Finite Group Databasebeta

← C59×#9056 #28421165

Subgroups — C59×#9056 #28421165

Order 15104.

432 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3170C211yesmaximal, characteristic
2C59×#2962C211yesmaximal, characteristic
3C59×#1986C211yesmaximal, characteristic
4C59×#1503C211yesmaximal, characteristic
5C59×#1481C211yesmaximal, characteristic
6C59×#1279C211yesmaximal, characteristic
7C59×#1278C211yesmaximal, characteristic
8C59×#533C2211yescharacteristic
9C59×#533C2211yescharacteristic
10C59×#500C411yescharacteristic, complemented
11C59×#494C411yescharacteristic, complemented
12C59×#487C411yescharacteristic, complemented
13C59×#487C411yescharacteristic, complemented
14C59×#467—22no—
15C59×#446—22no—
16C59×#446—21no—
17C8×Q8×C59—21no—
18C4×Q16×C59—22no—
19C59×#431C2211yescharacteristic
20C59×#424—21no—
21C2×C4⋊C8×C59C2211yescharacteristic
22C59×#405—21no—
23C4×M16×C59C2211yescharacteristic
24C2×C4×C8×C59C2211yescharacteristic
25C4×C4⋊C4×C59C2211yescharacteristic
26C59×#339—21no—
27C59×#339—21no—
28C59×#328—22no—
29C59×Q8⋊C8—21no—
30C59×Q8⋊C8—21no—
31C2×Q16×C59—22no—
32C2×M16×C59D411yescharacteristic
33C2×M16×C59D411yescharacteristic
34C22×C8×C59—21no—
35C22×C8×C59—21no—
36C4⋊Q8×C59C2×C411yescharacteristic
37C4⋊Q8×C59C2×C411yescharacteristic
38C4⋊Q8×C59C2×C411yescharacteristic
39C4⋊Q8×C59C2×C411yescharacteristic
40#125×C59—21no—
41#125×C59—21no—
42C22⋊Q8×C59—21no—
43C22⋊Q8×C59—21no—
44C22⋊Q8×C59—21no—
45C22⋊Q8×C59D412yes—
46C4×Q8×C59—21no—
47C4×Q8×C59—21no—
48C4×Q8×C59—21no—
49C4×Q8×C59D412yes—
50#117×C59—21no—
51#117×C59—21no—
52C2×C4⋊C4×C59—21no—
53C2×C42×C59—21no—
54C2×C42×C59C2311yescharacteristic
55#107×C59—22no—
56C4⋊C8×C59—22no—
57C4⋊C8×C59—22no—
58C4⋊C8×C59—21no—
59C4⋊C8×C59—21no—
60C4⋊C8×C59D411yescharacteristic
61C4⋊C8×C59D411yescharacteristic
62#103×C59—22no—
63#103×C59—22no—
64C22⋊C8×C59—21no—
65#97×C59—22no—
66#97×C59—21no—
67C4×C8×C59—22no—
68C4×C8×C59—21no—
69C4×C8×C59—21no—
70C4×C8×C59—21no—
71C4×C8×C59C2×C411yescharacteristic
72C4×C8×C59C2×C411yescharacteristic
73#95×C59—21no—
74C2×Q8×C59—21no—
75C2×Q8×C59—21no—
76C2×Q8×C59—21no—
77C2×Q8×C59C22⋊C412yes—
78C22×C4×C59—41no—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59#4111yescharacteristic
82Q16×C59—44no—
83M16×C59—21no—
84M16×C59—21no—
85M16×C59—21no—
86M16×C59Q1612yescomplemented
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59—21no—
90C2×C8×C59—21no—
91C2×C8×C59—21no—
92C2×C8×C59—21no—
93C2×C8×C59—42no—
94C2×C8×C59—41no—
95C2×C8×C59—41no—
96C2×C8×C59—41no—
97C4⋊C4×C59—21no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—22no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59C22⋊C412yes—
111C22⋊C4×C59—22no—
112C22⋊C4×C59—41no—
113C42×C59—22no—
114C42×C59—41no—
115C42×C59—41no—
116C42×C59—41no—
117C42×C59C22×C411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C42×C59#4111yescharacteristic
121C23×C59#12111yescharacteristic
122Q8×C59—42no—
123Q8×C59—42no—
124Q8×C59—41no—
125Q8×C59—41no—
126Q8×C59—41no—
127Q8×C59—41no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—41no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59C2×C22⋊C411yescharacteristic
139C2×C4×C59C4×D411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59#12011yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12311yescharacteristic
144C2×C4×C59C2×Q1611yescharacteristic
145C2×C4×C59#13511yescharacteristic
146C2×C4×C59#13711yescharacteristic
147C2×C4×C59#13711yescharacteristic
148C8×C59—22no—
149C8×C59—22no—
150C8×C59—41no—
151C8×C59—41no—
152C8×C59—41no—
153C8×C59—41no—
154C8×C59—81no—
155#9056C5911yesmaximal, characteristic, direct product factor, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59#38611yescharacteristic
159C22×C59#46211yescharacteristic
160C22×C59#46411yescharacteristic
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—42no—
165C4×C59—42no—
166C4×C59—82no—
167C4×C59—81no—
168C4×C59—81no—
169C4×C59C4×Q1611yescharacteristic
170C4×C59#44111yescharacteristic
171C4×C59#44311yescharacteristic
172C4×C59#44411yescharacteristic
173C4×C59#45111yescharacteristic
174C4×C59#45211yescharacteristic
175#3170C2×C5911yescharacteristic
176#2962C2×C5911yescharacteristic
177#1986C2×C5911yescharacteristic
