Finite Group Databasebeta

← #21435623

Subgroups — #21435623

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×#14273751C211yesmaximal, characteristic, complemented
2C2×#14273746C211yesmaximal, characteristic, complemented
3#16838056C211yesmaximal, characteristic, complemented
4#16838047C211yesmaximal, characteristic, complemented
5#16837796C211yesmaximal, characteristic, complemented
6#16837661C211yesmaximal, characteristic, complemented
7#16837632C211yesmaximal, characteristic, complemented
8#16837617C211yesmaximal, characteristic, complemented
9#16837607C211yesmaximal, characteristic, complemented
10#16837565C211yesmaximal, characteristic, complemented
11#16837565C211yesmaximal, characteristic, complemented
12#16837554C211yesmaximal, characteristic, complemented
13#16837548C211yesmaximal, characteristic, complemented
14#16837475C211yesmaximal, characteristic, complemented
15C59×#442C211yesmaximal, characteristic, complemented
16C2×#11745687C2211yescharacteristic
17C2×#11745687C2211yescharacteristic
18C2×#11745687C2211yescharacteristic, complemented
19C2×D4×D59C2211yescharacteristic
20#14273751—24no—
21#14273746—24no—
22C2×Dic236C2211yescharacteristic, complemented
23C2×Dic236C2211yescharacteristic, complemented
24C2×#11745660C2211yescharacteristic
25C2×#11745660C2211yescharacteristic
26C2×#11745670C2211yescharacteristic
27C2×#11745670C2211yescharacteristic
28C2×#11745670C2211yescharacteristic, complemented
29C2×#11745669C2211yescharacteristic
30C2×#11745659C2211yescharacteristic
31C2×C8×D59C2211yescharacteristic, complemented
32#14273727C2211yescharacteristic, complemented
33#14273715C2211yescharacteristic
34#14273711—21no—
35#14273711—21no—
36#14273711—21no—
37#14273707C2211yescharacteristic
38#14273707C2211yescharacteristic
39#14273707C2211yescharacteristic, complemented
40#14273702C2211yescharacteristic
41#14273701—21no—
42#14273701—21no—
43#14273697—21no—
44C4×D236C2211yescharacteristic
45C2×C4⋊Dic59—21no—
46C2×(C2×D59)⋊C4—21no—
47C4⋊(C59⋊C8)C2211yescharacteristic, complemented
48#14273650C2211yescharacteristic, complemented
49#14273650C2211yescharacteristic, complemented
50#14273650C2211yescharacteristic, complemented
51#14273647C2211yescharacteristic
52(C2×D59)⋊C8C2211yescharacteristic
53(C2×D59)⋊C8C2211yescharacteristic, complemented
54C2×D8×C59C2211yescharacteristic
55C4×D4×C59C2211yescharacteristic
56C4×D4×C59C2211yescharacteristic, complemented
57#106×C59C2211yescharacteristic, complemented
58#102×C59C2211yescharacteristic
59#102×C59C2211yescharacteristic, complemented
60#97×C59C2211yescharacteristic, complemented
61C23×D59—21no—
62C2×D4×C59C2311yescharacteristic
63C2×D4×C59C2311yescharacteristic
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C2×C22⋊D59—21no—
67C2×C22⋊D59—21no—
68C2×C22⋊D59—21no—
69C2×C22⋊D59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73#11745687—22no—
74#11745687—22no—
75#11745687—22no—
76#11745687—41no—
77#11745687—41no—
78#11745687—41no—
79D4×D59—22no—
80D4×D59—41no—
81C2×D236—21no—
82C2×D236D411yescharacteristic
83C2×D236C2311yescharacteristic
84C2×C4×D59C2311yescharacteristic
85C2×C4×D59C2311yescharacteristic
86C2×Dic118C2311yescharacteristic
87C2×Dic118C2311yescharacteristic
88C2×Dic118C2311yescharacteristic
89D8×C59D414yescomplemented
90C2×C8×C59C2311yescharacteristic
91C2×C8×C59C2311yescharacteristic
92C4⋊C4×C59C2311yescharacteristic
93C4⋊C4×C59C2311yescharacteristic
94C22⋊C4×C59—21no—
95C22⋊C4×C59—21no—
