Finite Group Databasebeta

← #21435599

Subgroups — #21435599

Order 7552.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273759C211yesmaximal, characteristic, complemented
2C22×#11745671C211yesmaximal, characteristic, complemented
3Q8×C22⋊D59C211yesmaximal, characteristic, complemented
4#16838043C211yesmaximal, characteristic
5#16837795C211yesmaximal, characteristic, complemented
6#16837651C211yesmaximal, characteristic, complemented
7#16837641C211yesmaximal, characteristic, complemented
8#16837626C211yesmaximal, characteristic, complemented
9#16837606C211yesmaximal, characteristic, complemented
10#16837563C211yesmaximal, characteristic, complemented
11#16837555C211yesmaximal, characteristic, complemented
12#16837542C211yesmaximal, characteristic, complemented
13#16837505C211yesmaximal, characteristic, complemented
14#16837470C211yesmaximal, characteristic, complemented
15C59×#451C211yesmaximal, characteristic, complemented
16C2×#11745689C2211yescharacteristic
17C2×Q8×D59C2211yescharacteristic, complemented
18C22×D236C2211yescharacteristic
19#14273759—24no—
20C2×#11745672C2211yescharacteristic, complemented
21C2×#11745660C2211yescharacteristic
22C2×#11745660C2211yescharacteristic
23C2×#11745671—24no—
24C2×#11745671C2211yescharacteristic
25C2×#11745671C2211yescharacteristic
26C2×#11745671C2211yescharacteristic
27C2×#11745671C2211yescharacteristic
28C2×#11745659C2211yescharacteristic, complemented
29C22×C59⋊C8C2211yescharacteristic
30#14273725—21no—
31#14273725C2211yescharacteristic, complemented
32#14273716C2211yescharacteristic
33#14273712—21no—
34#14273706—21no—
35#14273706C2211yescharacteristic
36#14273703—21no—
37#14273703C2211yescharacteristic, complemented
38#14273702—21no—
39#14273702C2211yescharacteristic
40#14273697—21no—
41Q8×Dic59C2211yescharacteristic, complemented
42#14273689C2211yescharacteristic
43C4×C22⋊D59—21no—
44C4×C22⋊D59C2211yescharacteristic
45C22⋊C4×D59—21no—
46#14273660C2211yescharacteristic, complemented
47Dic59⋊C8C2211yescharacteristic, complemented
48#14273651C2211yescharacteristic, complemented
49#14273650C2211yescharacteristic, complemented
50#14273649C2211yescharacteristic
51#14273649C2211yescharacteristic
52(C2×D59)⋊C8C2211yescharacteristic, complemented
53#14273637C2211yescharacteristic, complemented
54C22×Q8×C59C2211yescharacteristic
55C2×Q16×C59C2211yescharacteristic, complemented
56C2×Q16×C59C2211yescharacteristic, complemented
57C22⋊Q8×C59C2211yescharacteristic
58#103×C59C2211yescharacteristic, complemented
59#103×C59C2211yescharacteristic, complemented
60C22⋊C8×C59C2211yescharacteristic
61C23×D59—21no—
62C2×Q8×C59—21no—
63C2×Q8×C59—22no—
64C2×Q8×C59—41no—
65C2×Q8×C59C2311yescharacteristic
66C2×Q8×C59C2311yescharacteristic
67C2×Q8×C59C2311yescharacteristic
68C22×C4×C59—21no—
69C22×C4×C59C2311yescharacteristic
70C2×C22⋊D59D411yescharacteristic
71#11745689—22no—
72#11745689—41no—
73Q8×D59—22no—
74Q8×D59—41no—
75C2×D236—22no—
76C2×D236—21no—
77C2×D236—21no—
78C2×D236—21no—
79C2×D236—41no—
80C2×D236C2311yescharacteristic
81C2×D236C2311yescharacteristic
82C2×C4×D59—21no—
83C2×C4×D59—21no—
84C2×C4×D59—21no—
85C2×C4×D59C2311yescharacteristic
86C2×Dic118—21no—
87C2×Dic118C2311yescharacteristic
88Q16×C59—22no—
89Q16×C59—41no—
90C2×C8×C59C2311yescharacteristic
91C2×C8×C59C2311yescharacteristic
92C4⋊C4×C59—21no—
93C4⋊C4×C59C2311yescharacteristic
94C22⋊C4×C59—21no—
95C22⋊Dic59—21no—
96C22⋊Dic59D411yescharacteristic
97#11745672—22no—
98#11745671—22no—
