Finite Group Databasebeta

← #28492222

Subgroups — #28492222

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429014C211yesmaximal, characteristic
2#21424252C211yesmaximal, characteristic, complemented
3#21424093C211yesmaximal, characteristic
4C2×Q8⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
5C2×#16837129C211yesmaximal, characteristic
6#21421980C211yesmaximal, characteristic, complemented
7C59×#1803C211yesmaximal, characteristic, complemented
8C2×#14273703—21no—
9C2×C4×D236C2211yescharacteristic
10C2×#14273658—22no—
11C2×C4⋊(C59⋊C8)C2211yescharacteristic
12C2×#14273647—21no—
13C2×C4×C59⋊C8C2211yescharacteristic
14#16837324—22no—
15#16837304—21no—
16#16837300—21no—
17#16837272C2211yescharacteristic
18#16837259—21no—
19#16837164—21no—
20Q8⋊(C59⋊C8)—22no—
21#16837129—22no—
22C2×C4×Q8×C59C2211yescharacteristic
23C2×C4⋊C8×C59C2211yescharacteristic
24C2×#103×C59—21no—
25C59×#384C2211yescharacteristic
26C59×#340—21no—
27C22×D236D411yescharacteristic
28C22×C4×D59—21no—
29C22×C59⋊C8—21no—
30C22×C59⋊C8—21no—
31#14273703—42no—
32#14273703—42no—
33C4×D236—21no—
34C4×D236—21no—
35C4×D236—21no—
36C4×D236—21no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59D411yescharacteristic
40C2×C4⋊Dic59D412yes—
41C2×Dic59⋊C4—22no—
42C2×(C2×D59)⋊C4—22no—
43C2×(C2×D59)⋊C4—21no—
44C2×(C2×D59)⋊C4—22no—
45#14273658—44no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48#14273647—42no—
49C4×C59⋊C8—21no—
50C4×C59⋊C8D412yes—
51C59⋊#95—22no—
52C22×Q8×C59D411yescharacteristic
53C22×C8×C59—21no—
54C4×Q8×C59—21no—
55C4×Q8×C59—21no—
56C4×Q8×C59—21no—
57C4×Q8×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C42×C59—21no—
64C2×C42×C59C2311yescharacteristic
65C4⋊C8×C59—21no—
66C4⋊C8×C59—21no—
67#103×C59—42no—
68C23×D59—21no—
69C2×Q8×C59—21no—
70C2×Q8×C59—21no—
71C2×Q8×C59—21no—
72C2×Q8×C59—21no—
73C2×Q8×C59—21no—
74C2×Q8×C59SD1612yes—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—41no—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×Dic59—21no—
82C22×Dic59—22no—
83C2×D236—21no—
84C2×D236—21no—
85C2×D236—21no—
86C2×D236—21no—
87C2×D236—21no—
88C2×D236SD1612yes—
89C2×C4×D59—41no—
90C2×C4×D59—41no—
91C2×C4×D59—41no—
92C2×C4×D59—41no—
93C2×C4×D59—41no—
94C2×C4×D59—41no—
95C2×C8×C59—41no—
96C2×C8×C59—41no—
97C2×C8×C59—41no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—42no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C42×C59—41no—
108C42×C59—41no—
109C42×C59C2×D411yescharacteristic
110C42×C59#4111yescharacteristic
111C42×C59#4111yescharacteristic
112C42×C59#4111yescharacteristic
113(C2×D59)⋊C4—44no—
114(C2×D59)⋊C4—41no—
115(C2×D59)⋊C4—41no—
116(C2×D59)⋊C4—82no—
117C4⋊Dic59—21no—
118C4⋊Dic59—21no—
119C4⋊Dic59—41no—
120C4⋊Dic59—41no—
121C4⋊Dic59—41no—
122C4⋊Dic59—41no—
123C4⋊Dic59—42no—
124Dic59⋊C4—44no—
125C2×C59⋊C8—22no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—41no—
129C2×C59⋊C8—41no—
130C2×C59⋊C8—41no—
131C23×C59#12011yescharacteristic
132C22×D59—22no—
133C22×D59—41no—
134C22×D59—41no—
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
139Q8×C59—22no—
140Q8×C59—22no—
141Q8×C59—41no—
142Q8×C59—41no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—22no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—81no—
152C2×C4×C59—81no—
153C2×C4×C59—81no—
