Finite Group Databasebeta

← #28484777

Subgroups — #28484777

Order 15104.

532 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433551C211yesmaximal, characteristic
2#41×C59⋊C8C211yesmaximal, characteristic, complemented
3#21422590C211yesmaximal, characteristic, complemented
4#21422546C211yesmaximal, characteristic, complemented
5#21421991C211yesmaximal, characteristic, complemented
6#21421937C211yesmaximal, characteristic, complemented
7C59×#1806C211yesmaximal, characteristic, complemented
8C2×#14273747—21no—
9#16837952C2211yescharacteristic
10#16837624—21no—
11#16837617—21no—
12#16837612C411yescharacteristic, complemented
13#16837607C411yescharacteristic, complemented
14#16837602C411yescharacteristic, complemented
15#16837597C411yescharacteristic, complemented
16Q8×C59⋊C8—21no—
17D4×C59⋊C8—22no—
18D4×C59⋊C8—21no—
19#16837455—21no—
20#16837455C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22#16837396—21no—
23C2×C4×C59⋊C8—21no—
24C2×C4×C59⋊C8C2211yescharacteristic
25#16837135—22no—
26#16837129—21no—
27#16837129—21no—
28C4×#41×C59C2211yescharacteristic
29C59×#423C2211yescharacteristic
30C2×#104×C59—21no—
31C4×C4⋊C4×C59C2211yescharacteristic
32C59×#339—21no—
33C2×#11745685—21no—
34#14273747—42no—
35#14273747—42no—
36C2×#11745672—21no—
37C2×#11745670—21no—
38C2×#11745671—21no—
39C2×#11745669—21no—
40C22×C59⋊C8—22no—
41C22×C59⋊C8—21no—
42C22×C59⋊C8—21no—
43#14273727C2×C411yescharacteristic
44#14273724C2×C411yescharacteristic
45#14273715—21no—
46#14273701—21no—
47#14273701—21no—
48C4×D236—21no—
49C4×D236—21no—
50C4⋊(C59⋊C8)—22no—
51C4⋊(C59⋊C8)—21no—
52C4⋊(C59⋊C8)—21no—
53C4⋊(C59⋊C8)C2×C411yescharacteristic
54C4⋊(C59⋊C8)C2×C411yescharacteristic
55#14273650—21no—
56#14273650—21no—
57#14273647—21no—
58#14273647—21no—
59C22⋊(C59⋊C8)—22no—
60C22⋊(C59⋊C8)D412yes—
61C4×C59⋊C8—22no—
62C4×C59⋊C8—21no—
63C4×C59⋊C8—21no—
64C4×C59⋊C8D411yescharacteristic
65C4×C59⋊C8D411yescharacteristic
66C2×#41×C59D411yescharacteristic
67C2×M16×C59—21no—
68C4×Q8×C59C2×C411yescharacteristic
69C4×Q8×C59D411yescharacteristic
70C4×D4×C59—21no—
71C4×D4×C59—21no—
72C4×D4×C59C2×C411yescharacteristic
73C4×D4×C59D411yescharacteristic
74#117×C59—21no—
75#117×C59D411yescharacteristic
76C2×C4⋊C4×C59—21no—
77C2×C42×C59—21no—
78C2×C42×C59—21no—
79C2×C42×C59C2311yescharacteristic
80C4⋊C8×C59—21no—
81C4⋊C8×C59—21no—
82#104×C59—42no—
83#95×C59—21no—
84#41×C59—21no—
85#41×C59—21no—
86#41×C59D812yes—
87#41×C59SD1612yes—
88C2×Q8×C59C22⋊C411yescharacteristic
89C2×D4×C59—21no—
90C2×D4×C59C22⋊C411yescharacteristic
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—41no—
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59#4111yescharacteristic
98C2×C22⋊D59—41no—
99#11745685—41no—
100#11745685—41no—
101#11745685—41no—
102#11745685—41no—
103C2×D236—21no—
104C2×D236—21no—
105C2×D236—21no—
106C2×C4×D59—41no—
107C2×Dic118—21no—
108M16×C59—41no—
109M16×C59—41no—
110C2×C8×C59—41no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—22no—
114C4⋊C4×C59—41no—
115C4⋊C4×C59C22⋊C411yescharacteristic
116C4⋊C4×C59C22⋊C411yescharacteristic
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—21no—
120C42×C59—21no—
121C42×C59—21no—
122C42×C59—21no—
123C42×C59—41no—
124C42×C59—41no—
125C42×C59C22×C411yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59C2×D411yescharacteristic
128C42×C59#4111yescharacteristic
129#11745672—42no—
130#11745671—42no—
131#11745670—42no—
132#11745669—42no—
133(C2×D59)⋊C4—41no—
134(C2×D59)⋊C4—41no—
135C4⋊Dic59—21no—
136C4⋊Dic59—21no—
137C2×C59⋊C8—21no—