99#11745671—22no—
100#11745671—22no—
101#11745671D414yescomplemented
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
108C4⋊Dic59—21no—
109C4⋊Dic59C2311yescharacteristic
110Dic59⋊C4—21no—
111Dic59⋊C4—21no—
112Dic59⋊C4—21no—
113Dic59⋊C4—21no—
114Dic59⋊C4D411yescharacteristic
115C4×Dic59—21no—
116C4×Dic59C2311yescharacteristic
117C2×C59⋊C8—22no—
118C2×C59⋊C8C2311yescharacteristic
119C2×C59⋊C8C2311yescharacteristic
120#11745660—22no—
121#11745660—41no—
122#11745659—22no—
123C23×C59C2×D411yescharacteristic
124C22×D59—21no—
125C22×D59—21no—
126C22×D59—42no—
127C22×D59—42no—
128C22×D59—81no—
129C22×D59C2×D411yescharacteristic
130Q8×C59—21no—
131Q8×C59—21no—
132Q8×C59—21no—
133Q8×C59—21no—
134Q8×C59—21no—
135Q8×C59—42no—
136Q8×C59C2×D412yes—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—42no—
144C2×C4×C59C2×D411yescharacteristic
145C2×C4×C59C2411yescharacteristic
146C22⋊D59—22no—
147C2×Dic59—21no—
148C2×Dic59—21no—
149C2×Dic59C2×D411yescharacteristic
150C2×Dic59C2×D411yescharacteristic
151C2×Dic59C2×D411yescharacteristic
152D236—21no—
153D236—21no—
154D236—21no—
155D236—42no—
156D236—42no—
157D236—81no—
158D236C2×D412yes—
159C4×D59—21no—
160C4×D59—21no—
161C4×D59—42no—
162C4×D59—42no—
163C4×D59—81no—
164Dic118—21no—
165Dic118—21no—
166Dic118—42no—
167C8×C59—21no—
168C8×C59—21no—
169C59⋊C8—21no—
170C59⋊C8C2×D412yes—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59#13712yescomplemented
174C22×C59C22×D411yescharacteristic
175C2×D59—21no—
176C2×D59—22no—
177C2×D59—21no—
178C2×D59—21no—
179C2×D59—41no—
180C2×D59—41no—
181C2×D59—41no—
182C2×D59—41no—
183C2×D59—41no—
184C2×D59—84no—
185C4×C59—22no—
186C4×C59—21no—
187C4×C59—41no—
188C4×C59—41no—
189C4×C59—41no—
190C4×C59C22×D411yescharacteristic
191C4×C59H(5,𝔽2)11yescharacteristic
192Dic59—21no—
193Dic59—21no—
194Dic59—21no—
195Dic59—41no—
196Dic59—41no—
197#3266—591nomaximal, Sylow
198C2×C59—22no—
199C2×C59D4211yescharacteristic
200C2×C59C2×#13711yescharacteristic
201C2×C59#57711yescharacteristic
202D59—41no—
203D59—42no—
204D59—81no—
205C2×#137—591no—
206C22×SD16—591no—
207D4×Q8—591no—
208#546—591no—
209#497—591no—
210#468—591no—
211#467—591no—
212#464—591no—
213#461—591no—
214#452—591no—
215#451D5911yescharacteristic, complemented
216#450—591no—
217#448—591no—
218#440—591no—
219#435—591no—
220C59#326611yescharacteristic, complemented, Sylow
221C2×#41—591no—
222C22×Q8—591no—
223C22×Q8C2×D5911yescharacteristic
224C22×D4—591no—
225#137—1184no—
226C2×Q16—591no—
227C2×Q16C2×D5911yescharacteristic, complemented
228C2×Q16C2×D5911yescharacteristic, complemented
229C2×SD16—591no—
230C2×SD16—591no—
231C2×SD16—591no—
232C2×SD16—591no—
233C2×SD16—591no—
234C2×SD16—591no—
235C2×SD16—1184no—
236C2×M16—591no—
237C22×C8—591no—
238C4⋊Q8—591no—
239C4⋊Q8—1181no—
240#124—591no—
241#123—1181no—
242C22⋊Q8—591no—
243C22⋊Q8—591no—
244C22⋊Q8—1181no—
245C22⋊Q8—1181no—
246C22⋊Q8C2×D5911yescharacteristic
247#121—591no—
248#121—1181no—
249#120—1181no—
250C4×Q8—591no—
251C4×D4—591no—
252C4×D4—591no—
253C4×D4—1181no—
254C2×C22⋊C4—1181no—
255#107—591no—
256C4⋊C8—591no—
257#103—591no—
258#103—591no—
259#103C2×D5911yescharacteristic, complemented
260#103C2×D5911yescharacteristic, complemented
261#102—591no—
262#102—591no—
263C22⋊C8—591no—
264C22⋊C8C2×D5911yescharacteristic
265#97—591no—
266C24—1181no—
267#41—1182no—
268#41—2361no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—591no—
273C2×Q8—1181no—
274C2×Q8—1182no—
275C2×Q8—2361no—
276C2×Q8C22×D5911yescharacteristic