154C2×C4×C59#12111yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12311yescharacteristic
158C2×C4×C59#12311yescharacteristic
159C2×C4×C59#12411yescharacteristic
160C2×C4×C59C2×SD1611yescharacteristic
161C2×C4×C59C2×SD1611yescharacteristic
162C2×C4×C59#13511yescharacteristic
163C2×C4×C59#13511yescharacteristic
164C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C59#13711yescharacteristic
167C2×C4×C59#13711yescharacteristic
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—42no—
172C2×Dic59—42no—
173C2×Dic59—42no—
174D236—22no—
175D236—22no—
176D236—41no—
177D236—41no—
178C4×D59—81no—
179C4×D59—81no—
180C4×D59—81no—
181C4×D59—81no—
182C8×C59—81no—
183C59⋊C8—24no—
184C59⋊C8—81no—
185#10435—591nomaximal, Sylow
186C22×C59#39411yescharacteristic
187C22×C59#44811yescharacteristic
188C22×C59#44911yescharacteristic
189C22×C59#45011yescharacteristic
190C22×C59#45211yescharacteristic
191C22×C59#45311yescharacteristic
192C22×C59#45611yescharacteristic
193C2×D59—42no—
194C2×D59—42no—
195C2×D59—42no—
196C2×D59—81no—
197C2×D59—81no—
198C2×D59—81no—
199C2×D59—81no—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—42no—
203C4×C59—81no—
204C4×C59—161no—
205C4×C59#46511yescharacteristic
206C4×C59#46511yescharacteristic
207C4×C59#46911yescharacteristic
208C4×C59#47011yescharacteristic
209C4×C59#48111yescharacteristic
210C4×C59#48211yescharacteristic
211C4×C59#48311yescharacteristic
212C4×C59#48511yescharacteristic
213Dic59—82no—
214Dic59—81no—
215#2538—591no—
216#1844—591no—
217#1803D5911yescharacteristic, complemented
218#1802—591no—
219C2×Q8⋊C8—591no—
220C2×D4⋊C8—591no—
221#1296—591no—
222C2×C59#162211yescharacteristic
223C2×C59#163111yescharacteristic
224C2×C59#165111yescharacteristic
225C2×C59#165211yescharacteristic
226C2×C59#203411yescharacteristic
227C2×C59#203511yescharacteristic
228C2×C59#203911yescharacteristic
229D59—82no—
230C2×C22⋊Q8—1181no—
231C2×C4×Q8C2×D5911yescharacteristic
232C2×C4×D4—591no—
233C2×#106—1182no—
234C2×C4⋊C8—591no—
235C2×C4⋊C8C2×D5911yescharacteristic
236C2×#103—21no—
237C2×#102—1181no—
238C2×C4×C8—591no—
239#396—1182no—
240#390—1181no—
241#389—1181no—
242#384—591no—
243#384C2×D5911yescharacteristic
244#380—1181no—
245#340—21no—
246#340—1181no—
247Q8⋊C8—1182no—
248D4⋊C8—1182no—
249C59#1043511yescharacteristic, complemented, Sylow
250C22×Q8C22⋊D5911yescharacteristic
251C22×D4—591no—
252C23×C4—1181no—
253C22×C8—21no—
254C22×C8—1181no—
255C22×C8—1181no—
256C22⋊Q8—2362no—
257C22⋊Q8—2362no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—1181no—
263C4×D4—1181no—
264C4×D4—1181no—
265C4×D4—1181no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—591no—
271C2×C4⋊C4—592no—
272C2×C4⋊C4—1182no—
273C2×C4⋊C4—1181no—
274C2×C4⋊C4—1181no—
275C2×C4⋊C4C22⋊D5911yescharacteristic
276C2×C22⋊C4—1182no—
277C2×C22⋊C4—1182no—
278C2×C22⋊C4—1181no—
279C2×C42—21no—
280C2×C42C22×D5911yescharacteristic
281#106—2364no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—1181no—
285C4⋊C8—1181no—
286#103—42no—
287#102—2362no—
288C4×C8—592no—
289C4×C8—1181no—
290#95—1182no—
291C24—1181no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8#1174567112yes—
298C2×D4—592no—
299C2×D4—1181no—
300C2×D4—1181no—
301C2×D4—1181no—
302C2×D4—1181no—
303C2×D4—1181no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—1181no—
308C22×C4—1182no—