138C2×C59⋊C8—21no—
139C2×C59⋊C8—21no—
140C2×C59⋊C8—22no—
141C2×C59⋊C8—21no—
142C2×C59⋊C8—42no—
143C2×C59⋊C8—44no—
144C2×C59⋊C8—41no—
145C2×C59⋊C8—41no—
146C23×C59—21no—
147C23×C59#12111yescharacteristic
148C22×D59—41no—
149Q8×C59—21no—
150Q8×C59#10212yes—
151D4×C59—22no—
152D4×C59—22no—
153D4×C59—21no—
154D4×C59#10212yes—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—22no—
159C2×C4×C59—22no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—41no—
163C2×C4×C59—42no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—42no—
167C2×C4×C59—81no—
168C2×C4×C59—81no—
169C2×C4×C59C2×C22⋊C411yescharacteristic
170C2×C4×C59C4×D411yescharacteristic
171C2×C4×C59C4×D411yescharacteristic
172C2×C4×C59#12011yescharacteristic
173C2×C4×C59#12111yescharacteristic
174C2×C4×C59#12311yescharacteristic
175C2×C4×C59C2×D811yescharacteristic
176C2×C4×C59C2×SD1611yescharacteristic
177C2×C4×C59#13511yescharacteristic
178C2×C4×C59#13511yescharacteristic
179C22⋊D59—81no—
180C22⋊D59—81no—
181C2×Dic59—41no—
182D236—41no—
183D236—41no—
184D236—41no—
185D236—41no—
186D236—81no—
187C4×D59—81no—
188C4×D59—81no—
189Dic118—41no—
190Dic118—41no—
191C8×C59—81no—
192C59⋊C8—22no—
193C59⋊C8—42no—
194C59⋊C8—41no—
195C59⋊C8—41no—
196C59⋊C8—41no—
197#9208—591nomaximal, Sylow
198C22×C59—21no—
199C22×C59—21no—
200C22×C59—22no—
201C22×C59—41no—
202C22×C59—41no—
203C22×C59#38611yescharacteristic
204C22×C59#46511yescharacteristic
205C22×C59#46611yescharacteristic
206C2×D59—81no—
207C2×D59—81no—
208C2×D59—81no—
209C4×C59—21no—
210C4×C59—22no—
211C4×C59—21no—
212C4×C59—21no—
213C4×C59—41no—
214C4×C59—41no—
215C4×C59—82no—
216C4×C59C2×#10211yescharacteristic
217C4×C59#41611yescharacteristic
218C4×C59#44311yescharacteristic
219C4×C59#44411yescharacteristic
220C4×C59#46511yescharacteristic
221C4×C59#46611yescharacteristic
222Dic59—81no—
223#3035—591no—
224C8×#41—591no—
225#1806D5911yescharacteristic, complemented
226#1502—591no—
227#1492—591no—
228#1301—591no—
229#1279—591no—
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—42no—
233C2×C59#187611yescharacteristic
234C2×C59#189311yescharacteristic
235C2×C59#190611yescharacteristic
236D59—161no—
237C2×#135—1181no—
238#532—591no—
239C4×#41C2×D5911yescharacteristic
240#464—1181no—
241#463—1181no—
242#462—591no—
243#461—591no—
244#460—591no—
245#459—591no—
246C8×Q8—1181no—
247C8×D4—1182no—
248C8×D4—1181no—
249#431—591no—
250#431—1181no—
251#423C2×D5911yescharacteristic
252C2×C4⋊C8—591no—
253C2×#104—21no—
254#416—1181no—
255C2×C4×C8—591no—
256C2×C4×C8—1181no—
257C4×C4⋊C4C2×D5911yescharacteristic
258#339—21no—
259#327—1182no—
260D4⋊C8—1181no—
261D4⋊C8—1181no—
262C59#920811yescharacteristic, complemented, Sylow
263C2×#41—1181no—
264C2×#41C22⋊D5911yescharacteristic
265#135—2362no—
266#135—2362no—
267C2×Q16—1181no—
268C2×SD16—1181no—
269C2×SD16—1181no—
270C2×D8—1181no—
271C2×M16—21no—
272C22×C8—1182no—
273C22×C8—1181no—
274C22×C8—1181no—
275C4⋊Q8—591no—
276#127—591no—
277#124—1181no—
278#121—1181no—
279#121—1181no—
280C4×Q8C4×D5911yescharacteristic
281C4×Q8D23611yescharacteristic
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—1181no—
285C4×D4—1181no—
286C4×D4C4×D5911yescharacteristic
287C4×D4D23611yescharacteristic
288#117—21no—
289#117C22⋊D5911yescharacteristic
290C2×C4⋊C4—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22×D5911yescharacteristic
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—591no—
297C4⋊C8—591no—
298C4⋊C8—1182no—
299C4⋊C8—1181no—
300C4⋊C8—1181no—
301#104—42no—
302#103—1181no—
303#103—1181no—
