Finite Group Databasebeta

← C59×#33808 #28445917

Subgroups — C59×#33808 #28445917

Order 15104.

442 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3177C211yesmaximal, characteristic
2C59×#3164C211yesmaximal, characteristic
3C59×#2255C211yesmaximal, characteristic, complemented
4C59×#2253C211yesmaximal, characteristic, complemented
5C59×#2245C211yesmaximal, characteristic, complemented
6C59×#2241C211yesmaximal, characteristic, complemented
7C59×#2237C211yesmaximal, characteristic
8C59×#2237C211yesmaximal, characteristic
9C59×#2236C211yesmaximal, characteristic
10C59×#2235C211yesmaximal, characteristic
11C59×#2229C211yesmaximal, characteristic, complemented
12C59×#2229C211yesmaximal, characteristic, complemented
13C59×#2227C211yesmaximal, characteristic, complemented
14C59×#2225C211yesmaximal, characteristic, complemented
15C59×#2164C211yesmaximal, characteristic
16C59×#544—21no—
17C59×#541—21no—
18C59×#501—21no—
19C59×#501—21no—
20C59×#500C2211yescharacteristic
21C59×#500C2211yescharacteristic
22C59×#499C2211yescharacteristic
23C59×#499C2211yescharacteristic
24C59×#499C2211yescharacteristic
25C59×#494C2211yescharacteristic
26C59×#491—21no—
27C59×#470—21no—
28C59×#467C2211yescharacteristic
29C59×#467C2211yescharacteristic
30C59×#441—21no—
31C4×Q16×C59C2211yescharacteristic
32C4×Q16×C59C2211yescharacteristic
33C59×#428—21no—
34C59×#427C2211yescharacteristic
35C2×#107×C59C2211yescharacteristic
36C59×#405C2211yescharacteristic
37C59×#367C2211yescharacteristic
38C59×#367C2211yescharacteristic
39C59×#367C2211yescharacteristic
40C59×#367C2211yescharacteristic
41C59×#366C2211yescharacteristic
42C59×#366C2211yescharacteristic
43C59×#366C2211yescharacteristic
44C59×#366C2211yescharacteristic
45C59×C4⋊C16C2211yescharacteristic
46C59×C4⋊C16C2211yescharacteristic
47C59×#358C2211yescharacteristic
48C59×#358C2211yescharacteristic
49C59×#358C2211yescharacteristic
50C59×#358C2211yescharacteristic
51C59×#358C2211yescharacteristic
52C59×#358C2211yescharacteristic
53C59×#358C2211yescharacteristic
54C59×#358C2211yescharacteristic
55C59×C22⋊C16C2211yescharacteristic
56C59×C22⋊C16C2211yescharacteristic
57C59×#346C2211yescharacteristic
58C4×C16×C59C2211yescharacteristic
59C2×Q16×C59—21no—
60C2×Q16×C59C2311yescharacteristic
61C2×Q16×C59C2311yescharacteristic
62C2×M16×C59D411yescharacteristic
63C22×C8×C59C2311yescharacteristic
64C4⋊Q8×C59—21no—
65C4⋊Q8×C59—21no—
66C4⋊Q8×C59—21no—
67C4⋊Q8×C59—41no—
68#126×C59—41no—
69#126×C59—41no—
70#125×C59—21no—
71#125×C59—21no—
72#125×C59—21no—
73#125×C59—41no—
74#125×C59—41no—
75#125×C59—41no—
76C22⋊Q8×C59—21no—
77C22⋊Q8×C59—21no—
78C22⋊Q8×C59—41no—
79C22⋊Q8×C59—41no—
80C4×Q8×C59—21no—
81C4×Q8×C59—21no—
82#117×C59—21no—
83#117×C59D411yescharacteristic
84C2×C4⋊C4×C59—21no—
85C2×C16×C59C2311yescharacteristic
86C2×C16×C59C2311yescharacteristic
87C2×C16×C59C2311yescharacteristic
88C2×C16×C59C2311yescharacteristic
