21no—
25#14556107C2211yescharacteristic
26#14556102C2211yescharacteristic
27#14556101—21no—
28#14556096—21no—
29C4×D284C2211yescharacteristic
30C22⋊C4×D71—21no—
31#14556061C2211yescharacteristic, complemented
32#14556061C2211yescharacteristic, complemented
33#14556059C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556051C2211yescharacteristic, complemented
36#14556051C2211yescharacteristic, complemented
37#14556051C2211yescharacteristic, complemented
38#14556049C2211yescharacteristic, complemented
39#14556048C2211yescharacteristic
40#14556048C2211yescharacteristic
41#14556048C2211yescharacteristic
42#14556048C2211yescharacteristic
43#14556048C2211yescharacteristic
44#14556048C2211yescharacteristic
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic, complemented
47C4×C71⋊C8C2211yescharacteristic, complemented
48#126×C71—21no—
49#125×C71C2211yescharacteristic, complemented
50#123×C71—21no—
51#121×C71—21no—
52#121×C71C2211yescharacteristic
53#121×C71C2211yescharacteristic, complemented
54C4×D4×C71—21no—
55C4×D4×C71C2211yescharacteristic
56C4×D4×C71C2211yescharacteristic, complemented
57#117×C71C2211yescharacteristic
58C2×C4⋊C4×C71C2211yescharacteristic, complemented
59C23×D71—21no—
60#11765865—22no—
61#11765865—41no—
62C2×Dic142C2311yescharacteristic
63C2×D284—22no—
64C2×D284—21no—
65C2×D284—41no—
66C2×D284C2311yescharacteristic
67C2×D284C2311yescharacteristic
68C2×C22⋊D71—21no—
69C2×C22⋊D71—21no—
70C2×C4×D71—21no—
71C2×C4×D71—21no—
72#11765853—41no—
73#11765850—22no—
74#11765850—41no—
75#11765849—41no—
76#11765849—41no—
77C2×C71⋊C8C2311yescharacteristic
78C2×C71⋊C8C2311yescharacteristic
79C2×C71⋊C8C2311yescharacteristic
80C2×C71⋊C8C2311yescharacteristic
81C4⋊Dic71C2311yescharacteristic
82Dic71⋊C4—21no—
83C22⋊Dic71—21no—
84(C2×D71)⋊C4—21no—
85(C2×D71)⋊C4—21no—
86(C2×D71)⋊C4—21no—
87(C2×D71)⋊C4—21no—
88C2×D4×C71—21no—
89C2×D4×C71—21no—
90C2×D4×C71D411yescharacteristic
91C2×D4×C71C2311yescharacteristic
92C22×C4×C71—21no—
93C22×C4×C71—21no—
94C22×C4×C71C2311yescharacteristic
95C22×C4×C71C2311yescharacteristic
96C4⋊C4×C71—21no—
97C4⋊C4×C71—21no—
98C4⋊C4×C71—21no—
99C4⋊C4×C71D411yescharacteristic
100C4⋊C4×C71C2311yescharacteristic
101C4⋊C4×C71C2311yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71D411yescharacteristic
108C22⋊C4×C71D411yescharacteristic
109C42×C71—21no—
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C23×C71C2×D411yescharacteristic
113C23×C71C2×D411yescharacteristic
114C22×D71—21no—
115C22×D71—21no—
116C22×D71—42no—
117C22×D71—42no—
118C22×D71—81no—
119D4×C71—21no—
120D4×C71—21no—
121D4×C71—41no—
122D4×C71—42no—
123D4×C71—81no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—81no—
132C2×C4×C71—81no—
133C2×C4×C71C2×D411yescharacteristic
134C2×C4×C71C2×D411yescharacteristic
135C2×C4×C71C2×D411yescharacteristic
136C2×C4×C71C2×D411yescharacteristic
137C2×C4×C71C2411yescharacteristic
138C22⋊D71—42no—
139C22⋊D71—81no—
140C2×Dic71—21no—
141C2×Dic71—21no—
142D284—21no—
143D284—21no—
144D284—21no—
145D284—42no—
146D284#4112yes—
147C4×D71—42no—
148C4×D71—81no—
149Dic142—21no—
150Dic142—21no—
151C71⋊C8—21no—
152C71⋊C8—21no—
153C71⋊C8—21no—
154C71⋊C8—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—21no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—41no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71C22×D411yescharacteristic
165C2×D71—22no—
166C2×D71—41no—
167C2×D71—41no—
168C2×D71—41no—
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—84no—
172C4×C71—21no—
173C4×C71—21no—
174C4×C71—21no—
175C4×C71—21no—
176C4×C71—41no—
177C4×C71—41no—
178C4×C71—41no—
179C4×C71C2×#4111yescharacteristic
180C4×C71H(5,𝔽2)11yescharacteristic
181Dic71—41no—
182Dic71—41no—
183C2×C71—41no—
184C2×C71—41no—
185C2×C71—81no—
186C2×C71#54611yescharacteristic
187C2×C71#57511yescharacteristic
188C2×C71#57611yescharacteristic
189D71—42no—
190D71—81no—
191#3322—711nomaximal, Sylow
192C71#332211yescharacteristic, complemented, Sylow
193#553D7111yescharacteristic, complemented
194#546—711no—
195#488—711no—
196#485—711no—
197#480—711no—
198#478—711no—
199#469—711no—
200#468—711no—
201#463—711no—
202#449—711no—
203#447—711no—
204C4×D8—711no—
205#436—711no—
206#418—711no—
207#417—711no—
208C2×#41—711no—
