Finite Group Databasebeta

← C59×#3322 #21421640

Subgroups — C59×#3322 #21421640

Order 7552.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#553C211yesmaximal, characteristic, complemented
2C59×#546C211yesmaximal, characteristic, complemented
3C59×#488C211yesmaximal, characteristic, complemented
4C59×#485C211yesmaximal, characteristic, complemented
5C59×#480C211yesmaximal, characteristic, complemented
6C59×#478C211yesmaximal, characteristic, complemented
7C59×#469C211yesmaximal, characteristic, complemented
8C59×#468C211yesmaximal, characteristic, complemented
9C59×#463C211yesmaximal, characteristic, complemented
10C59×#449C211yesmaximal, characteristic, complemented
11C59×#447C211yesmaximal, characteristic, complemented
12C4×D8×C59C211yesmaximal, characteristic, complemented
13C59×#436C211yesmaximal, characteristic, complemented
14C59×#418C211yesmaximal, characteristic, complemented
15C59×#417C211yesmaximal, characteristic, complemented
16C2×#41×C59C2211yescharacteristic
17C22×D4×C59C2211yescharacteristic
18C2×SD16×C59C2211yescharacteristic, complemented
19C2×D8×C59C2211yescharacteristic
20C2×D8×C59C2211yescharacteristic
21C2×M16×C59C2211yescharacteristic
22C2×M16×C59C2211yescharacteristic, complemented
23#126×C59—21no—
24#125×C59C2211yescharacteristic, complemented
25#124×C59C2211yescharacteristic
26#123×C59—21no—
27#123×C59—21no—
28C22⋊Q8×C59C2211yescharacteristic
29#121×C59—21no—
30#121×C59—21no—
31#121×C59C2211yescharacteristic
32#121×C59C2211yescharacteristic
33#121×C59C2211yescharacteristic, complemented
34#120×C59—21no—
35C4×D4×C59—21no—
36C4×D4×C59C2211yescharacteristic
37C4×D4×C59C2211yescharacteristic
38C4×D4×C59C2211yescharacteristic, complemented
39#117×C59C2211yescharacteristic
40C2×C4⋊C4×C59C2211yescharacteristic, complemented
41C2×C22⋊C4×C59—21no—
42#107×C59C2211yescharacteristic, complemented
43#107×C59C2211yescharacteristic, complemented
44#106×C59C2211yescharacteristic, complemented
45C4⋊C8×C59C2211yescharacteristic, complemented
46#103×C59C2211yescharacteristic, complemented
47#103×C59C2211yescharacteristic, complemented
48#103×C59C2211yescharacteristic, complemented
49#102×C59C2211yescharacteristic
50#102×C59C2211yescharacteristic
51#102×C59C2211yescharacteristic
52#102×C59C2211yescharacteristic
53#102×C59C2211yescharacteristic
54#102×C59C2211yescharacteristic
55#102×C59C2211yescharacteristic, complemented
56C22⋊C8×C59C2211yescharacteristic
57C22⋊C8×C59C2211yescharacteristic, complemented
58C4×C8×C59C2211yescharacteristic, complemented
59C24×C59—21no—
60#41×C59—22no—
61#41×C59—41no—
62C2×Q8×C59C2311yescharacteristic
63C2×D4×C59—22no—
64C2×D4×C59—21no—
65C2×D4×C59—21no—
66C2×D4×C59—21no—
67C2×D4×C59—21no—
68C2×D4×C59—21no—
69C2×D4×C59—41no—
70C2×D4×C59D411yescharacteristic
71C2×D4×C59C2311yescharacteristic
72C2×D4×C59C2311yescharacteristic
73C2×D4×C59C2311yescharacteristic
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59C2311yescharacteristic
79C22×C4×C59C2311yescharacteristic
80SD16×C59—41no—
81D8×C59—22no—
82D8×C59—41no—
83M16×C59—41no—
84M16×C59—41no—
85C2×C8×C59C2311yescharacteristic
86C2×C8×C59C2311yescharacteristic
87C2×C8×C59C2311yescharacteristic
88C2×C8×C59C2311yescharacteristic
89C4⋊C4×C59—21no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—21no—
93C4⋊C4×C59D411yescharacteristic
94C4⋊C4×C59C2311yescharacteristic
95C4⋊C4×C59C2311yescharacteristic
96C4⋊C4×C59C2311yescharacteristic
97C4⋊C4×C59C2311yescharacteristic
98C22⋊C4×C59—21no—
99C22⋊C4×C59—21no—
100C22⋊C4×C59—21no—
101C22⋊C4×C59—21no—
102C22⋊C4×C59—21no—
103C22⋊C4×C59—21no—
104C22⋊C4×C59—21no—
105C22⋊C4×C59—21no—
106C22⋊C4×C59—21no—
107C22⋊C4×C59D411yescharacteristic
108C22⋊C4×C59D411yescharacteristic
109C42×C59—21no—
110C42×C59C2311yescharacteristic