96C42×C59C2311yescharacteristic
97#11745670—22no—
98#11745670—22no—
99#11745670—22no—
100#11745669—22no—
101(C2×D59)⋊C4—21no—
102(C2×D59)⋊C4—21no—
103(C2×D59)⋊C4—21no—
104(C2×D59)⋊C4—21no—
105(C2×D59)⋊C4—21no—
106(C2×D59)⋊C4—21no—
107(C2×D59)⋊C4D411yescharacteristic
108(C2×D59)⋊C4D411yescharacteristic
109C4⋊Dic59—21no—
110C4⋊Dic59—21no—
111C4⋊Dic59—21no—
112C4⋊Dic59—21no—
113C4⋊Dic59—21no—
114C4⋊Dic59D411yescharacteristic
115C2×C59⋊C8C2311yescharacteristic
116C2×C59⋊C8C2311yescharacteristic
117Dic236—22no—
118Dic236—41no—
119#11745660—22no—
120#11745660—41no—
121#11745659—22no—
122C8×D59—22no—
123C23×C59—21no—
124C23×C59—21no—
125C22×D59—21no—
126C22×D59—42no—
127C22×D59—42no—
128C22×D59—81no—
129C22×D59C2×D411yescharacteristic
130C22×D59C2×D411yescharacteristic
131D4×C59—21no—
132D4×C59—21no—
133D4×C59C2×D412yes—
134D4×C59C2×D412yes—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—42no—
138C2×C4×C59—81no—
139C2×C4×C59C2×D411yescharacteristic
140C2×C4×C59C2×D411yescharacteristic
141C2×C4×C59C2411yescharacteristic
142C22⋊D59—42no—
143C22⋊D59—42no—
144C22⋊D59—42no—
145C22⋊D59—42no—
146C2×Dic59—21no—
147C2×Dic59—21no—
148C2×Dic59—21no—
149C2×Dic59—42no—
150C2×Dic59—42no—
151C2×Dic59—42no—
152C2×Dic59C2×D411yescharacteristic
153C2×Dic59C2×D411yescharacteristic
154D236—21no—
155D236—21no—
156D236—41no—
157D236—81no—
158C4×D59—21no—
159C4×D59—21no—
160C4×D59—21no—
161C4×D59—21no—
162Dic118—21no—
163Dic118—21no—
164Dic118—21no—
165Dic118—21no—
166Dic118—21no—
167Dic118—21no—
168C8×C59—21no—
169C8×C59C2×D412yes—
170C59⋊C8—21no—
171C59⋊C8—21no—
172C22×C59—22no—
173C22×C59—22no—
174C22×C59—41no—
175C22×C59—41no—
176C22×C59—41no—
177C22×C59—41no—
178C22×C59C22×D411yescharacteristic
179C2×D59—21no—
180C2×D59—21no—
181C2×D59—21no—
182C2×D59—21no—
183C2×D59—21no—
184C2×D59—21no—
185C2×D59—41no—
186C2×D59—41no—
187C2×D59—41no—
188C2×D59—84no—
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59C22×D411yescharacteristic
194C4×C59H(5,𝔽2)11yescharacteristic
195Dic59—21no—
196Dic59—21no—
197Dic59—41no—
198Dic59—41no—
199Dic59—41no—
200#3267—591nomaximal, Sylow
201C2×C59—42no—
202C2×C59—42no—
203C2×C59D4211yescharacteristic
204C2×C59#57511yescharacteristic
205C2×C59#57711yescharacteristic
206D59—41no—
207D59—41no—
208D59—81no—
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#450—591no—
221#449—591no—
222#442D5911yescharacteristic, complemented
223#435—591no—
224C59#326711yescharacteristic, complemented, Sylow
225C2×#41—591no—
226C2×#41—591no—
227C2×#41—591no—
228C22×D4—591no—
229AGL(1,ℤ/8)—1184no—
230#135—1184no—
231C2×Q16—591no—
232C2×Q16—591no—
233C2×SD16—591no—
234C2×SD16—591no—
235C2×SD16—591no—
236C2×SD16—591no—
237C2×SD16—591no—
238C2×D8—591no—
239C2×D8C2×D5911yescharacteristic
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⋊Q8—591no—
250#121—591no—
251#121—1181no—
252#121—1181no—
253#120—1181no—
254C4×D4—591no—
255C4×D4C2×D5911yescharacteristic
256C4×D4C2×D5911yescharacteristic, complemented
257C2×C4⋊C4—1181no—
258C2×C22⋊C4—1181no—
259#106C2×D5911yescharacteristic, complemented
260C4⋊C8—591no—
261#103—591no—
262#103—591no—
263#103—591no—
264#102—591no—
265#102C2×D5911yescharacteristic
266#102C2×D5911yescharacteristic, complemented
267C22⋊C8—591no—
268C22⋊C8—591no—
269#97C2×D5911yescharacteristic, complemented
270C24—1181no—
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×Q8—591no—
280C2×D4—591no—
281C2×D4—591no—
282C2×D4—1181no—
283C2×D4—1181no—
