Finite Group Databasebeta

← #28484500

Subgroups — #28484500

Order 15104.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433110C211yesmaximal, characteristic, complemented
2#21423906C211yesmaximal, characteristic, complemented
3#21422688C211yesmaximal, characteristic, complemented
4#21422604C211yesmaximal, characteristic, complemented
5#21421988C211yesmaximal, characteristic, complemented
6#21421967C211yesmaximal, characteristic, complemented
7C59×#3115C211yesmaximal, characteristic
8#16837533—21no—
9#16837474—22no—
10D4×C59⋊C8—21no—
11#16837459—21no—
12#16837459C2211yescharacteristic
13C2×#14273654—21no—
14C4×#11745664C2211yescharacteristic
15C4×#11745664C2211yescharacteristic
16C2×#14273638—21no—
17#16837248C2211yescharacteristic
18#16837161—21no—
19#16837142—21no—
20#16837142—21no—
21#16837136—22no—
22C59×#532C2211yescharacteristic
23C4×#41×C59C2211yescharacteristic
24C59×#482—21no—
25C59×#475C411yescharacteristic, complemented
26C59×#474C411yescharacteristic, complemented
27C59×#460C411yescharacteristic, complemented
28C59×#459C411yescharacteristic, complemented
29C59×#449—21no—
30C59×#442—21no—
31C59×#440—21no—
32C59×#432C2211yescharacteristic
33C2×#11745664—21no—
34C2×#11745664—21no—
35C22×C59⋊C8—22no—
36C2×C4×Dic59—21no—
37C4⋊(C59⋊C8)—22no—
38C4⋊(C59⋊C8)—21no—
39#14273654—24no—
40C22⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)D412yes—
42#14273638—22no—
43#14273638—21no—
44#14273638—21no—
45#14273638D411yescharacteristic
46#14273638D411yescharacteristic
47C4×C59⋊C8—21no—
48C4×C59⋊C8—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C2×#41×C59—21no—
52C2×#41×C59D411yescharacteristic
53C2×SD16×C59—21no—
54C2×D8×C59—21no—
55C4⋊Q8×C59C2×C411yescharacteristic
56#127×C59C2×C411yescharacteristic
57#124×C59—21no—
58#121×C59—21no—
59#121×C59—21no—
60C4×Q8×C59C2×C411yescharacteristic
61C4×Q8×C59D411yescharacteristic
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59—21no—
66C4×D4×C59C2×C411yescharacteristic
67C4×D4×C59D411yescharacteristic
68#117×C59—21no—
69#117×C59D411yescharacteristic
70C2×C42×C59—21no—
71C2×C42×C59C2311yescharacteristic
72#107×C59—21no—
73#106×C59—21no—
74C4⋊C8×C59C2×C411yescharacteristic
75C4⋊C8×C59C2×C411yescharacteristic
76#103×C59—21no—
77#102×C59—21no—
78#102×C59—21no—
79#102×C59—21no—
80C22⋊C8×C59—21no—
81#97×C59—21no—
82#41×C59—22no—
83#41×C59—41no—
84#41×C59—82no—
85C2×Q8×C59—21no—
86C2×Q8×C59C22⋊C411yescharacteristic
87C2×D4×C59—21no—
88C2×D4×C59—21no—
89C2×D4×C59—21no—
90C2×D4×C59—21no—
91C2×D4×C59—41no—
92C2×D4×C59C22⋊C411yescharacteristic
93C22×C4×C59—21no—
94C22×C4×C59—21no—
95C22×C4×C59—21no—
96C22×C4×C59—41no—
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59#4111yescharacteristic
100C22×Dic59—41no—
101SD16×C59—42no—
102D8×C59—42no—
103C2×C8×C59—21no—
104C2×C8×C59—21no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59C22⋊C411yescharacteristic
110C4⋊C4×C59C22⋊C411yescharacteristic
111C22⋊C4×C59—21no—
112C22⋊C4×C59—21no—
113C22⋊C4×C59—21no—
114C22⋊C4×C59—41no—
115C22⋊C4×C59—41no—
116C42×C59—21no—
117C42×C59—21no—
118C42×C59—21no—
119C42×C59C22×C411yescharacteristic
120C42×C59C2×D411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59#4111yescharacteristic
123C4×Dic59—22no—
124C4×Dic59—41no—
125#11745664—22no—
126#11745664—42no—
127#11745664—41no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—21no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—44no—
132C2×C59⋊C8—41no—
133C23×C59—21no—
134C23×C59—41no—
135C23×C59#12111yescharacteristic
136Q8×C59—21no—
137Q8×C59—21no—
138Q8×C59—81no—
139D4×C59—21no—
140D4×C59—42no—
141D4×C59—41no—
142D4×C59—41no—
143D4×C59—81no—
144D4×C59—81no—
145D4×C59—82no—
146D4×C59#10412yes—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—42no—
155C2×C4×C59—42no—
156C2×C4×C59—41no—
157C2×C4×C59—81no—
158C2×C4×C59—82no—
159C2×C4×C59C2×C22⋊C411yescharacteristic
160C2×C4×C59C4×D411yescharacteristic
161C2×C4×C59C4×D411yescharacteristic
162C2×C4×C59#12011yescharacteristic
163C2×C4×C59#12111yescharacteristic
164C2×C4×C59#12311yescharacteristic
165C2×Dic59—42no—
166C2×Dic59—81no—
167C2×Dic59—81no—
168C8×C59—41no—
169C8×C59—41no—
170C59⋊C8—22no—
171C59⋊C8—42no—
172C59⋊C8—41no—
173C59⋊C8—42no—
174#9161—591nomaximal, Sylow
175C22×C59—21no—
176C22×C59—22no—
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59—81no—
182C22×C59—81no—
183C22×C59—81no—
184C22×C59#38611yescharacteristic
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59—81no—
