Finite Group Databasebeta

← C59×#8905 #28421014

Subgroups — C59×#8905 #28421014

Order 15104.

392 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#2925C211yesmaximal, characteristic
2C59×#1806C211yesmaximal, characteristic
3C59×#1571C211yesmaximal, characteristic, complemented
4C59×#1566C211yesmaximal, characteristic, complemented
5C59×#1476C211yesmaximal, characteristic
6C59×#1303C211yesmaximal, characteristic, complemented
7C59×#1279C211yesmaximal, characteristic, complemented
8C4×#41×C59C2211yescharacteristic
9C59×#433—22no—
10C59×#432—21no—
11C59×#431—21no—
12C59×#431C2211yescharacteristic
13C59×#423C2211yescharacteristic
14C59×#423C2211yescharacteristic
15C2×C4⋊C8×C59C2211yescharacteristic
16C2×#104×C59—21no—
17C59×#408—21no—
18C59×#405—21no—
19C2×C4×C8×C59C2211yescharacteristic
20C4×C4⋊C4×C59C2211yescharacteristic
21C59×#339—21no—
22C59×#335—21no—
23C59×#335—21no—
24C59×#334—22no—
25C59×Q8⋊C8C411yescharacteristic, complemented
26C59×Q8⋊C8C411yescharacteristic, complemented
27C59×D4⋊C8C411yescharacteristic, complemented
28C59×D4⋊C8C411yescharacteristic, complemented
29C2×#41×C59D411yescharacteristic
30C2×M16×C59—21no—
31C2×M16×C59—21no—
32C22×C8×C59—21no—
33C22×C8×C59—21no—
34C4×Q8×C59C2×C411yescharacteristic
35C4×Q8×C59Q811yescharacteristic
36C4×D4×C59—21no—
37C4×D4×C59—21no—
38C4×D4×C59C2×C411yescharacteristic
39C4×D4×C59Q811yescharacteristic
40#117×C59—21no—
41#117×C59D411yescharacteristic
42C2×C4⋊C4×C59—21no—
43C2×C42×C59—21no—
44C2×C42×C59—21no—
45C2×C42×C59C2311yescharacteristic
46C4⋊C8×C59—21no—
47C4⋊C8×C59—21no—
48C4⋊C8×C59—21no—
49C4⋊C8×C59—21no—
50C4⋊C8×C59C2×C411yescharacteristic
51C4⋊C8×C59C2×C411yescharacteristic
52C4⋊C8×C59D412yes—
53C4⋊C8×C59D411yescharacteristic
54C4⋊C8×C59D411yescharacteristic
55#104×C59—42no—
56C22⋊C8×C59—22no—
57C22⋊C8×C59—22no—
58C22⋊C8×C59—21no—
59#97×C59—22no—
60C4×C8×C59—22no—
61C4×C8×C59—21no—
62C4×C8×C59—21no—
63C4×C8×C59C2×C411yescharacteristic
64C4×C8×C59C2×C411yescharacteristic
65#95×C59—21no—
66#41×C59—21no—
67#41×C59—21no—
68#41×C59—21no—
69#41×C59—21no—
70C2×Q8×C59C4⋊C411yescharacteristic
71C2×D4×C59—21no—
72C2×D4×C59C4⋊C411yescharacteristic
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—41no—
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C22×C4×C59#4111yescharacteristic
80M16×C59—41no—
81M16×C59—41no—
82M16×C59—41no—
83M16×C59—41no—
84C2×C8×C59—21no—
85C2×C8×C59—21no—
86C2×C8×C59—21no—
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59—22no—
90C2×C8×C59—21no—
91C2×C8×C59—42no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C2×C8×C59—41no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—22no—
98C4⋊C4×C59—41no—
99C4⋊C4×C59C4⋊C411yescharacteristic
100C4⋊C4×C59C4⋊C411yescharacteristic
101C22⋊C4×C59—21no—
102C22⋊C4×C59—21no—
103C22⋊C4×C59—21no—
104C42×C59—21no—
105C42×C59—21no—
106C42×C59—21no—
107C42×C59—41no—
108C42×C59—41no—
109C42×C59C22×C411yescharacteristic
110C42×C59C2×D411yescharacteristic
111C42×C59C2×D411yescharacteristic
112C42×C59C2×Q811yescharacteristic
113C23×C59—21no—
114C23×C59#12311yescharacteristic
115Q8×C59—21no—
116Q8×C59—21no—
117D4×C59—21no—
118D4×C59—21no—
119D4×C59—41no—
120D4×C59—41no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—42no—
132C2×C4×C59—81no—
133C2×C4×C59—81no—
134C2×C4×C59—81no—
135C2×C4×C59C2×C4⋊C411yescharacteristic
136C2×C4×C59C4×D411yescharacteristic
137C2×C4×C59C4×D411yescharacteristic
138C2×C4×C59#12011yescharacteristic
139C2×C4×C59C22⋊Q811yescharacteristic
140C2×C4×C59C22⋊Q811yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C2×C4×C59#13511yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C59#13711yescharacteristic
145C8×C59—22no—
146C8×C59—41no—
147C8×C59—41no—
148C8×C59—41no—
149C8×C59—41no—
150C8×C59—41no—
151C8×C59—42no—
152C8×C59—81no—
153#8905C5911yesmaximal, characteristic, direct product factor, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—41no—
159C22×C59#38511yescharacteristic
160C22×C59#48211yescharacteristic
161C22×C59#48511yescharacteristic
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—82no—
169C4×C59#42711yescharacteristic
170C4×C59#42811yescharacteristic
171C4×C59#44311yescharacteristic
172C4×C59#44411yescharacteristic
173C4×C59#44911yescharacteristic
174C4×C59#45211yescharacteristic
175#2925C2×C5911yescharacteristic
176#1806C2×C5911yescharacteristic
177#1571C2×C5911yescharacteristic, complemented
178#1566C2×C5911yescharacteristic, complemented
