Finite Group Databasebeta

← #22599947

Subgroups — #22599947

Order 9088.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556154C211yesmaximal, characteristic, complemented
2C2×#14556153C211yesmaximal, characteristic, complemented
3#17986645C211yesmaximal, characteristic, complemented
4#17986381C211yesmaximal, characteristic, complemented
5#17986247C211yesmaximal, characteristic, complemented
6#17986243C211yesmaximal, characteristic, complemented
7#17986214C211yesmaximal, characteristic, complemented
8#17986211C211yesmaximal, characteristic, complemented
9#17986161C211yesmaximal, characteristic, complemented
10#17986159C211yesmaximal, characteristic, complemented
11#17986133C211yesmaximal, characteristic, complemented
12#17986132C211yesmaximal, characteristic, complemented
13C4×#11765850C211yesmaximal, characteristic, complemented
14#17986077C211yesmaximal, characteristic, complemented
15#546×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×D284C2211yescharacteristic, complemented
18#14556154—24no—
19#14556153—24no—
20C2×#11765855C2211yescharacteristic, complemented
21C2×#11765853C2211yescharacteristic, complemented
22C2×#11765853C2211yescharacteristic, complemented
23C2×#11765852C2211yescharacteristic, complemented
24C2×#11765852C2211yescharacteristic, complemented
25C2×#11765850C2211yescharacteristic
26C2×#11765850C2211yescharacteristic
27C2×#11765850C2211yescharacteristic
28C2×#11765850C2211yescharacteristic
29C2×#11765849C2211yescharacteristic, complemented
30C2×#11765849C2211yescharacteristic, complemented
31#14556116C2211yescharacteristic, complemented
32#14556113—21no—
33#14556107C2211yescharacteristic, complemented
34#14556102C2211yescharacteristic
35#14556101—21no—
36#14556096—21no—
37C4×D284C2211yescharacteristic, complemented
38C22⋊C4×D71—21no—
39#14556061C2211yescharacteristic, complemented
40C4⋊(C71⋊C8)C2211yescharacteristic, complemented
41#14556052C2211yescharacteristic, complemented
42#14556051C2211yescharacteristic, complemented
43#14556049C2211yescharacteristic, complemented
44#14556049C2211yescharacteristic, complemented
45#14556048C2211yescharacteristic, complemented
46#14556048C2211yescharacteristic, complemented
47C22⋊(C71⋊C8)C2211yescharacteristic, complemented
48C22⋊(C71⋊C8)C2211yescharacteristic, complemented
49C4×C71⋊C8C2211yescharacteristic, complemented
50C2×#41×C71C2211yescharacteristic
51C22×D4×C71C2211yescharacteristic, complemented
52#124×C71C2211yescharacteristic, complemented
53#123×C71—21no—
54C22⋊Q8×C71C2211yescharacteristic, complemented
55#121×C71—21no—
56#121×C71C2211yescharacteristic
57#120×C71—21no—
58C4×D4×C71C2211yescharacteristic
59C4×D4×C71C2211yescharacteristic, complemented
60C2×C22⋊C4×C71—21no—
61C23×D71—21no—
62#11765865—22no—
63#11765865—41no—
64C2×Dic142C2311yescharacteristic
65C2×D284—22no—
66C2×D284—21no—
67C2×D284—41no—
68C2×D284C2311yescharacteristic
69C2×D284C2311yescharacteristic
70C2×C22⋊D71—21no—
71C2×C22⋊D71—21no—
72C2×C4×D71—21no—
73C2×C4×D71—21no—
74#11765855—41no—
75#11765853—22no—
76#11765853—22no—
77#11765852—22no—
78#11765852—22no—
79#11765850—22no—
80#11765850—22no—
81#11765850—41no—
82#11765850D414yescomplemented
83#11765849—22no—
84#11765849—22no—
85C2×C71⋊C8C2311yescharacteristic
86C2×C71⋊C8C2311yescharacteristic
87C2×C71⋊C8C2311yescharacteristic
88C2×C71⋊C8C2311yescharacteristic
89C4⋊Dic71C2311yescharacteristic
90Dic71⋊C4—21no—
91C22⋊Dic71—21no—
92(C2×D71)⋊C4—21no—
93(C2×D71)⋊C4—21no—
94(C2×D71)⋊C4—21no—
95(C2×D71)⋊C4—21no—
96C24×C71—21no—
97#41×C71—22no—
98#41×C71—41no—
99C2×Q8×C71C2311yescharacteristic
100C2×D4×C71—22no—
101C2×D4×C71—21no—
102C2×D4×C71—21no—
103C2×D4×C71—21no—
104C2×D4×C71—41no—
105C2×D4×C71D411yescharacteristic