89#107×C59—21no—
90#107×C59—21no—
91#107×C59—21no—
92#107×C59C2311yescharacteristic
93#107×C59C2311yescharacteristic
94#107×C59C2311yescharacteristic
95#107×C59C2311yescharacteristic
96#107×C59C2311yescharacteristic
97#107×C59C2311yescharacteristic
98#106×C59—21no—
99#106×C59—21no—
100#106×C59—21no—
101#106×C59—21no—
102#103×C59—21no—
103#103×C59—21no—
104#103×C59—21no—
105#103×C59—21no—
106#97×C59D411yescharacteristic
107#97×C59D411yescharacteristic
108C4×C8×C59C2311yescharacteristic
109C4×C8×C59C2311yescharacteristic
110C2×Q8×C59—21no—
111C2×Q8×C59—21no—
112C2×Q8×C59—41no—
113C2×Q8×C59—41no—
114C22×C4×C59—41no—
115C22×C4×C59C2×D411yescharacteristic
116Q16×C59—21no—
117Q16×C59—21no—
118Q16×C59—21no—
119Q16×C59—21no—
120Q16×C59—42no—
121M16×C59—22no—
122C2×C8×C59—21no—
123C2×C8×C59—21no—
124C2×C8×C59C2×D411yescharacteristic
125C2×C8×C59C2×D411yescharacteristic
126C2×C8×C59C2×D411yescharacteristic
127C2×C8×C59C2411yescharacteristic
128C4⋊C4×C59—21no—
129C4⋊C4×C59—21no—
130C4⋊C4×C59—21no—
131C4⋊C4×C59—21no—
132C4⋊C4×C59—21no—
133C4⋊C4×C59—21no—
134C4⋊C4×C59—41no—
135C4⋊C4×C59—41no—
136C4⋊C4×C59—41no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59—41no—
140C4⋊C4×C59—41no—
141C4⋊C4×C59—41no—
142C4⋊C4×C59—41no—
143C4⋊C4×C59—41no—
144C4⋊C4×C59—41no—
145C4⋊C4×C59—41no—
146C4⋊C4×C59—41no—
147C4⋊C4×C59—41no—
148C4⋊C4×C59—41no—
149C4⋊C4×C59D811yescharacteristic
150C4⋊C4×C59D811yescharacteristic
151C22⋊C4×C59—41no—
152C22⋊C4×C59—41no—
153C22⋊C4×C59—41no—
154C22⋊C4×C59D811yescharacteristic
155C22⋊C4×C59D811yescharacteristic
156C42×C59—41no—
157C42×C59—41no—
158C42×C59—41no—
159C42×C59C2×D411yescharacteristic
160C42×C59C2×D411yescharacteristic
161C16×C59—21no—
162C16×C59—21no—
163C16×C59—21no—
164C16×C59—21no—
165C23×C59C2×D811yescharacteristic
166Q8×C59—41no—
167Q8×C59—41no—
168Q8×C59—41no—
169Q8×C59—41no—
170Q8×C59—161no—
171Q8×C59—161no—
172C2×C4×C59—21no—
173C2×C4×C59—21no—
174C2×C4×C59—41no—
175C2×C4×C59—41no—
176C2×C4×C59—41no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59—161no—
183C2×C4×C59C2×D811yescharacteristic
184C2×C4×C59C2×D811yescharacteristic
185C2×C4×C59C2×D811yescharacteristic
186C2×C4×C59C2×D811yescharacteristic
187C2×C4×C59C2×D811yescharacteristic
188C2×C4×C59C22×D411yescharacteristic
189C8×C59—21no—
190C8×C59—21no—
191C8×C59#13712yes—
192C8×C592−1+411yescharacteristic
193C8×C592−1+411yescharacteristic
194#33808C5911yesmaximal, characteristic, direct product factor, Sylow
195C22×C59—21no—
196C22×C59—21no—
197C22×C59—21no—
198C22×C59C22×D811yescharacteristic
199C4×C59—21no—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—21no—
204C4×C59—81no—
205C4×C59—81no—
206C4×C59—81no—
207C4×C59—81no—
208C4×C59—81no—
209C4×C59—81no—
210C4×C59—81no—
211C4×C59—81no—
212C4×C59#53611yescharacteristic
213C4×C59C2×#13711yescharacteristic
214#3177C2×C5911yescharacteristic
215#3164C2×C5911yescharacteristic