178#1503C2×C5911yescharacteristic
179#1481C2×C5911yescharacteristic
180#1279C2×C5911yescharacteristic
181#1278C2×C5911yescharacteristic
182C2×C59—21no—
183C2×C59—21no—
184C2×C59#187611yescharacteristic
185C2×C59#187811yescharacteristic
186C2×C59#188411yescharacteristic
187#533C22×C5911yescharacteristic
188#533C22×C5911yescharacteristic
189#500C4×C5911yescharacteristic, complemented
190#494C4×C5911yescharacteristic, complemented
191#487C4×C5911yescharacteristic, complemented
192#487C4×C5911yescharacteristic, complemented
193#467—22no—
194#446—22no—
195#446—21no—
196C8×Q8—21no—
197C4×Q16—22no—
198#431C22×C5911yescharacteristic
199#424—21no—
200C2×C4⋊C8C22×C5911yescharacteristic
201#405—21no—
202C4×M16C22×C5911yescharacteristic
203C2×C4×C8C22×C5911yescharacteristic
204C4×C4⋊C4C22×C5911yescharacteristic
205#339—21no—
206#339—21no—
207#328—22no—
208Q8⋊C8—21no—
209Q8⋊C8—21no—
210C59#905611yescharacteristic, direct product factor, Sylow
211C2×Q16—22no—
212C2×M16D4×C5911yescharacteristic
213C2×M16D4×C5911yescharacteristic
214C22×C8—21no—
215C22×C8—21no—
216C4⋊Q8C2×C4×C5911yescharacteristic
217C4⋊Q8C2×C4×C5911yescharacteristic
218C4⋊Q8C2×C4×C5911yescharacteristic
219C4⋊Q8C2×C4×C5911yescharacteristic
220#125—21no—
221#125—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8D4×C5912yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8D4×C5912yes—
230#117—21no—
231#117—21no—
232C2×C4⋊C4—21no—
233C2×C42—21no—
234C2×C42C23×C5911yescharacteristic
235#107—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8D4×C5911yescharacteristic
241C4⋊C8D4×C5911yescharacteristic
242#103—22no—
243#103—22no—
244C22⋊C8—21no—
245#97—22no—
246#97—21no—
247C4×C8—22no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8C2×C4×C5911yescharacteristic
252C4×C8C2×C4×C5911yescharacteristic
253#95—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8C22⋊C4×C5912yes—
258C22×C4—41no—
259C22×C4C2×D4×C5911yescharacteristic
260C22×C4C2×D4×C5911yescharacteristic
261C22×C4#41×C5911yescharacteristic
262Q16—44no—
263M16—21no—
264M16—21no—
265M16—21no—
266M16Q16×C5912yescomplemented
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4C22⋊C4×C5912yes—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C42—22no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42C22×C4×C5911yescharacteristic
298C42C2×D4×C5911yescharacteristic
299C42C2×D4×C5911yescharacteristic
300C42#41×C5911yescharacteristic
301C23#121×C5911yescharacteristic
302Q8—42no—
303Q8—42no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307Q8—41no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4C2×C22⋊C4×C5911yescharacteristic
319C2×C4C4×D4×C5911yescharacteristic
320C2×C4C4×D4×C5911yescharacteristic
321C2×C4#120×C5911yescharacteristic
322C2×C4#121×C5911yescharacteristic
323C2×C4#123×C5911yescharacteristic
324C2×C4C2×Q16×C5911yescharacteristic
325C2×C4#135×C5911yescharacteristic
326C2×C4#137×C5911yescharacteristic
327C2×C4#137×C5911yescharacteristic
328C8—22no—
329C8—22no—
330C8—41no—
331C8—41no—
332C8—41no—
333C8—41no—
334C8—81no—
335C22—21no—
336C22—21no—
337C22C59×#38611yescharacteristic
338C22C59×#46211yescharacteristic
339C22C59×#46411yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—42no—
344C4—42no—
345C4—82no—
346C4—81no—
347C4—81no—
348C4C4×Q16×C5911yescharacteristic
349C4C59×#44111yescharacteristic
350C4C59×#44311yescharacteristic
351C4C59×#44411yescharacteristic
352C4C59×#45111yescharacteristic
353C4C59×#45211yescharacteristic
354C2—21no—
355C2—21no—
356C2C59×#187611yescharacteristic
357C2C59×#187811yescharacteristic
358C2C59×#188411yescharacteristic
subgroupG/H
centerC2×C4×C59#120
commutatorC42C22×C4×C59
Frattini, cosocleC2×C42C23×C59
socleC22×C59#386
prime radicalC23×C59#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
10. C59×#500C4semidirect
11. C59×#494C4semidirect
12. C59×#487C4semidirect
13. C59×#487C4semidirect
86. M16×C59Q16semidirect
155. #9056C59direct (N × H)
189. #500C4×C59semidirect
190. #494C4×C59semidirect
191. #487C4×C59semidirect
192. #487C4×C59semidirect
210. C59#9056direct (N × H)
266. M16Q16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#31702
C59×#29622
C59×#19862
C59×#15032
C59×#14812
C59×#12792
C59×#12782
#905659