Finite Group Databasebeta

← #22600541

Subgroups — #22600541

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#17986695C211yesmaximal, characteristic, complemented
2#17986353C211yesmaximal, characteristic, complemented
3#17986337C211yesmaximal, characteristic, complemented
4#17986310C211yesmaximal, characteristic, complemented
5#17986299C211yesmaximal, characteristic, complemented
6#17986250C211yesmaximal, characteristic, complemented
7#17986247C211yesmaximal, characteristic, complemented
8#17986211C211yesmaximal, characteristic, complemented
9#17986144C211yesmaximal, characteristic, complemented
10#17986133C211yesmaximal, characteristic, complemented
11C4×#11765850C211yesmaximal, characteristic, complemented
12#17986077C211yesmaximal, characteristic, complemented
13#17986004C211yesmaximal, characteristic, complemented
14#17986000C211yesmaximal, characteristic, complemented
15#546×C71C211yesmaximal, characteristic, complemented
16C2×#11765852C2211yescharacteristic, complemented
17C2×#11765850C2211yescharacteristic
18C2×#11765850C2211yescharacteristic
19C2×#11765849C2211yescharacteristic
20C2×#11765849C2211yescharacteristic, complemented
21#14556123—21no—
22#14556120C2211yescharacteristic, complemented
23#14556110—21no—
24#14556102C2211yescharacteristic
25#14556102C2211yescharacteristic, complemented
26#14556100—21no—
27C4×D284C2211yescharacteristic, complemented
28(C22⋊D71)⋊C4—21no—
29#14556087C2211yescharacteristic
30C2×C4⋊Dic71C2211yescharacteristic, complemented
31#14556061C2211yescharacteristic, complemented
32#14556061C2211yescharacteristic, complemented
33#14556059C2211yescharacteristic, complemented
34C4⋊(C71⋊C8)C2211yescharacteristic, complemented
35#14556052C2211yescharacteristic, complemented
36#14556052C2211yescharacteristic, complemented
37#14556052C2211yescharacteristic, complemented
38#14556049C2211yescharacteristic
39#14556049C2211yescharacteristic
40#14556049C2211yescharacteristic
41#14556049C2211yescharacteristic
42#14556049C2211yescharacteristic
43#14556049C2211yescharacteristic
44#14556048C2211yescharacteristic, complemented
45C22⋊(C71⋊C8)C2211yescharacteristic
46C22⋊(C71⋊C8)C2211yescharacteristic, complemented
47C4×C71⋊C8C2211yescharacteristic, complemented
48C2×#41×C71C2211yescharacteristic
49C22×D4×C71C2211yescharacteristic
50#124×C71C2211yescharacteristic
51#123×C71—21no—
52C22⋊Q8×C71C2211yescharacteristic
53#121×C71—21no—
54#121×C71C2211yescharacteristic
55#120×C71—21no—
56C4×D4×C71C2211yescharacteristic
57C4×D4×C71C2211yescharacteristic
58C2×C22⋊C4×C71—21no—
59C2×D284C2311yescharacteristic
60C2×C22⋊D71—21no—
61C2×C22⋊D71—21no—
62C22×Dic71—21no—
63C2×C4×D71—21no—
64#11765852—41no—
65#11765850—22no—
66#11765850—41no—
67#11765849—41no—
68#11765849—41no—
69C2×C71⋊C8C2311yescharacteristic
70C2×C71⋊C8C2311yescharacteristic
71C2×C71⋊C8C2311yescharacteristic
72C2×C71⋊C8C2311yescharacteristic
73C4⋊Dic71—21no—
74C4⋊Dic71C2311yescharacteristic
75C4⋊Dic71C2311yescharacteristic
76C4⋊Dic71C2311yescharacteristic
77Dic71⋊C4—21no—
78Dic71⋊C4—21no—
79C22⋊Dic71—21no—
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82(C2×D71)⋊C4—21no—
83C4×Dic71—21no—
84C24×C71—21no—
85#41×C71—22no—
86#41×C71—41no—
87C2×Q8×C71C2311yescharacteristic
88C2×D4×C71—22no—
89C2×D4×C71—21no—
90C2×D4×C71—21no—
91C2×D4×C71—21no—
92C2×D4×C71—41no—
93C2×D4×C71D411yescharacteristic
94C2×D4×C71C2311yescharacteristic
95C2×D4×C71C2311yescharacteristic
96C22×C4×C71—21no—
97C22×C4×C71—21no—
98C22×C4×C71C2311yescharacteristic
99C22×C4×C71C2311yescharacteristic
100C4⋊C4×C71—21no—
101C4⋊C4×C71D411yescharacteristic
102C4⋊C4×C71C2311yescharacteristic
103C22⋊C4×C71—21no—
104C22⋊C4×C71—21no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—21no—
107C22⋊C4×C71—21no—
108C22⋊C4×C71D411yescharacteristic
109C22⋊C4×C71D411yescharacteristic
110C42×C71C2311yescharacteristic
111C23×C71—21no—
112C23×C71—21no—
113C23×C71—42no—
114C23×C71—42no—
115C23×C71—81no—
116C23×C71C2×D411yescharacteristic
117C23×C71C2×D411yescharacteristic
118C22×D71—21no—
119Q8×C71—21no—
120Q8×C71—21no—
121D4×C71—21no—
122D4×C71—21no—
123D4×C71—21no—
124D4×C71—41no—
125D4×C71—42no—
126D4×C71—42no—
127D4×C71—81no—
128D4×C71#4112yes—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—42no—
136C2×C4×C71—81no—
137C2×C4×C71C2×D411yescharacteristic
138C2×C4×C71C2×D411yescharacteristic
