Finite Group Databasebeta

← #21435619

Subgroups — #21435619

Order 7552.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273753C211yesmaximal, characteristic, complemented
2C2×#14273748C211yesmaximal, characteristic, complemented
3#16838054C211yesmaximal, characteristic, complemented
4#16838046C211yesmaximal, characteristic, complemented
5#16837793C211yesmaximal, characteristic, complemented
6#16837658C211yesmaximal, characteristic, complemented
7#16837630C211yesmaximal, characteristic, complemented
8#16837616C211yesmaximal, characteristic, complemented
9#16837605C211yesmaximal, characteristic, complemented
10#16837566C211yesmaximal, characteristic, complemented
11#16837565C211yesmaximal, characteristic, complemented
12#16837546C211yesmaximal, characteristic, complemented
13#16837511C211yesmaximal, characteristic, complemented
14#16837476C211yesmaximal, characteristic, complemented
15C59×#450C211yesmaximal, characteristic, complemented
16C2×#11745689C2211yescharacteristic, complemented
17C2×#11745685C2211yescharacteristic
18C2×#11745687C2211yescharacteristic
19#14273753—24no—
20#14273748—24no—
21C2×Dic236C2211yescharacteristic, complemented
22C2×#11745672C2211yescharacteristic, complemented
23C2×#11745660C2211yescharacteristic, complemented
24C2×#11745670C2211yescharacteristic
25C2×#11745670C2211yescharacteristic
26C2×#11745669C2211yescharacteristic
27C2×#11745669C2211yescharacteristic
28C2×#11745664C2211yescharacteristic
29C2×C8×D59C2211yescharacteristic, complemented
30#14273725C2211yescharacteristic, complemented
31#14273714C2211yescharacteristic
32#14273712—21no—
33#14273709—21no—
34#14273708—21no—
35#14273707C2211yescharacteristic, complemented
36#14273706C2211yescharacteristic
37#14273700—21no—
38#14273699—21no—
39#14273698C2211yescharacteristic
40#14273696—21no—
41D4×Dic59C2211yescharacteristic
42#14273689C2211yescharacteristic, complemented
43C4×C22⋊D59C2211yescharacteristic
44C4⋊C4×D59—21no—
45C2×C22⋊Dic59—21no—
46#14273658C2211yescharacteristic, complemented
47Dic59⋊C8C2211yescharacteristic, complemented
48#14273652C2211yescharacteristic, complemented
49#14273651C2211yescharacteristic, complemented
50#14273650C2211yescharacteristic, complemented
51#14273649C2211yescharacteristic, complemented
52(C2×D59)⋊C8C2211yescharacteristic, complemented
53#14273637C2211yescharacteristic, complemented
54C22×D4×C59C2211yescharacteristic
55C2×SD16×C59C2211yescharacteristic
56C2×SD16×C59C2211yescharacteristic
57C22⋊Q8×C59C2211yescharacteristic
58#102×C59C2211yescharacteristic
59#102×C59C2211yescharacteristic
60C22⋊C8×C59C2211yescharacteristic
61C24×C59—21no—
62C2×Q8×C59C2311yescharacteristic
63C2×D4×C59—22no—
64C2×D4×C59—21no—
65C2×D4×C59—41no—
66C2×D4×C59C2311yescharacteristic
67C2×D4×C59C2311yescharacteristic
68C22×C4×C59C2311yescharacteristic
69C2×C22⋊D59—21no—
70C2×C22⋊D59—21no—
71C2×C22⋊D59—21no—
72C2×C22⋊D59D411yescharacteristic
73C22×Dic59—21no—
74C22×Dic59—21no—
75#11745689—22no—
76#11745689—41no—
77#11745687—22no—
78#11745687—41no—
79#11745685—22no—
80#11745685—41no—
81C2×D236—21no—
82C2×D236C2311yescharacteristic
83C2×C4×D59—21no—
84C2×C4×D59—21no—
85C2×C4×D59—21no—
86C2×C4×D59C2311yescharacteristic
87C2×Dic118C2311yescharacteristic
88C2×Dic118C2311yescharacteristic
89SD16×C59—22no—
90SD16×C59—41no—
91C2×C8×C59C2311yescharacteristic
92C2×C8×C59C2311yescharacteristic
93C4⋊C4×C59—21no—
94C4⋊C4×C59C2311yescharacteristic
95C22⋊C4×C59—21no—
96C22⋊Dic59—21no—
97C22⋊Dic59—21no—
