Finite Group Databasebeta

← #21435700

Subgroups — #21435700

Order 7552.

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×#14273754C211yesmaximal, characteristic, complemented
2C2×#14273752C211yesmaximal, characteristic, complemented
3#16838043C211yesmaximal, characteristic, complemented
4#16838040C211yesmaximal, characteristic, complemented
5#16837787C211yesmaximal, characteristic, complemented
6#16837651C211yesmaximal, characteristic, complemented
7#16837650C211yesmaximal, characteristic, complemented
8#16837620C211yesmaximal, characteristic, complemented
9#16837619C211yesmaximal, characteristic, complemented
10#16837563C211yesmaximal, characteristic, complemented
11#16837538C211yesmaximal, characteristic, complemented
12#16837537C211yesmaximal, characteristic, complemented
13#16837482C211yesmaximal, characteristic, complemented
14#16837477C211yesmaximal, characteristic, complemented
15C59×#452C211yesmaximal, characteristic, complemented
16C2×#11745689C2211yescharacteristic
17C22×D236C2211yescharacteristic, complemented
18C2×D4×D59C2211yescharacteristic, complemented
19#14273754—24no—
20#14273752—24no—
21C2×#11745671C2211yescharacteristic, complemented
22C2×#11745671C2211yescharacteristic, complemented
23C2×#11745671C2211yescharacteristic, complemented
24C2×D472C2211yescharacteristic
25C2×D472C2211yescharacteristic
26C2×#11745669C2211yescharacteristic
27C2×#11745669C2211yescharacteristic
28C2×#11745664C2211yescharacteristic, complemented
29C2×#11745659C2211yescharacteristic, complemented
30#14273716C2211yescharacteristic, complemented
31#14273716C2211yescharacteristic, complemented
32#14273712—21no—
33#14273710—21no—
34#14273703C2211yescharacteristic, complemented
35#14273702—21no—
36#14273702C2211yescharacteristic
37#14273701C2211yescharacteristic
38#14273699—21no—
39#14273697—21no—
40#14273695—21no—
41D4×Dic59C2211yescharacteristic, complemented
42#14273689C2211yescharacteristic, complemented
43C4×C22⋊D59C2211yescharacteristic
44C2×(C2×D59)⋊C4—21no—
45C22⋊C4×D59—21no—
46#14273660C2211yescharacteristic, complemented
47Dic59⋊C8C2211yescharacteristic, complemented
48#14273651C2211yescharacteristic, complemented
49#14273649C2211yescharacteristic, complemented
50#14273649C2211yescharacteristic, complemented
51#14273647C2211yescharacteristic, complemented
52(C2×D59)⋊C8C2211yescharacteristic, complemented
53C8×Dic59C2211yescharacteristic, complemented
54C2×#41×C59C2211yescharacteristic
55C2×Q16×C59C2211yescharacteristic, complemented
56C2×SD16×C59C2211yescharacteristic, complemented
57C22⋊Q8×C59C2211yescharacteristic, complemented
58#103×C59C2211yescharacteristic, complemented
59#102×C59C2211yescharacteristic, complemented
60C22⋊C8×C59C2211yescharacteristic, complemented
61C23×D59—21no—
62C23×D59—21no—
63#41×C59—22no—
64#41×C59—41no—
65C2×Q8×C59C2311yescharacteristic
66C2×Q8×C59C2311yescharacteristic
67C2×D4×C59—21no—
68C2×D4×C59C2311yescharacteristic
69C22×C4×C59—21no—
70C22×C4×C59C2311yescharacteristic
71C2×C22⋊D59—21no—
72C2×C22⋊D59—21no—
73C2×C22⋊D59D411yescharacteristic
74C22×Dic59—21no—
75#11745689—22no—
76#11745689—41no—
77D4×D59—22no—
78D4×D59—41no—
79C2×D236—22no—
80C2×D236—21no—
81C2×D236—21no—
82C2×D236—21no—
83C2×D236—41no—
84C2×D236C2311yescharacteristic
85C2×D236C2311yescharacteristic
86C2×D236C2311yescharacteristic
87C2×C4×D59—21no—
88C2×C4×D59—21no—
89C2×C4×D59C2311yescharacteristic
90Q16×C59—41no—
91SD16×C59—22no—
92C2×C8×C59C2311yescharacteristic
93C2×C8×C59C2311yescharacteristic
94C4⋊C4×C59—21no—
95C4⋊C4×C59C2311yescharacteristic
96C22⋊C4×C59—21no—
97C22⋊Dic59—21no—
98C22⋊Dic59D411yescharacteristic
99#11745671—22no—
100#11745671—22no—
101#11745671—22no—
102#11745669—22no—
103#11745669D414yescomplemented
104(C2×D59)⋊C4—21no—