309C22×C4—2361no—
310C22×C4—2361no—
311C22×C4—2361no—
312C22×C4—2361no—
313C22×C4—2361no—
314C22×C4—2361no—
315C22×C4D4×D5911yescharacteristic
316C22×C4D4×D5911yescharacteristic
317C22×C4C2×C22⋊D5911yescharacteristic
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—1182no—
322C2×C8—1182no—
323C2×C8—1182no—
324C2×C8—2361no—
325C2×C8—2361no—
326C2×C8—2361no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—1181no—
337C4⋊C4—1181no—
338C4⋊C4—2362no—
339C4⋊C4—2364no—
340C4⋊C4—2361no—
341C4⋊C4—2361no—
342C4⋊C4—2361no—
343C4⋊C4—2361no—
344C22⋊C4—4722no—
345C22⋊C4—2364no—
346C22⋊C4—2361no—
347C22⋊C4—2361no—
348C42—41no—
349C42—41no—
350C42#1174568511yescharacteristic
351C42#1174568511yescharacteristic
352C42D4×D5911yescharacteristic
353C42#1174568711yescharacteristic
354C23—1182no—
355C23—2361no—
356C23—2361no—
357C23—2361no—
358C23—2361no—
359C23—2361no—
360C23—2361no—
361C23#1427369511yescharacteristic
362Q8—22no—
363Q8—22no—
364Q8—41no—
365Q8—41no—
366D4—1182no—
367D4—1182no—
368D4—2361no—
369D4—2361no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—4721no—
382C2×C4—4721no—
383C2×C4—4721no—
384C2×C4—4721no—
385C2×C4—2361no—
386C2×C4—2362no—
387C2×C4—2361no—
388C2×C4—2362no—
389C2×C4—2362no—
390C2×C4—2361no—
391C2×C4#1427369911yescharacteristic
392C2×C4#1427370011yescharacteristic
393C2×C4#1427370011yescharacteristic
394C2×C4#1427370911yescharacteristic
395C2×C4#1427371011yescharacteristic
396C2×C4#1427371311yescharacteristic
397C2×C4SD16×D5911yescharacteristic
398C2×C4C2×#1174567111yescharacteristic
399C2×C4#1427374711yescharacteristic
400C2×C4#1427375011yescharacteristic
401C2×C4#1427375211yescharacteristic
402C2×C4#1427375311yescharacteristic
403C2×C4#1427375611yescharacteristic
404C2×C4#1427375911yescharacteristic
405C8—81no—
406C8—1184no—
407C8—4721no—
408C22—4721no—
409C22—4721no—
410C22—4721no—
411C22—4721no—
412C22—2362no—
413C22—2362no—
414C22—2362no—
415C22#1683731711yescharacteristic
416C22#1683753911yescharacteristic
417C22#1683754511yescharacteristic
418C22#1683755211yescharacteristic
419C22#1683756311yescharacteristic
420C22#1683756811yescharacteristic
421C22#1683758311yescharacteristic
422C4—21no—
423C4—21no—
424C4—42no—
425C4—81no—
426C4—161no—
427C4—4722no—
428C4—4721no—
429C4#1683763311yescharacteristic
430C4#1683763411yescharacteristic
431C4#1683765311yescharacteristic
432C4#1683765911yescharacteristic
433C4#1683772011yescharacteristic
434C4#1683772611yescharacteristic
435C4#1683773111yescharacteristic
436C4#1683774211yescharacteristic
437C2—4722no—
438C2#2142316711yescharacteristic
439C2#2142322111yescharacteristic
440C2#2142335811yescharacteristic
441C2#2142336411yescharacteristic
442C2#2142528511yescharacteristic
443C2#2142529011yescharacteristic
444C2#2142531811yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1803C2
socleC23×C59#120
prime radicalC22×D236D4
hypercenter#1803D59
hypocenterC59#10435
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #21424252C2semidirect
4. C2×Q8⋊(C59⋊C8)C2semidirect
6. #21421980C2semidirect
7. C59×#1803C2semidirect
217. #1803D59semidirect
249. C59#10435semidirect

Maximal subgroups

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

subgroupindex
#214290142
#214242522
#214240932
C2×Q8⋊(C59⋊C8)2
C2×#168371292
#214219802
C59×#18032
#1043559