98C22⋊Dic59—21no—
99C22⋊Dic59—21no—
100C22⋊Dic59—21no—
101C22⋊Dic59D411yescharacteristic
102#11745672—22no—
103#11745670—22no—
104#11745670—22no—
105#11745669—22no—
106#11745669D414yescomplemented
107(C2×D59)⋊C4—21no—
108(C2×D59)⋊C4—21no—
109(C2×D59)⋊C4D411yescharacteristic
110C4⋊Dic59—21no—
111C4⋊Dic59C2311yescharacteristic
112Dic59⋊C4—21no—
113Dic59⋊C4—21no—
114Dic59⋊C4—21no—
115Dic59⋊C4D411yescharacteristic
116C4×Dic59C2311yescharacteristic
117#11745664—22no—
118C2×C59⋊C8C2311yescharacteristic
119C2×C59⋊C8C2311yescharacteristic
120Dic236—41no—
121#11745660—22no—
122C8×D59—22no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—42no—
126C23×C59—42no—
127C23×C59—81no—
128C23×C59C2×D411yescharacteristic
129C22×D59—21no—
130C22×D59C2×D411yescharacteristic
131Q8×C59—21no—
132Q8×C59—21no—
133D4×C59—21no—
134D4×C59—21no—
135D4×C59—21no—
136D4×C59—42no—
137D4×C59C2×D412yes—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59C2×D411yescharacteristic
143C2×C4×C59C2411yescharacteristic
144C22⋊D59—41no—
145C22⋊D59—42no—
146C22⋊D59—42no—
147C22⋊D59—81no—
148C2×Dic59—21no—
149C2×Dic59—21no—
150C2×Dic59—21no—
151C2×Dic59—42no—
152C2×Dic59—81no—
153C2×Dic59C2×D411yescharacteristic
154C2×Dic59C2×D411yescharacteristic
155C2×Dic59C2×D411yescharacteristic
156D236—21no—
157D236—42no—
158D236C2×D412yes—
159C4×D59—21no—
160C4×D59—21no—
161C4×D59—42no—
162C4×D59—42no—
163C4×D59—42no—
164Dic118—21no—
165Dic118—21no—
166Dic118—21no—
167Dic118—21no—
168C8×C59—21no—
169C8×C59—21no—
170C59⋊C8—21no—
171C59⋊C8C2×D412yes—
172C22×C59—22no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—41no—
177C22×C59—41no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59—84no—
182C22×C59C22×D411yescharacteristic
183C2×D59—21no—
184C2×D59—22no—
185C2×D59—21no—
186C2×D59—21no—
187C2×D59—41no—
188C2×D59—41no—
189C4×C59—21no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59C22×D411yescharacteristic
193C4×C59H(5,𝔽2)11yescharacteristic
194Dic59—21no—
195Dic59—21no—
196Dic59—21no—
197Dic59—41no—
198Dic59—41no—
199Dic59—41no—
200#3267—591nomaximal, Sylow
201C2×C59—41no—
202C2×C59—42no—
203C2×C59—81no—
204C2×C59D4211yescharacteristic
205C2×C59#57511yescharacteristic
206C2×C59#57711yescharacteristic
207D59—41no—
208D59—42no—
209C2×AGL(1,ℤ/8)—591no—
210C2×#135—591no—
211#547—591no—
212#546—591no—
213#497—591no—
214#470—591no—
215#465—591no—
216#463—591no—
217#461—591no—
218#452—591no—
219#452—591no—
220#450D5911yescharacteristic, complemented
221#449—591no—
222#442—591no—
223#435—591no—
224C59#326711yescharacteristic, complemented, Sylow
225C2×#41—591no—
226C2×#41—591no—
227C2×#41—591no—
228C22×D4C2×D5911yescharacteristic
229AGL(1,ℤ/8)—1184no—
230#135—1184no—
231C2×Q16—591no—
232C2×Q16—591no—
233C2×SD16—591no—
234C2×SD16—591no—
235C2×SD16—591no—
236C2×SD16C2×D5911yescharacteristic
237C2×SD16C2×D5911yescharacteristic
238C2×D8—591no—
239C2×D8—591no—
240C2×M16—591no—
241C22×C8—591no—
242C4⋊Q8—591no—
243#124—591no—
244#123—1181no—
245#123—1181no—
246#123—1181no—
247C22⋊Q8—591no—
248C22⋊Q8—591no—
249C22⋊Q8C2×D5911yescharacteristic
250#121—591no—
251#121—1181no—
252#121—1181no—
253#120—1181no—
254C4×D4—591no—
255C4×D4—591no—
256C4×D4—591no—
257C2×C4⋊C4—1181no—
258C2×C22⋊C4—1181no—
259#106—591no—
260C4⋊C8—591no—
261#103—591no—
262#103—591no—
263#103—591no—
264#102—591no—
265#102C2×D5911yescharacteristic
266#102C2×D5911yescharacteristic
267C22⋊C8—591no—
268C22⋊C8C2×D5911yescharacteristic
269#97—591no—
270C24—21no—