179#1476C2×C5911yescharacteristic
180#1303C2×C5911yescharacteristic, complemented
181#1279C2×C5911yescharacteristic, complemented
182C2×C59—21no—
183C2×C59—21no—
184C2×C59—81no—
185C2×C59#185411yescharacteristic
186C2×C59#186111yescharacteristic
187C2×C59#186611yescharacteristic
188C4×#41C22×C5911yescharacteristic
189#433—22no—
190#432—21no—
191#431—21no—
192#431C22×C5911yescharacteristic
193#423C22×C5911yescharacteristic
194#423C22×C5911yescharacteristic
195C2×C4⋊C8C22×C5911yescharacteristic
196C2×#104—21no—
197#408—21no—
198#405—21no—
199C2×C4×C8C22×C5911yescharacteristic
200C4×C4⋊C4C22×C5911yescharacteristic
201#339—21no—
202#335—21no—
203#335—21no—
204#334—22no—
205Q8⋊C8C4×C5911yescharacteristic, complemented
206Q8⋊C8C4×C5911yescharacteristic, complemented
207D4⋊C8C4×C5911yescharacteristic, complemented
208D4⋊C8C4×C5911yescharacteristic, complemented
209C59#890511yescharacteristic, direct product factor, Sylow
210C2×#41D4×C5911yescharacteristic
211C2×M16—21no—
212C2×M16—21no—
213C22×C8—21no—
214C22×C8—21no—
215C4×Q8C2×C4×C5911yescharacteristic
216C4×Q8Q8×C5911yescharacteristic
217C4×D4—21no—
218C4×D4—21no—
219C4×D4C2×C4×C5911yescharacteristic
220C4×D4Q8×C5911yescharacteristic
221#117—21no—
222#117D4×C5911yescharacteristic
223C2×C4⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42C23×C5911yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8C2×C4×C5911yescharacteristic
232C4⋊C8C2×C4×C5911yescharacteristic
233C4⋊C8D4×C5912yes—
234C4⋊C8D4×C5911yescharacteristic
235C4⋊C8D4×C5911yescharacteristic
236#104—42no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—21no—
240#97—22no—
241C4×C8—22no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8C2×C4×C5911yescharacteristic
245C4×C8C2×C4×C5911yescharacteristic
246#95—21no—
247#41—21no—
248#41—21no—
249#41—21no—
250#41—21no—
251C2×Q8C4⋊C4×C5911yescharacteristic
252C2×D4—21no—
253C2×D4C4⋊C4×C5911yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4C2×D4×C5911yescharacteristic
259C22×C4#41×C5911yescharacteristic
260C22×C4#41×C5911yescharacteristic
261M16—41no—
262M16—41no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—22no—
271C2×C8—21no—
272C2×C8—42no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4C4⋊C4×C5911yescharacteristic
281C4⋊C4C4⋊C4×C5911yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42C22×C4×C5911yescharacteristic
291C42C2×D4×C5911yescharacteristic
292C42C2×D4×C5911yescharacteristic
293C42C2×Q8×C5911yescharacteristic
294C23—21no—
295C23#123×C5911yescharacteristic
296Q8—21no—
297Q8—21no—
298D4—21no—
299D4—21no—
300D4—41no—
301D4—41no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4C2×C4⋊C4×C5911yescharacteristic
317C2×C4C4×D4×C5911yescharacteristic
318C2×C4C4×D4×C5911yescharacteristic
319C2×C4#120×C5911yescharacteristic
320C2×C4C22⋊Q8×C5911yescharacteristic
321C2×C4C22⋊Q8×C5911yescharacteristic
322C2×C4#135×C5911yescharacteristic
323C2×C4#135×C5911yescharacteristic
324C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
325C2×C4#137×C5911yescharacteristic
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—41no—
331C8—41no—
332C8—42no—
333C8—81no—
334C22—21no—
335C22—21no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22C59×#38511yescharacteristic
340C22C59×#48211yescharacteristic
341C22C59×#48511yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—82no—
349C4C59×#42711yescharacteristic
350C4C59×#42811yescharacteristic
351C4C59×#44311yescharacteristic
352C4C59×#44411yescharacteristic
353C4C59×#44911yescharacteristic
354C4C59×#45211yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2C59×#185411yescharacteristic
359C2C59×#186111yescharacteristic
360C2C59×#186611yescharacteristic
subgroupG/H
centerC2×C4×C59#120
commutatorC42C22×C4×C59
Frattini, cosocleC2×C42C23×C59
socleC22×C59#385
prime radicalC2×#41×C59D4
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
3. C59×#1571C2semidirect
4. C59×#1566C2semidirect
6. C59×#1303C2semidirect
7. C59×#1279C2semidirect
25. C59×Q8⋊C8C4semidirect
26. C59×Q8⋊C8C4semidirect
27. C59×D4⋊C8C4semidirect
28. C59×D4⋊C8C4semidirect
153. #8905C59direct (N × H)
177. #1571C2×C59semidirect
178. #1566C2×C59semidirect
180. #1303C2×C59semidirect
181. #1279C2×C59semidirect
205. Q8⋊C8C4×C59semidirect
206. Q8⋊C8C4×C59semidirect
207. D4⋊C8C4×C59semidirect
208. D4⋊C8C4×C59semidirect
209. C59#8905direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#29252
C59×#18062
C59×#15712
C59×#15662
C59×#14762
C59×#13032
C59×#12792
#890559