304#102—1181no—
305#102—1181no—
306C22⋊C8—592no—
307C22⋊C8—1182no—
308C4×C8—591no—
309C4×C8—591no—
310C4×C8—1182no—
311C4×C8—1181no—
312C4×C8—1181no—
313#95—21no—
314#41—21no—
315#41—21no—
316#41—2361no—
317#41—2361no—
318#41—2361no—
319#41—2361no—
320#41#1174566912yes—
321#41#1174567112yes—
322C2×Q8—1181no—
323C2×Q8(C2×D59)⋊C411yescharacteristic
324C2×D4—21no—
325C2×D4—1181no—
326C2×D4—1181no—
327C2×D4—1181no—
328C2×D4—2361no—
329C2×D4(C2×D59)⋊C411yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—2361no—
335C22×C4D4×D5911yescharacteristic
336C22×C4#1174568711yescharacteristic
337C22×C4C2×C22⋊D5911yescharacteristic
338Q16—2362no—
339SD16—2362no—
340SD16—2362no—
341D8—2362no—
342M16—41no—
343M16—41no—
344C2×C8—41no—
345C2×C8—1181no—
346C2×C8—1181no—
347C2×C8—1181no—
348C2×C8—1181no—
349C2×C8—1182no—
350C2×C8—2362no—
351C2×C8—2364no—
352C2×C8—2361no—
353C2×C8—2361no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—41no—
358C4⋊C4—1181no—
359C4⋊C4—1181no—
360C4⋊C4(C2×D59)⋊C411yescharacteristic
361C4⋊C4(C2×D59)⋊C411yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—2361no—
366C22⋊C4—2361no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42C2×C4×D5911yescharacteristic
373C42C2×D23611yescharacteristic
374C42D4×D5911yescharacteristic
375C42#1174568711yescharacteristic
376C23—21no—
377C23—2361no—
378C23#1427369911yescharacteristic
379Q8—21no—
380Q8—2361no—
381Q8—2361no—
382Q8#1427364712yes—
383D4—22no—
384D4—22no—
385D4—21no—
386D4—4721no—
387D4—4721no—
388D4—4721no—
389D4—2361no—
390D4—2361no—
391D4—2361no—
392D4—2361no—
393D4#1427364712yes—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—4721no—
409C2×C4—4721no—
410C2×C4—2361no—
411C2×C4C2×(C2×D59)⋊C411yescharacteristic
412C2×C4(C22⋊D59)⋊C411yescharacteristic
413C2×C4(C22⋊D59)⋊C411yescharacteristic
414C2×C4#1427369711yescharacteristic
415C2×C4#1427369911yescharacteristic
416C2×C4#1427371111yescharacteristic
417C2×C4C2×#1174566911yescharacteristic
418C2×C4C2×#1174567111yescharacteristic
419C2×C4#1427374711yescharacteristic
420C2×C4#1427374711yescharacteristic
421C8—81no—
422C8—1182no—
423C8—2361no—
424C8—2362no—
425C8—2361no—
426C8—2361no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—4721no—
433C22—4721no—
434C22—4721no—
435C22#1683727911yescharacteristic
436C22#1683763311yescharacteristic
437C22#1683763611yescharacteristic
438C4—21no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—4721no—
446C4C2×#1427364711yescharacteristic
447C4#1683739611yescharacteristic
448C4#1683751611yescharacteristic
449C4#1683752111yescharacteristic
450C4#1683763311yescharacteristic
451C4#1683763611yescharacteristic
452C2—21no—
453C2—21no—
454C2—42no—
455C2—9441no—
456C2#2142442111yescharacteristic
457C2#2142452211yescharacteristic
458C2#2142459511yescharacteristic
subgroupG/H
centerC2×C4#14273699
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1806C2
socleC22×C59#386
prime radical#21433551C2
cosocleC2×C42×C59C23
hypercenter#1806D59
hypocenterC59#9208
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #41×C59⋊C8C2semidirect
3. #21422590C2semidirect
4. #21422546C2semidirect
5. #21421991C2semidirect
6. #21421937C2semidirect
7. C59×#1806C2semidirect
12. #16837612C4semidirect
13. #16837607C4semidirect
14. #16837602C4semidirect
15. #16837597C4semidirect
225. #1806D59semidirect
262. C59#9208semidirect

Maximal subgroups

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

subgroupindex
#214335512
#41×C59⋊C82
#214225902
#214225462
#214219912
#214219372
C59×#18062
#920859