271#41—1182no—
272#41—1182no—
273#41—1182no—
274#41—2361no—
275#41—2361no—
276#41—2361no—
277C2×Q8—591no—
278C2×Q8—591no—
279C2×Q8C22×D5911yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—591no—
284C2×D4—591no—
285C2×D4—1181no—
286C2×D4—1181no—
287C2×D4—1181no—
288C2×D4—1181no—
289C2×D4C22×D5911yescharacteristic
290C2×D4C22×D5911yescharacteristic
291C22×C4—591no—
292C22×C4—1181no—
293C22×C4—1181no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4—1181no—
297C22×C4C22×D5911yescharacteristic
298Q16—1182no—
299Q16—2361no—
300SD16—22no—
301SD16—41no—
302SD16—1182no—
303SD16—1182no—
304SD16—1182no—
305D8—594no—
306D8—1182no—
307M16—1182no—
308C2×C8—591no—
309C2×C8—591no—
310C2×C8—1182no—
311C2×C8C22×D5911yescharacteristic
312C2×C8C22×D5911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—591no—
315C4⋊C4—591no—
316C4⋊C4—1181no—
317C4⋊C4—1181no—
318C4⋊C4—1181no—
319C4⋊C4—1181no—
320C4⋊C4C22×D5911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—591no—
323C22⋊C4—591no—
324C22⋊C4—1181no—
325C22⋊C4—1181no—
326C22⋊C4—1181no—
327C22⋊C4—1181no—
328C22⋊C4—1181no—
329C22⋊C4—1181no—
330C22⋊C4—1181no—
331C42—591no—
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—591no—
338C23—1181no—
339C23D4×D5911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—1181no—
343Q8—1181no—
344Q8—1181no—
345Q8—1181no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—592no—
351D4—1181no—
352D4—4721no—
353D4—2362no—
354D4—2361no—
355D4—2362no—
356D4—2362no—
357D4D4×D5912yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—591no—
363C2×C4—591no—
364C2×C4—591no—
365C2×C4—1181no—
366C2×C4—1181no—
367C2×C4—1181no—
368C2×C4—1181no—
369C2×C4—1181no—
370C2×C4—4721no—
371C2×C4—2362no—
372C2×C4—2362no—
373C2×C4—2362no—
374C2×C4—2362no—
375C2×C4D4×D5911yescharacteristic
376C2×C4C23×D5911yescharacteristic
377C8—21no—
378C8—21no—
379C8—592no—
380C8—1181no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—84no—
391C22—1181no—
392C22—1181no—
393C22—1181no—
394C22—1182no—
395C22—2361no—
396C22—2361no—
397C22C2×D4×D5911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—1181no—
402C4—1181no—
403C4—1181no—
404C4—2361no—
405C4—2361no—
406C4—2361no—
407C4C2×D4×D5911yescharacteristic
408C4#1427377211yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—2361no—
413C2—2362no—
414C2#1683803411yescharacteristic
415C2#1683819811yescharacteristic
416C2#1683820811yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#450C2
socleC22×C59C22×D4
hypercenter#450D59
hypocenterC59#3267
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#14273753C2semidirect
2. C2×#14273748C2semidirect
3. #16838054C2semidirect
4. #16838046C2semidirect
5. #16837793C2semidirect
6. #16837658C2semidirect
7. #16837630C2semidirect
8. #16837616C2semidirect
9. #16837605C2semidirect
10. #16837566C2semidirect
11. #16837565C2semidirect
12. #16837546C2semidirect
13. #16837511C2semidirect
14. #16837476C2semidirect
15. C59×#450C2semidirect
16. C2×#11745689C22semidirect
21. C2×Dic236C22semidirect
22. C2×#11745672C22semidirect
23. C2×#11745660C22semidirect
29. C2×C8×D59C22semidirect
30. #14273725C22semidirect
35. #14273707C22semidirect
42. #14273689C22semidirect
46. #14273658C22semidirect
47. Dic59⋊C8C22semidirect
48. #14273652C22semidirect
49. #14273651C22semidirect
50. #14273650C22semidirect
51. #14273649C22semidirect
52. (C2×D59)⋊C8C22semidirect
53. #14273637C22semidirect
106. #11745669D4semidirect
220. #450D59semidirect
224. C59#3267semidirect