284C2×D4—1182no—
285C2×D4—1181no—
286C2×D4—1181no—
287C2×D4—1181no—
288C2×D4—2361no—
289C2×D4C22×D5911yescharacteristic
290C2×D4C22×D5911yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—591no—
294C22×C4—591no—
295C22×C4—1181no—
296C22×C4—1181no—
297C22×C4—1181no—
298Q16—1182no—
299Q16—2361no—
300SD16—1182no—
301SD16—1182no—
302SD16—1182no—
303SD16—1182no—
304SD16—2361no—
305D8—1182no—
306D8D23614yescomplemented
307M16—1182no—
308C2×C8—591no—
309C2×C8—591no—
310C2×C8—1182no—
311C2×C8C22×D5911yescharacteristic
312C2×C8C22×D5911yescharacteristic
313C4⋊C4—591no—
314C4⋊C4—1181no—
315C4⋊C4—1181no—
316C4⋊C4—1181no—
317C4⋊C4—1181no—
318C4⋊C4—1181no—
319C4⋊C4C22×D5911yescharacteristic
320C4⋊C4C22×D5911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—591no—
324C22⋊C4—591no—
325C22⋊C4—1181no—
326C22⋊C4—1181no—
327C22⋊C4—1181no—
328C22⋊C4—1181no—
329C22⋊C4—1181no—
330C22⋊C4—1181no—
331C42C22×D5911yescharacteristic
332C23—21no—
333C23—21no—
334C23—591no—
335C23—591no—
336C23—1181no—
337C23—4721no—
338C23—2362no—
339C23—2362no—
340Q8—1181no—
341Q8—1181no—
342Q8—1181no—
343Q8—1181no—
344Q8—1181no—
345Q8—1181no—
346D4—21no—
347D4—21no—
348D4—1181no—
349D4—1181no—
350D4—4721no—
351D4—2361no—
352D4—2362no—
353D4—2362no—
354D4—2362no—
355D4—2362no—
356D4C2×D23612yes—
357D4C2×D23612yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—591no—
363C2×C4—591no—
364C2×C4—1181no—
365C2×C4—1181no—
366C2×C4—1181no—
367C2×C4—1181no—
368C2×C4—1181no—
369C2×C4—1181no—
370C2×C4—1181no—
371C2×C4—2362no—
372C2×C4—2362no—
373C2×C4—2362no—
374C2×C4D4×D5911yescharacteristic
375C2×C4D4×D5911yescharacteristic
376C2×C4C23×D5911yescharacteristic
377C8—21no—
378C8—1181no—
379C8—1181no—
380C8C2×D23612yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—1181no—
388C22—1181no—
389C22—1181no—
390C22—1181no—
391C22—1181no—
392C22—1181no—
393C22—4724no—
394C22—2361no—
395C22—2361no—
396C22—2361no—
397C22C2×D4×D5911yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—1181no—
403C4—1181no—
404C4—2361no—
405C4—2361no—
406C4—2361no—
407C4C22×D23611yescharacteristic
408C4#1427377311yescharacteristic
409C2—42no—
410C2—42no—
411C2—4721no—
412C2—2361no—
413C2—2361no—
414C2D4×D23611yescharacteristic
415C2#1683819811yescharacteristic
416C2#1683821211yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#442C2
socleC22×C59C22×D4
hypercenter#442D59
hypocenterC59#3267
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#14273751C2semidirect
2. C2×#14273746C2semidirect
3. #16838056C2semidirect
4. #16838047C2semidirect
5. #16837796C2semidirect
6. #16837661C2semidirect
7. #16837632C2semidirect
8. #16837617C2semidirect
9. #16837607C2semidirect
10. #16837565C2semidirect
11. #16837565C2semidirect
12. #16837554C2semidirect
13. #16837548C2semidirect
14. #16837475C2semidirect
15. C59×#442C2semidirect
18. C2×#11745687C22semidirect
22. C2×Dic236C22semidirect
23. C2×Dic236C22semidirect
28. C2×#11745670C22semidirect
31. C2×C8×D59C22semidirect
32. #14273727C22semidirect
39. #14273707C22semidirect
47. C4⋊(C59⋊C8)C22semidirect
48. #14273650C22semidirect
49. #14273650C22semidirect
50. #14273650C22semidirect
53. (C2×D59)⋊C8C22semidirect
56. C4×D4×C59C22semidirect
57. #106×C59C22semidirect
59. #102×C59C22semidirect
60. #97×C59C22semidirect
89. D8×C59D4semidirect
222. #442D59semidirect
224. C59#3267semidirect
256. C4×D4C2×D59semidirect
259. #106C2×D59semidirect
266. #102C2×D59semidirect
269. #97C2×D59semidirect
306. D8D236semidirect