106C2×D4×C71C2311yescharacteristic
107C2×D4×C71C2311yescharacteristic
108C22×C4×C71—21no—
109C22×C4×C71—21no—
110C22×C4×C71C2311yescharacteristic
111C22×C4×C71C2311yescharacteristic
112C4⋊C4×C71—21no—
113C4⋊C4×C71D411yescharacteristic
114C4⋊C4×C71C2311yescharacteristic
115C22⋊C4×C71—21no—
116C22⋊C4×C71—21no—
117C22⋊C4×C71—21no—
118C22⋊C4×C71—21no—
119C22⋊C4×C71—21no—
120C22⋊C4×C71D411yescharacteristic
121C22⋊C4×C71D411yescharacteristic
122C42×C71C2311yescharacteristic
123C23×C71—21no—
124C23×C71—21no—
125C23×C71—42no—
126C23×C71—42no—
127C23×C71—81no—
128C23×C71C2×D411yescharacteristic
129C23×C71C2×D411yescharacteristic
130C22×D71—21no—
131C22×D71—21no—
132C22×D71—42no—
133C22×D71—42no—
134C22×D71—81no—
135Q8×C71—21no—
136Q8×C71—21no—
137D4×C71—21no—
138D4×C71—21no—
139D4×C71—21no—
140D4×C71—41no—
141D4×C71—42no—
142D4×C71—42no—
143D4×C71—81no—
144D4×C71C2×D412yes—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—21no—
150C2×C4×C71—21no—
151C2×C4×C71—42no—
152C2×C4×C71—81no—
153C2×C4×C71C2×D411yescharacteristic
154C2×C4×C71C2×D411yescharacteristic
155C2×C4×C71C2×D411yescharacteristic
156C2×C4×C71C2×D411yescharacteristic
157C2×C4×C71C2411yescharacteristic
158C22⋊D71—42no—
159C22⋊D71—81no—
160C2×Dic71—21no—
161C2×Dic71—21no—
162D284—21no—
163D284—21no—
164D284—21no—
165D284—42no—
166D284C2×D412yes—
167C4×D71—42no—
168C4×D71—81no—
169Dic142—21no—
170Dic142—21no—
171C71⋊C8—21no—
172C71⋊C8—21no—
173C71⋊C8—21no—
174C71⋊C8C2×D412yes—
175C22×C71—21no—
176C22×C71—21no—
177C22×C71—22no—
178C22×C71—21no—
179C22×C71—21no—
180C22×C71—21no—
181C22×C71—21no—
182C22×C71—41no—
183C22×C71—41no—
184C22×C71—41no—
185C22×C71—41no—
186C22×C71—41no—
187C22×C71—84no—
188C22×C71C22×D411yescharacteristic
189C2×D71—22no—
190C2×D71—41no—
191C2×D71—41no—
192C2×D71—41no—
193C2×D71—41no—
194C2×D71—41no—
195C2×D71—84no—
196C4×C71—21no—
197C4×C71—21no—
198C4×C71—21no—
199C4×C71—21no—
200C4×C71—41no—
201C4×C71—41no—
202C4×C71C22×D411yescharacteristic
203C4×C71H(5,𝔽2)11yescharacteristic
204Dic71—41no—
205Dic71—41no—
206C2×C71—41no—
207C2×C71—41no—
208C2×C71—42no—
209C2×C71—81no—
210C2×C71D4211yescharacteristic
211C2×C71#57511yescharacteristic
212C2×C71#57611yescharacteristic
213D71—42no—
214D71—81no—
215#3273—711nomaximal, Sylow
216C71#327311yescharacteristic, complemented, Sylow
217C2×AGL(1,ℤ/8)—711no—
218C2×AGL(1,ℤ/8)—711no—
219#546—711no—
220#546D7111yescharacteristic, complemented
221#495—711no—
222#468—711no—
223#468—711no—
224#463—711no—
225#463—711no—
226#452—711no—
227#452—711no—
228#447—711no—
229#447—711no—
230C4×D8—711no—
231#436—711no—
232C2×#41—711no—
233C2×#41C2×D7111yescharacteristic
234C22×D4—711no—
235C22×D4C2×D7111yescharacteristic, complemented
236AGL(1,ℤ/8)—1424no—
237AGL(1,ℤ/8)—1424no—
238C2×Q16—711no—
239C2×SD16—711no—
240C2×SD16—711no—
241C2×SD16—711no—
242C2×SD16—711no—
243C2×D8—711no—
244C2×D8—711no—
245C2×D8—711no—
246C2×D8—711no—
247C2×M16—711no—
248C2×M16—711no—
249#124—711no—
250#124C2×D7111yescharacteristic, complemented
251#123—21no—
252#123—1421no—
253C22⋊Q8—711no—
254C22⋊Q8C2×D7111yescharacteristic, complemented
255#121—21no—
256#121—711no—
257#121—1421no—
258#121C2×D7111yescharacteristic
259#120—21no—
260#120—1421no—
261C4×D4—711no—
262C4×D4C2×D7111yescharacteristic
263C4×D4C2×D7111yescharacteristic, complemented
264C2×C22⋊C4—21no—
265C2×C22⋊C4—1421no—
266#107—711no—
267C4⋊C8—711no—
268#103—711no—
269#103—711no—
270#102—711no—
271#102—711no—
272#102—711no—
273#102—711no—
274C22⋊C8—711no—
275C22⋊C8—711no—
276C4×C8—711no—
277C24—21no—
278C24—1421no—
279#41—22no—
280#41—41no—
281#41—2841no—
282#41—1422no—
283C2×Q8—711no—
284C2×Q8C22×D7111yescharacteristic
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—2841no—