194C4×C59C2×#10411yescharacteristic
195C4×C59#42111yescharacteristic
196Dic59—82no—
197#3115D5911yescharacteristic
198#2970—591no—
199#1759—591no—
200#1525—591no—
201#1505—591no—
202#1300—591no—
203#1291—591no—
204C2×C59—41no—
205C2×C59—42no—
206C2×C59—161no—
207C2×C59#187111yescharacteristic
208C2×C59#190311yescharacteristic
209C2×C59#190611yescharacteristic
210#532C2×D5911yescharacteristic
211C4×#41C2×D5911yescharacteristic
212#482—21no—
213#475Dic5911yescharacteristic, complemented
214#474Dic5911yescharacteristic, complemented
215#460Dic5911yescharacteristic, complemented
216#459Dic5911yescharacteristic, complemented
217#449—21no—
218#446—1181no—
219#442—21no—
220#440—21no—
221#435—1182no—
222C8×D4—1181no—
223#432—591no—
224#432—1181no—
225#432C2×D5911yescharacteristic
226C2×#104—1181no—
227C4×M16—591no—
228C4×M16—591no—
229C2×#97—1181no—
230#376—591no—
231#339—1181no—
232#329—1181no—
233#329—1181no—
234#327—1182no—
235C59#916111yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41C22⋊D5911yescharacteristic
238C2×SD16—21no—
239C2×D8—21no—
240C2×M16—1181no—
241C2×M16—1181no—
242C22×C8—1182no—
243C4⋊Q8C2×Dic5911yescharacteristic
244#127C2×Dic5911yescharacteristic
245#124—21no—
246#121—21no—
247#121—21no—
248C4×Q8C2×Dic5911yescharacteristic
249C4×Q8C22⋊D5911yescharacteristic
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C2×Dic5911yescharacteristic
255C4×D4C22⋊D5911yescharacteristic
256#117—21no—
257#117C22⋊D5911yescharacteristic
258C2×C42—21no—
259C2×C42—1181no—
260C2×C42C22×D5911yescharacteristic
261#107—21no—
262#106—21no—
263C4⋊C8—1182no—
264C4⋊C8—1181no—
265C4⋊C8C2×Dic5911yescharacteristic
266C4⋊C8C2×Dic5911yescharacteristic
267#104—1184no—
268#103—21no—
269#102—21no—
270#102—21no—
271#102—21no—
272C22⋊C8—21no—
273C22⋊C8—592no—
274C22⋊C8—1182no—
275#97—21no—
276#97—591no—
277#97—591no—
278#97—1182no—
279#97—1181no—
280#97—1181no—
281C4×C8—1181no—
282C4×C8—1181no—
283#95—1181no—
284#95—1181no—
285#41—22no—
286#41—41no—
287#41—82no—
288C2×Q8—21no—
289C2×Q8C22⋊Dic5911yescharacteristic
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4C22⋊Dic5911yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—2361no—
301C22×C4D4×D5911yescharacteristic
302C22×C4#1174568711yescharacteristic
303C22×C4C2×C22⋊D5911yescharacteristic
304SD16—42no—
305D8—42no—
306M16—1182no—
307M16—2362no—
308M16—2361no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—1181no—
312C2×C8—1181no—
313C2×C8—1182no—
314C2×C8—2364no—
315C2×C8—2361no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4C22⋊Dic5911yescharacteristic
321C4⋊C4C22⋊Dic5911yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—1182no—
331C42—2361no—
332C42D4×D5911yescharacteristic
333C42#1174568711yescharacteristic
334C42C22×Dic5911yescharacteristic
335C42C2×C22⋊D5911yescharacteristic
336C23—21no—
337C23—41no—
338C23#1427369911yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—81no—
342D4—21no—
343D4—42no—
344D4—41no—
345D4—41no—
346D4—81no—
347D4—81no—
348D4—82no—
349D4#1427365412yes—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—4721no—
363C2×C4—4721no—
364C2×C4—2362no—
365C2×C4C2×C22⋊Dic5911yescharacteristic
366C2×C4D4×Dic5911yescharacteristic
367C2×C4D4×Dic5911yescharacteristic
368C2×C4#1427369511yescharacteristic
369C2×C4#1427369911yescharacteristic
370C2×C4#1427370811yescharacteristic
371C8—41no—
372C8—41no—
373C8—1182no—
374C8—2362no—
375C8—2361no—
376C8—2362no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1683728211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—4722no—
397C4C2×#1427365411yescharacteristic
398C4#1683741911yescharacteristic
399C2—41no—
400C2—42no—
401C2—161no—
402C2#2142439911yescharacteristic
403C2#2142457811yescharacteristic
404C2#2142459911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3115C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3115D59
hypocenterC59#9161
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #21433110C2semidirect
2. #21423906C2semidirect
3. #21422688C2semidirect
4. #21422604C2semidirect
5. #21421988C2semidirect
6. #21421967C2semidirect
25. C59×#475C4semidirect
26. C59×#474C4semidirect
27. C59×#460C4semidirect
28. C59×#459C4semidirect
213. #475Dic59semidirect
214. #474Dic59semidirect
215. #460Dic59semidirect
216. #459Dic59semidirect
235. C59#9161semidirect

Maximal subgroups

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

subgroupindex
#214331102
#214239062
#214226882
#214226042
#214219882
#214219672
C59×#31152
#916159