216#2255C2×C5911yescharacteristic, complemented
217#2253C2×C5911yescharacteristic, complemented
218#2245C2×C5911yescharacteristic, complemented
219#2241C2×C5911yescharacteristic, complemented
220#2237C2×C5911yescharacteristic
221#2237C2×C5911yescharacteristic
222#2236C2×C5911yescharacteristic
223#2235C2×C5911yescharacteristic
224#2229C2×C5911yescharacteristic, complemented
225#2229C2×C5911yescharacteristic, complemented
226#2227C2×C5911yescharacteristic, complemented
227#2225C2×C5911yescharacteristic, complemented
228#2164C2×C5911yescharacteristic
229C2×C59—41no—
230C2×C59#322611yescharacteristic
231C2×C59#341611yescharacteristic
232C2×C59#341711yescharacteristic
233#544—21no—
234#541—21no—
235#501—21no—
236#501—21no—
237#500C22×C5911yescharacteristic
238#500C22×C5911yescharacteristic
239#499C22×C5911yescharacteristic
240#499C22×C5911yescharacteristic
241#499C22×C5911yescharacteristic
242#494C22×C5911yescharacteristic
243#491—21no—
244#470—21no—
245#467C22×C5911yescharacteristic
246#467C22×C5911yescharacteristic
247#441—21no—
248C4×Q16C22×C5911yescharacteristic
249C4×Q16C22×C5911yescharacteristic
250#428—21no—
251#427C22×C5911yescharacteristic
252C2×#107C22×C5911yescharacteristic
253#405C22×C5911yescharacteristic
254#367C22×C5911yescharacteristic
255#367C22×C5911yescharacteristic
256#367C22×C5911yescharacteristic
257#367C22×C5911yescharacteristic
258#366C22×C5911yescharacteristic
259#366C22×C5911yescharacteristic
260#366C22×C5911yescharacteristic
261#366C22×C5911yescharacteristic
262C4⋊C16C22×C5911yescharacteristic
263C4⋊C16C22×C5911yescharacteristic
264#358C22×C5911yescharacteristic
265#358C22×C5911yescharacteristic
266#358C22×C5911yescharacteristic
267#358C22×C5911yescharacteristic
268#358C22×C5911yescharacteristic
269#358C22×C5911yescharacteristic
270#358C22×C5911yescharacteristic
271#358C22×C5911yescharacteristic
272C22⋊C16C22×C5911yescharacteristic
273C22⋊C16C22×C5911yescharacteristic
274#346C22×C5911yescharacteristic
275C4×C16C22×C5911yescharacteristic
276C59#3380811yescharacteristic, direct product factor, Sylow
277C2×Q16—21no—
278C2×Q16C23×C5911yescharacteristic
279C2×Q16C23×C5911yescharacteristic
280C2×M16D4×C5911yescharacteristic
281C22×C8C23×C5911yescharacteristic
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—41no—
286#126—41no—
287#126—41no—
288#125—21no—
289#125—21no—
290#125—21no—
291#125—41no—
292#125—41no—
293#125—41no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—41no—
297C22⋊Q8—41no—
298C4×Q8—21no—
299C4×Q8—21no—
300#117—21no—
301#117D4×C5911yescharacteristic
302C2×C4⋊C4—21no—
303C2×C16C23×C5911yescharacteristic
304C2×C16C23×C5911yescharacteristic
305C2×C16C23×C5911yescharacteristic
306C2×C16C23×C5911yescharacteristic
307#107—21no—
308#107—21no—
309#107—21no—
310#107C23×C5911yescharacteristic
311#107C23×C5911yescharacteristic
312#107C23×C5911yescharacteristic
313#107C23×C5911yescharacteristic