291C2×D4—711no—
292C2×D4—711no—
293C2×D4—1422no—
294C2×D4—1421no—
295C2×D4—1421no—
296C2×D4—1421no—
297C2×D4C22⋊D7111yescharacteristic
298C2×D4C22×D7111yescharacteristic
299C2×D4C22×D7111yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—1421no—
303C22×C4—1421no—
304C22×C4C22×D7111yescharacteristic
305C22×C4C22×D7111yescharacteristic
306Q16—2841no—
307SD16—1422no—
308SD16—1422no—
309SD16—1422no—
310SD16—1422no—
311D8—2841no—
312D8—714no—
313D8—1422no—
314D8—1422no—
315M16—1422no—
316M16—1422no—
317C2×C8—711no—
318C2×C8—711no—
319C2×C8—711no—
320C2×C8—711no—
321C4⋊C4—21no—
322C4⋊C4—711no—
323C4⋊C4—1421no—
324C4⋊C4C22⋊D7111yescharacteristic
325C4⋊C4C22×D7111yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—1421no—
332C22⋊C4—1421no—
333C22⋊C4—1421no—
334C22⋊C4—1421no—
335C22⋊C4—1421no—
336C22⋊C4C22⋊D7111yescharacteristic
337C22⋊C4C22⋊D7111yescharacteristic
338C42C22×D7111yescharacteristic
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—2842no—
345C23—2842no—
346C23—5681no—
347C23—1421no—
348C23—1421no—
349C23C2×C22⋊D7111yescharacteristic
350C23C2×C22⋊D7111yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—1421no—
354Q8—1421no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—41no—
359D4—42no—
360D4—42no—
361D4—81no—
362D4—2842no—
363D4—2842no—
364D4—5681no—
365D4—712no—
366D4—1421no—
367D4—1421no—
368D4—1421no—
369D4D4×D7112yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—2842no—
379C2×C4—5681no—
380C2×C4—1421no—
381C2×C4—1421no—
382C2×C4C2×C22⋊D7111yescharacteristic
383C2×C4C2×C22⋊D7111yescharacteristic
384C2×C4C2×C22⋊D7111yescharacteristic
385C2×C4C2×C22⋊D7111yescharacteristic
386C2×C4C23×D7111yescharacteristic
387C8—712no—
388C8—1421no—
389C8—1421no—
390C8—1421no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—84no—
404C22—2841no—
405C22—2841no—
406C22—2841no—
407C22—2841no—
408C22—2841no—
409C22—5684no—
410C22—1422no—
411C22C22×C22⋊D7111yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—2841no—
419C4—2841no—
420C4C2×D4×D7111yescharacteristic
421C4#1455617311yescharacteristic
422C2—41no—
423C2—41no—
424C2—42no—
425C2—81no—
426C2—2842no—
427C2—5681no—
428C2D4×C22⋊D7111yescharacteristic
429C2#1798679711yescharacteristic
430C2#1798680211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#546×C71C2
socleC22×C71C22×D4
hypercenter#546D71
hypocenterC71#3273
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. C2×#14556154C2semidirect
2. C2×#14556153C2semidirect
3. #17986645C2semidirect
4. #17986381C2semidirect
5. #17986247C2semidirect
6. #17986243C2semidirect
7. #17986214C2semidirect
8. #17986211C2semidirect
9. #17986161C2semidirect
10. #17986159C2semidirect
11. #17986133C2semidirect
12. #17986132C2semidirect
13. C4×#11765850C2semidirect
14. #17986077C2semidirect
15. #546×C71C2semidirect
17. C22×D284C22semidirect
20. C2×#11765855C22semidirect
21. C2×#11765853C22semidirect
22. C2×#11765853C22semidirect
23. C2×#11765852C22semidirect
24. C2×#11765852C22semidirect
29. C2×#11765849C22semidirect
30. C2×#11765849C22semidirect
31. #14556116C22semidirect
33. #14556107C22semidirect
37. C4×D284C22semidirect
39. #14556061C22semidirect
40. C4⋊(C71⋊C8)C22semidirect
41. #14556052C22semidirect
42. #14556051C22semidirect
43. #14556049C22semidirect
44. #14556049C22semidirect
45. #14556048C22semidirect
46. #14556048C22semidirect
47. C22⋊(C71⋊C8)C22semidirect
48. C22⋊(C71⋊C8)C22semidirect
49. C4×C71⋊C8C22semidirect
51. C22×D4×C71C22semidirect
52. #124×C71C22semidirect
54. C22⋊Q8×C71C22semidirect
59. C4×D4×C71C22semidirect
82. #11765850D4semidirect
216. C71#3273semidirect
220. #546D71semidirect
235. C22×D4C2×D71semidirect
250. #124C2×D71semidirect
254. C22⋊Q8C2×D71semidirect
263. C4×D4C2×D71semidirect