Finite Group Databasebeta

← C59×#3201 #21421519

Subgroups — C59×#3201 #21421519

Order 7552.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#537C211yesmaximal, characteristic
2C59×#536C211yesmaximal, characteristic, complemented
3C59×#485C211yesmaximal, characteristic, complemented
4C59×#484C211yesmaximal, characteristic, complemented
5C59×#483C211yesmaximal, characteristic, complemented
6C59×#482C211yesmaximal, characteristic, complemented
7C59×#470C211yesmaximal, characteristic, complemented
8C59×#468C211yesmaximal, characteristic, complemented
9C59×#467C211yesmaximal, characteristic, complemented
10C59×#465C211yesmaximal, characteristic, complemented
11C59×#452C211yesmaximal, characteristic, complemented
12C59×#452C211yesmaximal, characteristic, complemented
13C59×#451C211yesmaximal, characteristic, complemented
14C59×#448C211yesmaximal, characteristic, complemented
15C59×#408C211yesmaximal, characteristic, complemented
16C2×#41×C59C2211yescharacteristic
17C2×#41×C59C2211yescharacteristic
18C22×Q8×C59C2211yescharacteristic
19C2×Q16×C59C2211yescharacteristic
20C2×Q16×C59C2211yescharacteristic
21C2×SD16×C59C2211yescharacteristic
22C2×SD16×C59C2211yescharacteristic
23C2×M16×C59C2211yescharacteristic
24C22×C8×C59C2211yescharacteristic
25C4⋊Q8×C59—21no—
26#124×C59—21no—
27#123×C59—21no—
28#123×C59—21no—
29C22⋊Q8×C59—21no—
30C22⋊Q8×C59C2211yescharacteristic
31C22⋊Q8×C59C2211yescharacteristic
32C22⋊Q8×C59C2211yescharacteristic
33C22⋊Q8×C59C2211yescharacteristic
34C22⋊Q8×C59C2211yescharacteristic
35C22⋊Q8×C59C2211yescharacteristic
36#121×C59—21no—
37#121×C59—21no—
38#121×C59—21no—
39#121×C59C2211yescharacteristic
40#121×C59C2211yescharacteristic
41#117×C59C2211yescharacteristic
42C2×C4⋊C4×C59C2211yescharacteristic
43#107×C59C2211yescharacteristic
44#107×C59C2211yescharacteristic
45#106×C59C2211yescharacteristic
46#106×C59C2211yescharacteristic
47#103×C59C2211yescharacteristic
48#103×C59C2211yescharacteristic
49#103×C59C2211yescharacteristic
50#103×C59C2211yescharacteristic
51#103×C59C2211yescharacteristic
52#103×C59C2211yescharacteristic
53#102×C59C2211yescharacteristic
54#102×C59C2211yescharacteristic
55C22⋊C8×C59C2211yescharacteristic
56C22⋊C8×C59C2211yescharacteristic
57C22⋊C8×C59C2211yescharacteristic
58C22⋊C8×C59C2211yescharacteristic
59#41×C59—41no—
60#41×C59—41no—
61#41×C59—41no—
62#41×C59—41no—
63C2×Q8×C59—21no—
64C2×Q8×C59—41no—
65C2×Q8×C59—41no—
66C2×Q8×C59D411yescharacteristic
67C2×Q8×C59C2311yescharacteristic
68C2×Q8×C59C2311yescharacteristic
69C2×Q8×C59C2311yescharacteristic
70C2×D4×C59—21no—
71C2×D4×C59—21no—
72C2×D4×C59—21no—
73C2×D4×C59D411yescharacteristic
74C2×D4×C59D411yescharacteristic
75C2×D4×C59D411yescharacteristic
76C2×D4×C59C2311yescharacteristic
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59C2311yescharacteristic
81C22×C4×C59C2311yescharacteristic
82C22×C4×C59C2311yescharacteristic
83Q16×C59—41no—
84Q16×C59—41no—
85SD16×C59—41no—
86SD16×C59—41no—
87M16×C59—41no—
88C2×C8×C59—41no—
89C2×C8×C59C2311yescharacteristic
90C2×C8×C59C2311yescharacteristic
91C2×C8×C59C2311yescharacteristic
92C2×C8×C59C2311yescharacteristic
93C4⋊C4×C59—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59C2311yescharacteristic
101C4⋊C4×C59C2311yescharacteristic
102C4⋊C4×C59C2311yescharacteristic
103C4⋊C4×C59C2311yescharacteristic