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—42no—
115C23×C59—42no—
116C23×C59—81no—
117C23×C59C2×D411yescharacteristic
118C23×C59C2×D411yescharacteristic
119Q8×C59—21no—
120Q8×C59—21no—
121D4×C59—21no—
122D4×C59—21no—
123D4×C59—21no—
124D4×C59—21no—
125D4×C59—21no—
126D4×C59—41no—
127D4×C59—42no—
128D4×C59—42no—
129D4×C59—42no—
130D4×C59—81no—
131D4×C59—81no—
132D4×C59#4112yes—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—42no—
143C2×C4×C59—81no—
144C2×C4×C59—81no—
145C2×C4×C59—81no—
146C2×C4×C59C2×D411yescharacteristic
147C2×C4×C59C2×D411yescharacteristic
148C2×C4×C59C2×D411yescharacteristic
149C2×C4×C59C2×D411yescharacteristic
150C2×C4×C59C2411yescharacteristic
151C8×C59—21no—
152C8×C59—21no—
153C8×C59—21no—
154C8×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—22no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—84no—
171C22×C59C22×D411yescharacteristic
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59—41no—
177C4×C59—41no—
178C4×C59—41no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59C2×#4111yescharacteristic
182C4×C59H(5,𝔽2)11yescharacteristic
183#3322C5911yesmaximal, characteristic, direct product factor, Sylow
184C2×C59—41no—
185C2×C59—41no—
186C2×C59—42no—
187C2×C59—81no—
188C2×C59—81no—
189C2×C59#54611yescharacteristic
190C2×C59#57511yescharacteristic
191C2×C59#57611yescharacteristic
192#553C2×C5911yescharacteristic, complemented
193#546C2×C5911yescharacteristic, complemented
194#488C2×C5911yescharacteristic, complemented
195#485C2×C5911yescharacteristic, complemented
196#480C2×C5911yescharacteristic, complemented
197#478C2×C5911yescharacteristic, complemented
198#469C2×C5911yescharacteristic, complemented
199#468C2×C5911yescharacteristic, complemented
200#463C2×C5911yescharacteristic, complemented
201#449C2×C5911yescharacteristic, complemented
202#447C2×C5911yescharacteristic, complemented
203C4×D8C2×C5911yescharacteristic, complemented
204#436C2×C5911yescharacteristic, complemented
205#418C2×C5911yescharacteristic, complemented
206#417C2×C5911yescharacteristic, complemented
207C59#332211yescharacteristic, direct product factor, Sylow
208C2×#41C22×C5911yescharacteristic
209C22×D4C22×C5911yescharacteristic
210C2×SD16C22×C5911yescharacteristic, complemented
211C2×D8C22×C5911yescharacteristic
212C2×D8C22×C5911yescharacteristic
213C2×M16C22×C5911yescharacteristic
214C2×M16C22×C5911yescharacteristic, complemented
215#126—21no—
216#125C22×C5911yescharacteristic, complemented
217#124C22×C5911yescharacteristic
218#123—21no—
219#123—21no—
220C22⋊Q8C22×C5911yescharacteristic
221#121—21no—
222#121—21no—
223#121C22×C5911yescharacteristic
224#121C22×C5911yescharacteristic
225#121C22×C5911yescharacteristic, complemented
226#120—21no—
227C4×D4—21no—
228C4×D4C22×C5911yescharacteristic
229C4×D4C22×C5911yescharacteristic
230C4×D4C22×C5911yescharacteristic, complemented
231#117C22×C5911yescharacteristic
232C2×C4⋊C4C22×C5911yescharacteristic, complemented
233C2×C22⋊C4—21no—
234#107C22×C5911yescharacteristic, complemented
235#107C22×C5911yescharacteristic, complemented
236#106C22×C5911yescharacteristic, complemented
237C4⋊C8C22×C5911yescharacteristic, complemented
238#103C22×C5911yescharacteristic, complemented
239#103C22×C5911yescharacteristic, complemented
240#103C22×C5911yescharacteristic, complemented
241#102C22×C5911yescharacteristic
242#102C22×C5911yescharacteristic
243#102C22×C5911yescharacteristic
244#102C22×C5911yescharacteristic
245#102C22×C5911yescharacteristic
246#102C22×C5911yescharacteristic
247#102C22×C5911yescharacteristic, complemented
248C22⋊C8C22×C5911yescharacteristic
249C22⋊C8C22×C5911yescharacteristic, complemented
250C4×C8C22×C5911yescharacteristic, complemented