139C2×C4×C71C2×D411yescharacteristic
140C2×C4×C71C2×D411yescharacteristic
141C2×C4×C71C2411yescharacteristic
142C22⋊D71—42no—
143C22⋊D71—81no—
144C2×Dic71—21no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—81no—
148D284—21no—
149D284—21no—
150C4×D71—81no—
151C71⋊C8—21no—
152C71⋊C8—21no—
153C71⋊C8—21no—
154C71⋊C8—21no—
155C22×C71—21no—
156C22×C71—21no—
157C22×C71—22no—
158C22×C71—21no—
159C22×C71—21no—
160C22×C71—21no—
161C22×C71—21no—
162C22×C71—41no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—41no—
166C22×C71—41no—
167C22×C71—84no—
168C22×C71C22×D411yescharacteristic
169C2×D71—41no—
170C2×D71—41no—
171C2×D71—41no—
172C4×C71—21no—
173C4×C71—21no—
174C4×C71—21no—
175C4×C71—21no—
176C4×C71—41no—
177C4×C71—41no—
178C4×C71C2×#4111yescharacteristic
179C4×C71H(5,𝔽2)11yescharacteristic
180Dic71—41no—
181Dic71—41no—
182Dic71—41no—
183C2×C71—41no—
184C2×C71—41no—
185C2×C71—42no—
186C2×C71—81no—
187C2×C71#54611yescharacteristic
188C2×C71#57511yescharacteristic
189C2×C71#57611yescharacteristic
190D71—81no—
191#3322—711nomaximal, Sylow
192C71#332211yescharacteristic, complemented, Sylow
193#553—711no—
194#546D7111yescharacteristic, complemented
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×#41C2×D7111yescharacteristic
209C22×D4C2×D7111yescharacteristic
210C2×SD16—711no—
211C2×D8—711no—
212C2×D8—711no—
213C2×M16—711no—
214C2×M16—711no—
215#126—1421no—
216#125—711no—
217#124C2×D7111yescharacteristic
218#123—21no—
219#123—1421no—
220C22⋊Q8C2×D7111yescharacteristic
221#121—21no—
222#121—711no—
223#121—711no—
224#121—1421no—
225#121C2×D7111yescharacteristic
226#120—21no—
227C4×D4—711no—
228C4×D4—1421no—
229C4×D4C2×D7111yescharacteristic
230C4×D4C2×D7111yescharacteristic
231#117—711no—
232C2×C4⋊C4—711no—
233C2×C22⋊C4—21no—
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—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D7111yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—711no—
261C2×D4—1421no—
262C2×D4—1421no—
263C2×D4C22⋊D7111yescharacteristic
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—711no—
283C4⋊C4—711no—
284C4⋊C4—711no—
285C4⋊C4—1421no—
286C4⋊C4—1421no—
287C4⋊C4—1421no—
288C4⋊C4C22⋊D7111yescharacteristic
289C4⋊C4C22×D7111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—1421no—
296C22⋊C4—1421no—
297C22⋊C4—1421no—
298C22⋊C4—1421no—
299C22⋊C4C22⋊D7111yescharacteristic
300C22⋊C4C22⋊D7111yescharacteristic
301C42—1421no—
302C42C22×D7111yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—1421no—
309C23C2×C22⋊D7111yescharacteristic
310C23C2×C22⋊D7111yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—2842no—
321D4—5681no—
322D4—1421no—
323D4—1421no—
324D4#1176586412yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—5681no—
334C2×C4—5681no—
335C2×C4—1421no—
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—22no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—84no—
360C22—2841no—
361C22—2841no—
362C22—2841no—
363C22C22×C22⋊D7111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—2841no—
371C4—2841no—
372C4—2841no—
373C4C2×#1176586411yescharacteristic
374C4#1455617311yescharacteristic
375C2—41no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—5681no—
380C2#1798664311yescharacteristic
381C2#1798679711yescharacteristic
382C2#1798680211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#546×C71C2
socleC22×C71C22×D4
hypercenter#546D71
hypocenterC71#3322
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #17986695C2semidirect
2. #17986353C2semidirect
3. #17986337C2semidirect
4. #17986310C2semidirect
5. #17986299C2semidirect
6. #17986250C2semidirect
7. #17986247C2semidirect
8. #17986211C2semidirect
9. #17986144C2semidirect
10. #17986133C2semidirect
11. C4×#11765850C2semidirect
12. #17986077C2semidirect
13. #17986004C2semidirect
14. #17986000C2semidirect
15. #546×C71C2semidirect
16. C2×#11765852C22semidirect
20. C2×#11765849C22semidirect
22. #14556120C22semidirect
25. #14556102C22semidirect
27. C4×D284C22semidirect
30. C2×C4⋊Dic71C22semidirect
31. #14556061C22semidirect
32. #14556061C22semidirect
33. #14556059C22semidirect
34. C4⋊(C71⋊C8)C22semidirect
35. #14556052C22semidirect
36. #14556052C22semidirect
37. #14556052C22semidirect
44. #14556048C22semidirect
46. C22⋊(C71⋊C8)C22semidirect
47. C4×C71⋊C8C22semidirect
192. C71#3322semidirect
194. #546D71semidirect