104C22⋊C4×C59—21no—
105C22⋊C4×C59—21no—
106C22⋊C4×C59—21no—
107C22⋊C4×C59—21no—
108C22⋊C4×C59—21no—
109C22⋊C4×C59—21no—
110C22⋊C4×C59—21no—
111C22⋊C4×C59—21no—
112C22⋊C4×C59—21no—
113C42×C59—21no—
114C23×C59—21no—
115C23×C59C2×D411yescharacteristic
116C23×C59C2×D411yescharacteristic
117C23×C59C2×D411yescharacteristic
118Q8×C59—21no—
119Q8×C59—21no—
120Q8×C59—21no—
121Q8×C59—21no—
122Q8×C59—21no—
123Q8×C59—21no—
124Q8×C59—41no—
125Q8×C59—42no—
126D4×C59—21no—
127D4×C59—21no—
128D4×C59—41no—
129D4×C59—41no—
130D4×C59—41no—
131D4×C59—42no—
132D4×C59—81no—
133D4×C59—81no—
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—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59—81no—
150C2×C4×C59C2×D411yescharacteristic
151C2×C4×C59C2×D411yescharacteristic
152C2×C4×C59C2×D411yescharacteristic
153C2×C4×C59C2411yescharacteristic
154C8×C59—21no—
155C8×C59—21no—
156C8×C59—21no—
157C8×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59C22×D411yescharacteristic
171C4×C59—21no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—41no—
175C4×C59—41no—
176C4×C59—41no—
177C4×C59—41no—
178C4×C59—41no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59H(5,𝔽2)11yescharacteristic
182C4×C59H(5,𝔽2)11yescharacteristic
183#3201C5911yesmaximal, characteristic, direct product factor, Sylow
184C2×C59—41no—
185C2×C59—41no—
186C2×C59—41no—
187C2×C59—81no—
188C2×C59#53411yescharacteristic
189C2×C59#57711yescharacteristic
190C2×C59#57811yescharacteristic
191#537C2×C5911yescharacteristic
192#536C2×C5911yescharacteristic, complemented
193#485C2×C5911yescharacteristic, complemented
194#484C2×C5911yescharacteristic, complemented
195#483C2×C5911yescharacteristic, complemented
196#482C2×C5911yescharacteristic, complemented
197#470C2×C5911yescharacteristic, complemented
198#468C2×C5911yescharacteristic, complemented
199#467C2×C5911yescharacteristic, complemented
200#465C2×C5911yescharacteristic, complemented
201#452C2×C5911yescharacteristic, complemented
202#452C2×C5911yescharacteristic, complemented
203#451C2×C5911yescharacteristic, complemented
204#448C2×C5911yescharacteristic, complemented
205#408C2×C5911yescharacteristic, complemented
206C59#320111yescharacteristic, direct product factor, Sylow
207C2×#41C22×C5911yescharacteristic
208C2×#41C22×C5911yescharacteristic
209C22×Q8C22×C5911yescharacteristic
210C2×Q16C22×C5911yescharacteristic
211C2×Q16C22×C5911yescharacteristic
212C2×SD16C22×C5911yescharacteristic
213C2×SD16C22×C5911yescharacteristic
214C2×M16C22×C5911yescharacteristic
215C22×C8C22×C5911yescharacteristic
216C4⋊Q8—21no—
217#124—21no—
218#123—21no—
219#123—21no—
220C22⋊Q8—21no—
221C22⋊Q8C22×C5911yescharacteristic
222C22⋊Q8C22×C5911yescharacteristic
223C22⋊Q8C22×C5911yescharacteristic
224C22⋊Q8C22×C5911yescharacteristic
225C22⋊Q8C22×C5911yescharacteristic
226C22⋊Q8C22×C5911yescharacteristic
227#121—21no—
228#121—21no—
229#121—21no—
230#121C22×C5911yescharacteristic
231#121C22×C5911yescharacteristic
232#117C22×C5911yescharacteristic
233C2×C4⋊C4C22×C5911yescharacteristic
234#107C22×C5911yescharacteristic
235#107C22×C5911yescharacteristic
236#106C22×C5911yescharacteristic
237#106C22×C5911yescharacteristic
238#103C22×C5911yescharacteristic
239#103C22×C5911yescharacteristic
240#103C22×C5911yescharacteristic
241#103C22×C5911yescharacteristic
