Finite Group Databasebeta

← C59×#10221 #28422330

Subgroups — C59×#10221 #28422330

Order 15104.

570 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3227C211yesmaximal, characteristic, complemented
2C59×#3206C211yesmaximal, characteristic, complemented
3C59×#3150C211yesmaximal, characteristic
4C59×#1993C211yesmaximal, characteristic, complemented
5C59×#1517C211yesmaximal, characteristic, complemented
6C59×#1517C211yesmaximal, characteristic, complemented
7C59×#1281C211yesmaximal, characteristic, complemented
8C2×#135×C59—22no—
9C59×#532C2211yescharacteristic
10C59×#532C2211yescharacteristic
11C2×C4⋊Q8×C59C2211yescharacteristic
12C59×#498—21no—
13C59×#497—21no—
14C59×#497—21no—
15C59×#496—21no—
16C59×#496—21no—
17C59×#492—21no—
18C59×#487—21no—
19C59×#486—21no—
20C59×#481—22no—
21C59×#476—22no—
22C59×#461—22no—
23C59×#448—22no—
24C59×#431C2211yescharacteristic
25C59×#431C2211yescharacteristic
26C2×#97×C59C2211yescharacteristic
27C59×#382C2211yescharacteristic
28C59×#340—22no—
29C59×#327—22no—
30C59×#327—22no—
31C2×#41×C59—21no—
32C2×#41×C59—21no—
33C22×Q8×C59—21no—
34#135×C59—44no—
35#135×C59—44no—
36C2×Q16×C59—22no—
37C2×SD16×C59—22no—
38C2×SD16×C59—22no—
39C2×SD16×C59—24no—
40C2×D8×C59—22no—
41C22×C8×C59D412yes—
42C4⋊Q8×C59—21no—
43C4⋊Q8×C59—21no—
44C4⋊Q8×C59—21no—
45C4⋊Q8×C59D412yes—
46C4⋊Q8×C59D411yescharacteristic
47C4⋊Q8×C59D411yescharacteristic
48#127×C59D411yescharacteristic
49#127×C59D411yescharacteristic
50#124×C59—21no—
51#124×C59—21no—
52#121×C59—21no—
53#121×C59—21no—
54#121×C59—21no—
55#121×C59—21no—
56C4×D4×C59—21no—
57C4×D4×C59—21no—
58C4×D4×C59—21no—
59C4×D4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C42×C59C2311yescharacteristic
64#106×C59—24no—
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67#103×C59—24no—
68#102×C59—24no—
69C22⋊C8×C59D412yes—
70C22⋊C8×C59D412yes—
71#97×C59—21no—
72#97×C59—21no—
73C4×C8×C59—21no—
74C4×C8×C59—21no—
75#95×C59—21no—
76#95×C59—21no—
77#41×C59—42no—
78#41×C59—42no—
79#41×C59—42no—
80#41×C59—42no—
81C2×Q8×C59—22no—
82C2×Q8×C59—21no—
83C2×Q8×C59—21no—
84C2×Q8×C59—41no—
85C2×Q8×C59—82no—
86C2×D4×C59—21no—
87C2×D4×C59—21no—
88C2×D4×C59—21no—
89C2×D4×C59—21no—
90C2×D4×C59—21no—
91C2×D4×C59—21no—
92C2×D4×C59—41no—
93C2×D4×C59—41no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59—41no—
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59C2×D411yescharacteristic
101Q16×C59—44no—
102SD16×C59—44no—
103SD16×C59—44no—
104SD16×C59—84no—
105D8×C59—44no—
106C2×C8×C59—22no—
107C2×C8×C59—22no—
108C2×C8×C59—22no—
109C2×C8×C59—22no—
110C2×C8×C59SD1614yescomplemented
111C4⋊C4×C59—22no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—21no—
114C4⋊C4×C59—22no—
115C4⋊C4×C59—21no—
116C4⋊C4×C59—21no—
117C4⋊C4×C59—41no—
118C4⋊C4×C59—41no—
119C4⋊C4×C59—42no—
120C22⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C22⋊C4×C59—41no—
123C22⋊C4×C59—41no—
124C42×C59C2×D411yescharacteristic
125C42×C59C2×D411yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59#4111yescharacteristic
128C23×C59—41no—
129C23×C59—41no—
130C23×C59#12711yescharacteristic
131Q8×C59—44no—
132Q8×C59—42no—
133Q8×C59—42no—
134Q8×C59—82no—
135D4×C59—42no—
136D4×C59—42no—
137D4×C59—42no—
138D4×C59—42no—
139D4×C59—81no—
140D4×C59—81no—
141D4×C59—81no—
142D4×C59—81no—
143D4×C59—81no—
144D4×C59—81no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—42no—
150C2×C4×C59—82no—
151C2×C4×C59—82no—
152C2×C4×C59—82no—
153C2×C4×C59—81no—
154C2×C4×C59—81no—
155C2×C4×C59—81no—
156C2×C4×C59#12011yescharacteristic
157C2×C4×C59#12011yescharacteristic
158C2×C4×C59#12011yescharacteristic
159C2×C4×C59#12111yescharacteristic
160C2×C4×C59#12111yescharacteristic
161C2×C4×C59#12111yescharacteristic
162C2×C4×C59C2×SD1612yes—
163C2×C4×C59AGL(1,ℤ/8)12yes—
164C8×C59—24no—
165C8×C59—42no—
166C8×C59—42no—
167#10221C5911yesmaximal, characteristic, direct product factor, Sylow
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—81no—
176C22×C59#39211yescharacteristic
177C22×C59#49211yescharacteristic
178C22×C59#49611yescharacteristic
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—82no—
184C4×C59—81no—
