Finite Group Databasebeta

← #22599881

Subgroups — #22599881

Order 9088.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14556153C211yesmaximal, characteristic, complemented
2C2×#14556148C211yesmaximal, characteristic, complemented
3#17986645C211yesmaximal, characteristic, complemented
4#17986386C211yesmaximal, characteristic, complemented
5#17986243C211yesmaximal, characteristic, complemented
6#17986233C211yesmaximal, characteristic, complemented
7#17986221C211yesmaximal, characteristic, complemented
8#17986191C211yesmaximal, characteristic, complemented
9#17986161C211yesmaximal, characteristic, complemented
10#17986144C211yesmaximal, characteristic, complemented
11#17986142C211yesmaximal, characteristic, complemented
12#17986132C211yesmaximal, characteristic, complemented
13#17986097C211yesmaximal, characteristic, complemented
14#17986071C211yesmaximal, characteristic, complemented
15#550×C71C211yesmaximal, characteristic, complemented
16C2×#11765865C2211yescharacteristic
17C22×D284C2211yescharacteristic
18#14556153—24no—
19#14556148—24no—
20C2×#11765855C2211yescharacteristic, complemented
21C2×#11765853C2211yescharacteristic, complemented
22C2×#11765853C2211yescharacteristic, complemented
23C2×#11765852C2211yescharacteristic
24C2×#11765852C2211yescharacteristic
25C2×#11765850C2211yescharacteristic
26C2×#11765850C2211yescharacteristic
27C2×#11765850C2211yescharacteristic
28C2×#11765850C2211yescharacteristic
29C2×#11765849C2211yescharacteristic
30C22×C71⋊C8C2211yescharacteristic, complemented
31#14556116C2211yescharacteristic
32#14556113—21no—
33#14556107C2211yescharacteristic
34#14556102C2211yescharacteristic
35#14556101—21no—
36#14556096—21no—
37C4×D284C2211yescharacteristic
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
46#14556048C2211yescharacteristic
47C22⋊(C71⋊C8)C2211yescharacteristic
48C22⋊(C71⋊C8)C2211yescharacteristic, complemented
49#14556039C2211yescharacteristic, complemented
50C2×#41×C71C2211yescharacteristic
51C2×#41×C71C2211yescharacteristic, complemented
52#127×C71—21no—
53#127×C71C2211yescharacteristic, complemented
54#121×C71—21no—
55#121×C71—21no—
56#121×C71C2211yescharacteristic
57#121×C71C2211yescharacteristic, complemented
58C4×Q8×C71C2211yescharacteristic, complemented
59C4×D4×C71—21no—
60C4×D4×C71C2211yescharacteristic
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—22no—
75#11765853—22no—
76#11765853—41no—
77#11765852—22no—
78#11765852D414yescomplemented
79#11765850—22no—
80#11765850—22no—
81#11765850—22no—
82#11765850—41no—
83#11765849—22no—
84C2×C71⋊C8—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—
96#41×C71—22no—
97#41×C71—22no—
98#41×C71—41no—
99#41×C71—41no—
100C2×Q8×C71C2311yescharacteristic
101C2×D4×C71—21no—
102C2×D4×C71—21no—
103C2×D4×C71—21no—
104C2×D4×C71—21no—
105C2×D4×C71—21no—
106C2×D4×C71—21no—
107C2×D4×C71D411yescharacteristic
108C2×D4×C71C2311yescharacteristic
109C2×D4×C71C2311yescharacteristic
110C22×C4×C71—21no—
111C22×C4×C71—21no—
112C22×C4×C71C2311yescharacteristic
113C22×C4×C71C2311yescharacteristic
114C4⋊C4×C71—21no—
115C4⋊C4×C71D411yescharacteristic