242#103C22×C5911yescharacteristic
243#103C22×C5911yescharacteristic
244#102C22×C5911yescharacteristic
245#102C22×C5911yescharacteristic
246C22⋊C8C22×C5911yescharacteristic
247C22⋊C8C22×C5911yescharacteristic
248C22⋊C8C22×C5911yescharacteristic
249C22⋊C8C22×C5911yescharacteristic
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8D4×C5911yescharacteristic
258C2×Q8C23×C5911yescharacteristic
259C2×Q8C23×C5911yescharacteristic
260C2×Q8C23×C5911yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4D4×C5911yescharacteristic
265C2×D4D4×C5911yescharacteristic
266C2×D4D4×C5911yescharacteristic
267C2×D4C23×C5911yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C23×C5911yescharacteristic
272C22×C4C23×C5911yescharacteristic
273C22×C4C23×C5911yescharacteristic
274Q16—41no—
275Q16—41no—
276SD16—41no—
277SD16—41no—
278M16—41no—
279C2×C8—41no—
280C2×C8C23×C5911yescharacteristic
281C2×C8C23×C5911yescharacteristic
282C2×C8C23×C5911yescharacteristic
283C2×C8C23×C5911yescharacteristic
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4C23×C5911yescharacteristic
292C4⋊C4C23×C5911yescharacteristic
293C4⋊C4C23×C5911yescharacteristic
294C4⋊C4C23×C5911yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C42—21no—
305C23—21no—
306C23C2×D4×C5911yescharacteristic
307C23C2×D4×C5911yescharacteristic
308C23C2×D4×C5911yescharacteristic
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—21no—
315Q8—41no—
316Q8—42no—
317D4—21no—
318D4—21no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—42no—
323D4—81no—
324D4—81no—
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—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—81no—
341C2×C4C2×D4×C5911yescharacteristic
342C2×C4C2×D4×C5911yescharacteristic
343C2×C4C2×D4×C5911yescharacteristic
344C2×C4C24×C5911yescharacteristic
345C8—21no—
346C8—21no—
347C8—21no—
348C8—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22C22×D4×C5911yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4H(5,𝔽2)×C5911yescharacteristic
373C4H(5,𝔽2)×C5911yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—81no—
378C2C59×#53411yescharacteristic
379C2C59×#57711yescharacteristic
380C2C59×#57811yescharacteristic
subgroupG/H
center, socleC22×C59C22×D4
commutator, Frattini, cosocleC2×C4C24×C59
prime radicalC59×#537C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×#41×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#536C2semidirect
3. C59×#485C2semidirect
4. C59×#484C2semidirect
5. C59×#483C2semidirect
6. C59×#482C2semidirect
7. C59×#470C2semidirect
8. C59×#468C2semidirect
9. C59×#467C2semidirect
10. C59×#465C2semidirect
11. C59×#452C2semidirect
12. C59×#452C2semidirect
13. C59×#451C2semidirect
14. C59×#448C2semidirect
15. C59×#408C2semidirect
183. #3201C59direct (N × H)
192. #536C2×C59semidirect
193. #485C2×C59semidirect
194. #484C2×C59semidirect
195. #483C2×C59semidirect
196. #482C2×C59semidirect
197. #470C2×C59semidirect
198. #468C2×C59semidirect
199. #467C2×C59semidirect
200. #465C2×C59semidirect
201. #452C2×C59semidirect
202. #452C2×C59semidirect
203. #451C2×C59semidirect
204. #448C2×C59semidirect
205. #408C2×C59semidirect
206. C59#3201direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#5372
C59×#5362
C59×#4852
C59×#4842
C59×#4832
C59×#4822
C59×#4702
C59×#4682
C59×#4672
C59×#4652
C59×#4522
C59×#4512
C59×#4482
C59×#4082
#320159