314#107C23×C5911yescharacteristic
315#107C23×C5911yescharacteristic
316#106—21no—
317#106—21no—
318#106—21no—
319#106—21no—
320#103—21no—
321#103—21no—
322#103—21no—
323#103—21no—
324#97D4×C5911yescharacteristic
325#97D4×C5911yescharacteristic
326C4×C8C23×C5911yescharacteristic
327C4×C8C23×C5911yescharacteristic
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C22×C4—41no—
333C22×C4C2×D4×C5911yescharacteristic
334Q16—21no—
335Q16—21no—
336Q16—21no—
337Q16—21no—
338Q16—42no—
339M16—22no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8C2×D4×C5911yescharacteristic
343C2×C8C2×D4×C5911yescharacteristic
344C2×C8C2×D4×C5911yescharacteristic
345C2×C8C24×C5911yescharacteristic
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4D8×C5911yescharacteristic
368C4⋊C4D8×C5911yescharacteristic
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4D8×C5911yescharacteristic
373C22⋊C4D8×C5911yescharacteristic
374C42—41no—
375C42—41no—
376C42—41no—
377C42C2×D4×C5911yescharacteristic
378C42C2×D4×C5911yescharacteristic
379C16—21no—
380C16—21no—
381C16—21no—
382C16—21no—
383C23C2×D8×C5911yescharacteristic
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—41no—
388Q8—161no—
389Q8—161no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—161no—
401C2×C4C2×D8×C5911yescharacteristic
402C2×C4C2×D8×C5911yescharacteristic
403C2×C4C2×D8×C5911yescharacteristic
404C2×C4C2×D8×C5911yescharacteristic
405C2×C4C2×D8×C5911yescharacteristic
406C2×C4C22×D4×C5911yescharacteristic
407C8—21no—
408C8—21no—
409C8#137×C5912yes—
410C82−1+4×C5911yescharacteristic
411C82−1+4×C5911yescharacteristic
412C22—21no—
413C22—21no—
414C22—21no—
415C22C22×D8×C5911yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—81no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4C59×#53611yescharacteristic
430C4C2×#137×C5911yescharacteristic
431C2—41no—
432C2C59×#322611yescharacteristic
433C2C59×#341611yescharacteristic
434C2C59×#341711yescharacteristic
subgroupG/H
center, socleC22×C59C22×D8
commutator, Frattini, cosocleC2×C8C24×C59
prime radicalC23×C59C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C59 ◁ #117×C59 ◁ C59×#405 ◁ G

Complements

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

NH ≅ G/Ntype
3. C59×#2255C2semidirect
4. C59×#2253C2semidirect
5. C59×#2245C2semidirect
6. C59×#2241C2semidirect
11. C59×#2229C2semidirect
12. C59×#2229C2semidirect
13. C59×#2227C2semidirect
14. C59×#2225C2semidirect
194. #33808C59direct (N × H)
216. #2255C2×C59semidirect
217. #2253C2×C59semidirect
218. #2245C2×C59semidirect
219. #2241C2×C59semidirect
224. #2229C2×C59semidirect
225. #2229C2×C59semidirect
226. #2227C2×C59semidirect
227. #2225C2×C59semidirect
276. C59#33808direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#31772
C59×#31642
C59×#22552
C59×#22532
C59×#22452
C59×#22412
C59×#22372
C59×#22362
C59×#22352
C59×#22292
C59×#22272
C59×#22252
C59×#21642
#3380859