116C4⋊C4×C71C2311yescharacteristic
117C22⋊C4×C71—21no—
118C22⋊C4×C71—21no—
119C22⋊C4×C71D411yescharacteristic
120C22⋊C4×C71D411yescharacteristic
121C42×C71—21no—
122C42×C71C2311yescharacteristic
123C23×C71—21no—
124C23×C71—21no—
125C23×C71C2×D411yescharacteristic
126C23×C71C2×D411yescharacteristic
127C22×D71—21no—
128C22×D71—21no—
129C22×D71—42no—
130C22×D71—42no—
131C22×D71—81no—
132Q8×C71—21no—
133Q8×C71C2×D412yes—
134D4×C71—21no—
135D4×C71—21no—
136D4×C71—21no—
137D4×C71—21no—
138D4×C71—41no—
139D4×C71—42no—
140D4×C71—42no—
141D4×C71—42no—
142D4×C71—81no—
143D4×C71—81no—
144D4×C71—81no—
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—42no—
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—21no—
178C22×C71—21no—
179C22×C71—21no—
180C22×C71—21no—
181C22×C71—41no—
182C22×C71—41no—
183C22×C71—41no—
184C22×C71—41no—
185C22×C71—41no—
186C22×C71—41no—
187C22×C71C22×D411yescharacteristic
188C2×D71—22no—
189C2×D71—41no—
190C2×D71—41no—
191C2×D71—41no—
192C2×D71—41no—
193C2×D71—41no—
194C2×D71—84no—
195C4×C71—21no—
196C4×C71—21no—
197C4×C71—22no—
198C4×C71—21no—
199C4×C71—21no—
200C4×C71—41no—
201C4×C71C22×D411yescharacteristic
202C4×C71H(5,𝔽2)11yescharacteristic
203Dic71—41no—
204Dic71—41no—
205C2×C71—41no—
206C2×C71—41no—
207C2×C71—81no—
208C2×C71—81no—
209C2×C71D4211yescharacteristic
210C2×C71#57511yescharacteristic
211C2×C71#57611yescharacteristic
212D71—42no—
213D71—81no—
214#3268—711nomaximal, Sylow
215C71#326811yescharacteristic, complemented, Sylow
216C2×AGL(1,ℤ/8)—711no—
217C2×#135—711no—
218#550D7111yescharacteristic, complemented
219#546—711no—
220#496—711no—
221#468—711no—
222#466—711no—
223#464—711no—
224#459—711no—
225#452—711no—
226#449—711no—
227#449—711no—
228#447—711no—
229#440—711no—
230#435—711no—
231C2×#41—711no—
232C2×#41C2×D7111yescharacteristic
233C2×#41C2×D7111yescharacteristic, complemented
234C22×D4—711no—
235AGL(1,ℤ/8)—1424no—
236#135—1424no—
237C2×Q16—711no—
238C2×SD16—711no—
239C2×SD16—711no—
240C2×SD16—711no—
241C2×SD16—711no—
242C2×D8—711no—
243C2×D8—711no—
244C2×D8—711no—
245C2×D8—711no—
246C2×M16—711no—
247C22×C8—711no—
248#127—21no—
249#127C2×D7111yescharacteristic, complemented
250#124—711no—
251#123—1421no—
252C22⋊Q8—711no—
253#121—21no—
254#121—21no—
255#121—711no—
256#121—1421no—
257#121C2×D7111yescharacteristic
258#121C2×D7111yescharacteristic, complemented
259#120—1421no—
260C4×Q8C2×D7111yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—711no—
263C4×D4C2×D7111yescharacteristic
264C2×C22⋊C4—1421no—
265#107—711no—
266C4⋊C8—711no—
267#103—711no—
268#103—711no—
269#102—711no—
270#102—711no—
271#102—711no—
272#102—711no—
273C22⋊C8—711no—
274C22⋊C8—711no—
275#97—711no—
276C24—1421no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—2841no—
282#41—1422no—
283C2×Q8—711no—
284C2×Q8C22×D7111yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2841no—
292C2×D4—711no—
293C2×D4—711no—
294C2×D4—1422no—
295C2×D4—1421no—
296C2×D4—1421no—
297C2×D4—1421no—
298C2×D4C22⋊D7111yescharacteristic
299C2×D4C22×D7111yescharacteristic
300C2×D4C22×D7111yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—1421no—