277C2×Q8C22×D5911yescharacteristic
278C2×Q8C22×D5911yescharacteristic
279C2×D4—591no—
280C2×D4—591no—
281C2×D4—591no—
282C2×D4—1182no—
283C2×D4—1181no—
284C2×D4—1181no—
285C2×D4—1181no—
286C2×D4—2361no—
287C22×C4—21no—
288C22×C4—591no—
289C22×C4—1181no—
290C22×C4—1181no—
291C22×C4—1181no—
292C22×C4C22×D5911yescharacteristic
293Q16—22no—
294Q16—41no—
295Q16—1182no—
296SD16—594no—
297SD16—1182no—
298SD16—1182no—
299SD16—1182no—
300SD16—1182no—
301SD16—2361no—
302M16—1182no—
303C2×C8—591no—
304C2×C8—591no—
305C2×C8—1182no—
306C2×C8C22×D5911yescharacteristic
307C2×C8C22×D5911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—591no—
310C4⋊C4—591no—
311C4⋊C4—1181no—
312C4⋊C4—1181no—
313C4⋊C4—1181no—
314C4⋊C4—1181no—
315C4⋊C4—1181no—
316C4⋊C4C22×D5911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—591no—
319C22⋊C4—591no—
320C22⋊C4—1181no—
321C22⋊C4—1181no—
322C22⋊C4—1181no—
323C22⋊C4—1181no—
324C22⋊C4—1181no—
325C22⋊C4—1181no—
326C42—591no—
327C42—1181no—
328C23—591no—
329C23—1181no—
330C23—1181no—
331C23—4721no—
332C23—2362no—
333C23—2362no—
334C23D4×D5911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—1181no—
342Q8—1181no—
343Q8—2362no—
344Q8D4×D5912yes—
345D4—592no—
346D4—1182no—
347D4—1181no—
348D4—1181no—
349D4—1181no—
350D4—4721no—
351D4—2362no—
352D4—2362no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—591no—
361C2×C4—591no—
362C2×C4—591no—
363C2×C4—1181no—
364C2×C4—1181no—
365C2×C4—1181no—
366C2×C4—1181no—
367C2×C4—4721no—
368C2×C4—2362no—
369C2×C4—2362no—
370C2×C4D4×D5911yescharacteristic
371C2×C4C23×D5911yescharacteristic
372C8—21no—
373C8—21no—
374C8—592no—
375C8—1181no—
376C22—21no—
377C22—21no—
378C22—1181no—
379C22—1181no—
380C22—1181no—
381C22—1182no—
382C22—4724no—
383C22—2361no—
384C22—2361no—
385C22—2361no—
386C22—2361no—
387C22—2361no—
388C22#1427375912yescomplemented
389C22C2×D4×D5911yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—1181no—
396C4—1181no—
397C4—1181no—
398C4—2361no—
399C4—2361no—
400C4C2×D4×D5911yescharacteristic
401C4#1427377211yescharacteristic
402C2—22no—
403C2—4721no—
404C2—2361no—
405C2—2362no—
406C2#1683803411yescharacteristic
407C2C2×#1427375911yescharacteristic
408C2#1683821111yescharacteristic
subgroupG/H
centerC22C2×D4×D59
commutatorC2×C4×C59C24
FrattiniC2×C4C23×D59
FittingC59×#451C2
socleC22×C59C22×D4
hypercenter#451D59
hypocenterC59#3266
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#14273759C2semidirect
2. C22×#11745671C2semidirect
3. Q8×C22⋊D59C2semidirect
5. #16837795C2semidirect
6. #16837651C2semidirect
7. #16837641C2semidirect
8. #16837626C2semidirect
9. #16837606C2semidirect
10. #16837563C2semidirect
11. #16837555C2semidirect
12. #16837542C2semidirect
13. #16837505C2semidirect
14. #16837470C2semidirect
15. C59×#451C2semidirect
17. C2×Q8×D59C22semidirect
20. C2×#11745672C22semidirect
28. C2×#11745659C22semidirect
31. #14273725C22semidirect
37. #14273703C22semidirect
41. Q8×Dic59C22semidirect
46. #14273660C22semidirect
47. Dic59⋊C8C22semidirect
48. #14273651C22semidirect
49. #14273650C22semidirect
52. (C2×D59)⋊C8C22semidirect
53. #14273637C22semidirect
55. C2×Q16×C59C22semidirect
56. C2×Q16×C59C22semidirect
58. #103×C59C22semidirect
59. #103×C59C22semidirect
101. #11745671D4semidirect
173. C22×C59#137semidirect
215. #451D59semidirect
220. C59#3266semidirect
227. C2×Q16C2×D59semidirect
228. C2×Q16C2×D59semidirect
259. #103C2×D59semidirect
260. #103C2×D59semidirect
388. C22#14273759semidirect