105(C2×D59)⋊C4—21no—
106(C2×D59)⋊C4—21no—
107(C2×D59)⋊C4—21no—
108(C2×D59)⋊C4—21no—
109(C2×D59)⋊C4—21no—
110(C2×D59)⋊C4—21no—
111(C2×D59)⋊C4—21no—
112(C2×D59)⋊C4D411yescharacteristic
113C4⋊Dic59C2311yescharacteristic
114Dic59⋊C4—21no—
115Dic59⋊C4D411yescharacteristic
116C4×Dic59C2311yescharacteristic
117#11745664—22no—
118C2×C59⋊C8C2311yescharacteristic
119C2×C59⋊C8C2311yescharacteristic
120D472—22no—
121D472—41no—
122#11745659—22no—
123C23×C59—21no—
124C23×C59C2×D411yescharacteristic
125C22×D59—21no—
126C22×D59—21no—
127C22×D59—21no—
128C22×D59—42no—
129C22×D59—42no—
130C22×D59—42no—
131C22×D59—42no—
132C22×D59—81no—
133C22×D59—81no—
134C22×D59C2×D411yescharacteristic
135Q8×C59—21no—
136Q8×C59—21no—
137Q8×C59—21no—
138Q8×C59—21no—
139D4×C59—21no—
140D4×C59—42no—
141D4×C59C2×D412yes—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—42no—
148C2×C4×C59C2×D411yescharacteristic
149C2×C4×C59C2411yescharacteristic
150C22⋊D59—41no—
151C22⋊D59—42no—
152C22⋊D59—81no—
153C2×Dic59—21no—
154C2×Dic59—81no—
155C2×Dic59C2×D411yescharacteristic
156C2×Dic59C2×D411yescharacteristic
157C2×Dic59C2×D411yescharacteristic
158D236—21no—
159D236—21no—
160D236—21no—
161D236—21no—
162D236—21no—
163D236—42no—
164D236—42no—
165D236—81no—
166D236C2×D412yes—
167C4×D59—21no—
168C4×D59—21no—
169C4×D59—42no—
170C4×D59—81no—
171C8×C59—21no—
172C8×C59—21no—
173C59⋊C8—21no—
174C59⋊C8C2×D412yes—
175C22×C59—22no—
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59C22×D411yescharacteristic
182C2×D59—21no—
183C2×D59—22no—
184C2×D59—21no—
185C2×D59—21no—
186C2×D59—41no—
187C2×D59—41no—
188C2×D59—41no—
189C2×D59—41no—
190C2×D59—41no—
191C2×D59—41no—
192C2×D59—41no—
193C2×D59—41no—
194C2×D59—84no—
195C2×D59—84no—
196C4×C59—21no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—41no—
200C4×C59C22×D411yescharacteristic
201C4×C59H(5,𝔽2)11yescharacteristic
202Dic59—21no—
203Dic59—21no—
204Dic59—21no—
205Dic59—41no—
206#3273—591nomaximal, Sylow
207C2×C59—41no—
208C2×C59—42no—
209C2×C59D4211yescharacteristic
210C2×C59#57511yescharacteristic
211C2×C59#57611yescharacteristic
212D59—41no—
213D59—42no—
214D59—81no—
215D59—81no—
216C2×AGL(1,ℤ/8)—591no—
217C2×AGL(1,ℤ/8)—591no—
218#546—591no—
219#546—591no—
220#495—591no—
221#468—591no—
222#468—591no—
223#463—591no—
224#463—591no—
225#452—591no—
226#452D5911yescharacteristic, complemented
227#447—591no—
228#447—591no—
229C4×D8—591no—
230#436—591no—
231C59#327311yescharacteristic, complemented, Sylow
232C2×#41—591no—
233C2×#41C2×D5911yescharacteristic
234C22×D4—591no—
235C22×D4—591no—
236AGL(1,ℤ/8)—1184no—
237AGL(1,ℤ/8)—1184no—
238C2×Q16C2×D5911yescharacteristic, complemented
239C2×SD16—591no—
240C2×SD16—591no—
241C2×SD16—591no—
242C2×SD16C2×D5911yescharacteristic, complemented
243C2×D8—591no—
244C2×D8—591no—
245C2×D8—591no—
246C2×D8—591no—
247C2×M16—591no—
248C2×M16—591no—
249#124—591no—
250#124—591no—
251#123—1181no—
252#123—1181no—
253C22⋊Q8—591no—
254C22⋊Q8C2×D5911yescharacteristic, complemented
255#121—591no—
256#121—591no—
257#121—1181no—
258#121—1181no—
259#120—1181no—
260#120—1181no—
261C4×D4—591no—
262C4×D4—591no—
263C4×D4—591no—
264C2×C22⋊C4—1181no—
265C2×C22⋊C4—1181no—
266#107—591no—
267C4⋊C8—591no—
268#103—591no—
269#103C2×D5911yescharacteristic, complemented
270#102—591no—
271#102—591no—
272#102—591no—
273#102C2×D5911yescharacteristic, complemented
274C22⋊C8—591no—
275C22⋊C8C2×D5911yescharacteristic, complemented
276C4×C8—591no—
277C24—1181no—
278C24—1181no—
279#41—22no—
280#41—41no—
281#41—1182no—
282#41—2361no—
283C2×Q8C22×D5911yescharacteristic
284C2×Q8C22×D5911yescharacteristic
285C2×D4—21no—
286C2×D4—591no—
287C2×D4—591no—
288C2×D4—591no—
289C2×D4—591no—
290C2×D4—1182no—
291C2×D4—1181no—
292C2×D4—1181no—