304C22×C4—1421no—
305C22×C4C22×D7111yescharacteristic
306C22×C4C22×D7111yescharacteristic
307Q16—1422no—
308SD16—2841no—
309SD16—714no—
310SD16—1422no—
311SD16—1422no—
312D8—2841no—
313D8—1422no—
314D8—1422no—
315D8—1422no—
316M16—1422no—
317C2×C8—711no—
318C2×C8—711no—
319C2×C8—711no—
320C2×C8—711no—
321C2×C8—1422no—
322C4⋊C4—21no—
323C4⋊C4—711no—
324C4⋊C4—1421no—
325C4⋊C4C22⋊D7111yescharacteristic
326C4⋊C4C22×D7111yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—1421no—
330C22⋊C4—1421no—
331C22⋊C4—1421no—
332C22⋊C4—1421no—
333C22⋊C4—1421no—
334C22⋊C4C22⋊D7111yescharacteristic
335C22⋊C4C22⋊D7111yescharacteristic
336C42—21no—
337C42C22×D7111yescharacteristic
338C23—21no—
339C23—21no—
340C23—2842no—
341C23—2842no—
342C23—5681no—
343C23—1421no—
344C23—1421no—
345C23C2×C22⋊D7111yescharacteristic
346C23C2×C22⋊D7111yescharacteristic
347Q8—21no—
348Q8—1421no—
349Q8—1421no—
350Q8D4×D7112yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—2842no—
363D4—2842no—
364D4—5681no—
365D4—712no—
366D4—1421no—
367D4—1421no—
368D4—1421no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—2842no—
378C2×C4—5681no—
379C2×C4—1421no—
380C2×C4—1421no—
381C2×C4C2×C22⋊D7111yescharacteristic
382C2×C4C2×C22⋊D7111yescharacteristic
383C2×C4C2×C22⋊D7111yescharacteristic
384C2×C4C2×C22⋊D7111yescharacteristic
385C2×C4C23×D7111yescharacteristic
386C8—712no—
387C8—1421no—
388C8—1421no—
389C8—1421no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—2841no—
403C22—2841no—
404C22—2841no—
405C22—2841no—
406C22—2841no—
407C22—5684no—
408C22—1422no—
409C22C22×C22⋊D7111yescharacteristic
410C4—21no—
411C4—21no—
412C4—22no—
413C4—21no—
414C4—21no—
415C4—41no—
416C4—2841no—
417C4—2841no—
418C4C2×D4×D7111yescharacteristic
419C4#1455617311yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—2842no—
425C2—5681no—
426C2D4×C22⋊D7111yescharacteristic
427C2#1798679711yescharacteristic
428C2#1798680211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#550×C71C2
socleC22×C71C22×D4
hypercenter#550D71
hypocenterC71#3268
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#14556153C2semidirect
2. C2×#14556148C2semidirect
3. #17986645C2semidirect
4. #17986386C2semidirect
5. #17986243C2semidirect
6. #17986233C2semidirect
7. #17986221C2semidirect
8. #17986191C2semidirect
9. #17986161C2semidirect
10. #17986144C2semidirect
11. #17986142C2semidirect
12. #17986132C2semidirect
13. #17986097C2semidirect
14. #17986071C2semidirect
15. #550×C71C2semidirect
20. C2×#11765855C22semidirect
21. C2×#11765853C22semidirect
22. C2×#11765853C22semidirect
30. C22×C71⋊C8C22semidirect
39. #14556061C22semidirect
40. C4⋊(C71⋊C8)C22semidirect
41. #14556052C22semidirect
42. #14556051C22semidirect
43. #14556049C22semidirect
44. #14556049C22semidirect
48. C22⋊(C71⋊C8)C22semidirect
49. #14556039C22semidirect
51. C2×#41×C71C22semidirect
53. #127×C71C22semidirect
57. #121×C71C22semidirect
58. C4×Q8×C71C22semidirect
78. #11765852D4semidirect
215. C71#3268semidirect
218. #550D71semidirect
233. C2×#41C2×D71semidirect
249. #127C2×D71semidirect
258. #121C2×D71semidirect
260. C4×Q8C2×D71semidirect