209C22×D4—711no—
210C2×SD16—711no—
211C2×D8—711no—
212C2×D8—711no—
213C2×M16—711no—
214C2×M16—711no—
215#126—21no—
216#125C2×D7111yescharacteristic, complemented
217#124—711no—
218#123—21no—
219#123—1421no—
220C22⋊Q8—711no—
221#121—21no—
222#121—711no—
223#121—1421no—
224#121C2×D7111yescharacteristic
225#121C2×D7111yescharacteristic, complemented
226#120—1421no—
227C4×D4—21no—
228C4×D4—711no—
229C4×D4C2×D7111yescharacteristic
230C4×D4C2×D7111yescharacteristic, complemented
231#117C2×D7111yescharacteristic
232C2×C4⋊C4C2×D7111yescharacteristic, complemented
233C2×C22⋊C4—1421no—
234#107—711no—
235#107—711no—
236#106—711no—
237C4⋊C8—711no—
238#103—711no—
239#103—711no—
240#103—711no—
241#102—711no—
242#102—711no—
243#102—711no—
244#102—711no—
245#102—711no—
246#102—711no—
247#102—711no—
248C22⋊C8—711no—
249C22⋊C8—711no—
250C4×C8—711no—
251C24—1421no—
252#41—2841no—
253#41—1422no—
254C2×Q8—711no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2841no—
258C2×D4—711no—
259C2×D4—711no—
260C2×D4—1422no—
261C2×D4—1421no—
262C2×D4—1421no—
263C2×D4—1421no—
264C2×D4C22⋊D7111yescharacteristic
265C2×D4C22×D7111yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1421no—
269C22×C4—1421no—
270C22×C4C22×D7111yescharacteristic
271C22×C4C22×D7111yescharacteristic
272SD16—2841no—
273D8—2841no—
274D8—1422no—
275M16—2841no—
276M16—2841no—
277C2×C8—711no—
278C2×C8—711no—
279C2×C8—711no—
280C2×C8—711no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—711no—
285C4⋊C4—1421no—
286C4⋊C4C22⋊D7111yescharacteristic
287C4⋊C4C22×D7111yescharacteristic
288C4⋊C4C22×D7111yescharacteristic
289C4⋊C4C22×D7111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—1421no—
295C22⋊C4—1421no—
296C22⋊C4—1421no—
297C22⋊C4—1421no—
298C22⋊C4—1421no—
299C22⋊C4C22⋊D7111yescharacteristic
300C22⋊C4C22⋊D7111yescharacteristic
301C42—21no—
302C42C22×D7111yescharacteristic
303C23—21no—
304C23—2842no—
305C23—2842no—
306C23—5681no—
307C23—1421no—
308C23—1421no—
309C23C2×C22⋊D7111yescharacteristic
310C23C2×C22⋊D7111yescharacteristic
311Q8—1421no—
312Q8—1421no—
313D4—21no—
314D4—21no—
315D4—41no—
316D4—42no—
317D4—81no—
318D4—2842no—
319D4—2842no—
320D4—5681no—
321D4—712no—
322D4—1421no—
323D4—1421no—
324D4—1421no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—2842no—
335C2×C4—5681no—
336C2×C4—1421no—
337C2×C4—1421no—
338C2×C4C2×C22⋊D7111yescharacteristic
339C2×C4C2×C22⋊D7111yescharacteristic
340C2×C4C2×C22⋊D7111yescharacteristic
341C2×C4C2×C22⋊D7111yescharacteristic
342C2×C4C23×D7111yescharacteristic
343C8—1421no—
344C8—1421no—
345C8—1421no—
346C8—1421no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2841no—
357C22—2841no—
358C22—2841no—
359C22—2841no—
360C22—2841no—
361C22—5684no—
362C22—1422no—
363C22C22×C22⋊D7111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—2841no—
372C4—2841no—
373C4#41×D7111yescharacteristic
374C4#1455617411yescharacteristic
375C2—41no—
376C2—41no—
377C2—81no—
378C2—2842no—
379C2—5681no—
380C2#1798663711yescharacteristic
381C2#1798679711yescharacteristic
382C2#1798680211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#553×C71C2
socleC22×C71C22×D4
hypercenter#553D71
hypocenterC71#3322
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #17986645C2semidirect
2. #17986356C2semidirect
3. #17986340C2semidirect
4. #17986312C2semidirect
5. #17986296C2semidirect
6. #17986250C2semidirect
7. #17986243C2semidirect
8. #17986214C2semidirect
9. #17986142C2semidirect
10. #17986132C2semidirect
11. C4×#11765850C2semidirect
12. #17986077C2semidirect
13. #17986003C2semidirect
14. #17985998C2semidirect
15. #553×C71C2semidirect
18. C2×#11765853C22semidirect
22. C2×#11765849C22semidirect
31. #14556061C22semidirect
32. #14556061C22semidirect
33. #14556059C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556051C22semidirect
36. #14556051C22semidirect
37. #14556051C22semidirect
38. #14556049C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. C4×C71⋊C8C22semidirect
49. #125×C71C22semidirect
53. #121×C71C22semidirect
56. C4×D4×C71C22semidirect
58. C2×C4⋊C4×C71C22semidirect
192. C71#3322semidirect
193. #553D71semidirect
216. #125C2×D71semidirect
225. #121C2×D71semidirect
230. C4×D4C2×D71semidirect
232. C2×C4⋊C4C2×D71semidirect