293C2×D4—1181no—
294C2×D4—1182no—
295C2×D4—1181no—
296C2×D4—1181no—
297C2×D4—2361no—
298C2×D4—2361no—
299C2×D4C22×D5911yescharacteristic
300C22×C4—21no—
301C22×C4—591no—
302C22×C4—1181no—
303C22×C4—1181no—
304C22×C4—1181no—
305C22×C4C22×D5911yescharacteristic
306Q16—41no—
307SD16—22no—
308SD16—1182no—
309SD16—1182no—
310SD16—1182no—
311D8—594no—
312D8—1182no—
313D8—1182no—
314D8—2361no—
315M16—1182no—
316M16—1182no—
317C2×C8—591no—
318C2×C8—591no—
319C2×C8C22×D5911yescharacteristic
320C2×C8C22×D5911yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—591no—
323C4⋊C4—591no—
324C4⋊C4—1181no—
325C4⋊C4C22×D5911yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—591no—
328C22⋊C4—591no—
329C22⋊C4—1181no—
330C22⋊C4—1181no—
331C22⋊C4—1181no—
332C22⋊C4—1181no—
333C22⋊C4—1181no—
334C22⋊C4—1181no—
335C22⋊C4—1181no—
336C22⋊C4—1181no—
337C22⋊C4—1181no—
338C42—591no—
339C23—21no—
340C23—591no—
341C23—1181no—
342C23—1181no—
343C23—1181no—
344C23—4721no—
345C23—4721no—
346C23—2362no—
347C23—2362no—
348C23—2362no—
349C23—2362no—
350C23D4×D5911yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—592no—
358D4—1181no—
359D4—1181no—
360D4—1181no—
361D4—1181no—
362D4—1181no—
363D4—4721no—
364D4—4721no—
365D4—2362no—
366D4—2362no—
367D4—2361no—
368D4—2362no—
369D4D4×D5912yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—591no—
377C2×C4—591no—
378C2×C4—591no—
379C2×C4—1181no—
380C2×C4—1181no—
381C2×C4—1181no—
382C2×C4—4721no—
383C2×C4—4721no—
384C2×C4—2362no—
385C2×C4D4×D5911yescharacteristic
386C2×C4C23×D5911yescharacteristic
387C8—21no—
388C8—21no—
389C8—592no—
390C8—1181no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—1181no—
398C22—1181no—
399C22—1181no—
400C22—1182no—
401C22—4724no—
402C22—4724no—
403C22—2361no—
404C22—2361no—
405C22—2361no—
406C22—2361no—
407C22—2361no—
408C22—2361no—
409C22—2361no—
410C22—2361no—
411C22C2×D4×D5911yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—1181no—
417C4—1181no—
418C4—1181no—
419C4—2361no—
420C4C2×D4×D5911yescharacteristic
421C4#1427377211yescharacteristic
422C2—41no—
423C2—42no—
424C2—4721no—
425C2—4721no—
426C2—2361no—
427C2—2362no—
428C2#1683803411yescharacteristic
429C2#1683820111yescharacteristic
430C2#1683820411yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#452C2
socleC22×C59C22×D4
hypercenter#452D59
hypocenterC59#3273
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#14273754C2semidirect
2. C2×#14273752C2semidirect
3. #16838043C2semidirect
4. #16838040C2semidirect
5. #16837787C2semidirect
6. #16837651C2semidirect
7. #16837650C2semidirect
8. #16837620C2semidirect
9. #16837619C2semidirect
10. #16837563C2semidirect
11. #16837538C2semidirect
12. #16837537C2semidirect
13. #16837482C2semidirect
14. #16837477C2semidirect
15. C59×#452C2semidirect
17. C22×D236C22semidirect
18. C2×D4×D59C22semidirect
21. C2×#11745671C22semidirect
22. C2×#11745671C22semidirect
23. C2×#11745671C22semidirect
28. C2×#11745664C22semidirect
29. C2×#11745659C22semidirect
30. #14273716C22semidirect
31. #14273716C22semidirect
34. #14273703C22semidirect
41. D4×Dic59C22semidirect
42. #14273689C22semidirect
46. #14273660C22semidirect
47. Dic59⋊C8C22semidirect
48. #14273651C22semidirect
49. #14273649C22semidirect
50. #14273649C22semidirect
51. #14273647C22semidirect
52. (C2×D59)⋊C8C22semidirect
53. C8×Dic59C22semidirect
55. C2×Q16×C59C22semidirect
56. C2×SD16×C59C22semidirect
57. C22⋊Q8×C59C22semidirect
58. #103×C59C22semidirect
59. #102×C59C22semidirect
60. C22⋊C8×C59C22semidirect
103. #11745669D4semidirect
226. #452D59semidirect
231. C59#3273semidirect
238. C2×Q16C2×D59semidirect
242. C2×SD16C2×D59semidirect
254. C22⋊Q8C2×D59semidirect
269. #103C2×D59semidirect
273. #102C2×D59semidirect
275. C22⋊C8C2×D59semidirect