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C23×C5911yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4D4×C5911yescharacteristic
263C2×D4C23×C5911yescharacteristic
264C2×D4C23×C5911yescharacteristic
265C2×D4C23×C5911yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4C23×C5911yescharacteristic
271C22×C4C23×C5911yescharacteristic
272SD16—41no—
273D8—22no—
274D8—41no—
275M16—41no—
276M16—41no—
277C2×C8C23×C5911yescharacteristic
278C2×C8C23×C5911yescharacteristic
279C2×C8C23×C5911yescharacteristic
280C2×C8C23×C5911yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4D4×C5911yescharacteristic
286C4⋊C4C23×C5911yescharacteristic
287C4⋊C4C23×C5911yescharacteristic
288C4⋊C4C23×C5911yescharacteristic
289C4⋊C4C23×C5911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4D4×C5911yescharacteristic
300C22⋊C4D4×C5911yescharacteristic
301C42—21no—
302C42C23×C5911yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23C2×D4×C5911yescharacteristic
310C23C2×D4×C5911yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—41no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
324D4#41×C5912yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4C2×D4×C5911yescharacteristic
339C2×C4C2×D4×C5911yescharacteristic
340C2×C4C2×D4×C5911yescharacteristic
341C2×C4C2×D4×C5911yescharacteristic
342C2×C4C24×C5911yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—84no—
363C22C22×D4×C5911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4C2×#41×C5911yescharacteristic
374C4H(5,𝔽2)×C5911yescharacteristic
375C2—41no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—81no—
380C2C59×#54611yescharacteristic
381C2C59×#57511yescharacteristic
382C2C59×#57611yescharacteristic
subgroupG/H
center, socleC22×C59C22×D4
commutator, Frattini, cosocleC2×C4C24×C59
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C4×D4×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#553C2semidirect
2. C59×#546C2semidirect
3. C59×#488C2semidirect
4. C59×#485C2semidirect
5. C59×#480C2semidirect
6. C59×#478C2semidirect
7. C59×#469C2semidirect
8. C59×#468C2semidirect
9. C59×#463C2semidirect
10. C59×#449C2semidirect
11. C59×#447C2semidirect
12. C4×D8×C59C2semidirect
13. C59×#436C2semidirect
14. C59×#418C2semidirect
15. C59×#417C2semidirect
18. C2×SD16×C59C22semidirect
22. C2×M16×C59C22semidirect
24. #125×C59C22semidirect
33. #121×C59C22semidirect
38. C4×D4×C59C22semidirect
40. C2×C4⋊C4×C59C22semidirect
42. #107×C59C22semidirect
43. #107×C59C22semidirect
44. #106×C59C22semidirect
45. C4⋊C8×C59C22semidirect
46. #103×C59C22semidirect
47. #103×C59C22semidirect
48. #103×C59C22semidirect
55. #102×C59C22semidirect
57. C22⋊C8×C59C22semidirect
58. C4×C8×C59C22semidirect
183. #3322C59direct (N × H)
192. #553C2×C59semidirect
193. #546C2×C59semidirect
194. #488C2×C59semidirect
195. #485C2×C59semidirect
196. #480C2×C59semidirect
197. #478C2×C59semidirect
198. #469C2×C59semidirect
199. #468C2×C59semidirect
200. #463C2×C59semidirect
201. #449C2×C59semidirect
202. #447C2×C59semidirect
203. C4×D8C2×C59semidirect
204. #436C2×C59semidirect
205. #418C2×C59semidirect
206. #417C2×C59semidirect
207. C59#3322direct (N × H)
210. C2×SD16C22×C59semidirect
214. C2×M16C22×C59semidirect
216. #125C22×C59semidirect
225. #121C22×C59semidirect
230. C4×D4C22×C59semidirect
232. C2×C4⋊C4C22×C59semidirect
234. #107C22×C59semidirect
235. #107C22×C59semidirect
236. #106C22×C59semidirect
237. C4⋊C8C22×C59semidirect
238. #103C22×C59semidirect
239. #103C22×C59semidirect
240. #103C22×C59semidirect
247. #102C22×C59semidirect
249. C22⋊C8C22×C59semidirect
250. C4×C8C22×C59semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C59×#5532
C59×#5462
C59×#4882
C59×#4852
C59×#4802
C59×#4782
C59×#4692
C59×#4682
C59×#4632
C59×#4492
C59×#4472
C4×D8×C592
C59×#4362
C59×#4182
C59×#4172
#332259