185C4×C59—81no—
186C4×C59—161no—
187C4×C59#45012yes—
188C4×C59#46012yes—
189#3227C2×C5911yescharacteristic, complemented
190#3206C2×C5911yescharacteristic, complemented
191#3150C2×C5911yescharacteristic
192#1993C2×C5911yescharacteristic, complemented
193#1517C2×C5911yescharacteristic, complemented
194#1517C2×C5911yescharacteristic, complemented
195#1281C2×C5911yescharacteristic, complemented
196C2×C59—21no—
197C2×C59—21no—
198C2×C59—161no—
199C2×C59—161no—
200C2×C59#200811yescharacteristic
201C2×C59#200911yescharacteristic
202C2×C59#201411yescharacteristic
203C2×#135—22no—
204#532C22×C5911yescharacteristic
205#532C22×C5911yescharacteristic
206C2×C4⋊Q8C22×C5911yescharacteristic
207#498—21no—
208#497—21no—
209#497—21no—
210#496—21no—
211#496—21no—
212#492—21no—
213#487—21no—
214#486—21no—
215#481—22no—
216#476—22no—
217#461—22no—
218#448—22no—
219#431C22×C5911yescharacteristic
220#431C22×C5911yescharacteristic
221C2×#97C22×C5911yescharacteristic
222#382C22×C5911yescharacteristic
223#340—22no—
224#327—22no—
225#327—22no—
226C59#1022111yescharacteristic, direct product factor, Sylow
227C2×#41—21no—
228C2×#41—21no—
229C22×Q8—21no—
230#135—44no—
231#135—44no—
232C2×Q16—22no—
233C2×SD16—22no—
234C2×SD16—22no—
235C2×SD16—24no—
236C2×D8—22no—
237C22×C8D4×C5912yes—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8D4×C5912yes—
242C4⋊Q8D4×C5911yescharacteristic
243C4⋊Q8D4×C5911yescharacteristic
244#127D4×C5911yescharacteristic
245#127D4×C5911yescharacteristic
246#124—21no—
247#124—21no—
248#121—21no—
249#121—21no—
250#121—21no—
251#121—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C42C23×C5911yescharacteristic
260#106—24no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263#103—24no—
264#102—24no—
265C22⋊C8D4×C5912yes—
266C22⋊C8D4×C5912yes—
267#97—21no—
268#97—21no—
269C4×C8—21no—
270C4×C8—21no—
271#95—21no—
272#95—21no—
273#41—42no—
274#41—42no—
275#41—42no—
276#41—42no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—82no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×D4×C5911yescharacteristic
295C22×C4C2×D4×C5911yescharacteristic
296C22×C4C2×D4×C5911yescharacteristic
297Q16—44no—
298SD16—44no—
299SD16—44no—
300SD16—84no—
301D8—44no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8SD16×C5914yescomplemented
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C42C2×D4×C5911yescharacteristic
321C42C2×D4×C5911yescharacteristic
322C42C2×D4×C5911yescharacteristic
323C42#41×C5911yescharacteristic
324C23—41no—
325C23—41no—
326C23#127×C5911yescharacteristic
327Q8—44no—
328Q8—42no—
329Q8—42no—
330Q8—82no—
331D4—42no—
332D4—42no—
333D4—42no—
334D4—42no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—81no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4#120×C5911yescharacteristic
353C2×C4#120×C5911yescharacteristic
354C2×C4#120×C5911yescharacteristic
355C2×C4#121×C5911yescharacteristic
356C2×C4#121×C5911yescharacteristic
357C2×C4#121×C5911yescharacteristic
358C2×C4C2×SD16×C5912yes—
359C2×C4AGL(1,ℤ/8)×C5912yes—
360C8—24no—
361C8—42no—
362C8—42no—
363C22—21no—
364C22—21no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C59×#39211yescharacteristic
372C22C59×#49211yescharacteristic
373C22C59×#49611yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—81no—
380C4—81no—
381C4—161no—
382C4C59×#45012yes—
383C4C59×#46012yes—
384C2—21no—
385C2—21no—
386C2—161no—
387C2—161no—
388C2C59×#200811yescharacteristic
389C2C59×#200911yescharacteristic
390C2C59×#201411yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC59×#3150C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
1. C59×#3227C2semidirect
2. C59×#3206C2semidirect
4. C59×#1993C2semidirect
5. C59×#1517C2semidirect
6. C59×#1517C2semidirect
7. C59×#1281C2semidirect
110. C2×C8×C59SD16semidirect
167. #10221C59direct (N × H)
189. #3227C2×C59semidirect
190. #3206C2×C59semidirect
192. #1993C2×C59semidirect
193. #1517C2×C59semidirect
194. #1517C2×C59semidirect
195. #1281C2×C59semidirect
226. C59#10221direct (N × H)
306. C2×C8SD16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#32272
C59×#32062
C59×#31502
C59×#19932
C59×#15172
C59×#12812
#1022159