Finite Group Databasebeta

← #28499718

Subgroups — #28499718

Order 15104.

522 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424084C211yesmaximal, characteristic, complemented
2#21422417C211yesmaximal, characteristic, complemented
3#21422096C212yesmaximal, complemented
4#21422093C212yesmaximal, complemented
5C59×#2543C211yesmaximal, characteristic
6#16837455—21no—
7C2×#14273642—21no—
8#16837262—21no—
9#16837169C2212yes—
10#16837126—21no—
11#16837126C2212yes—
12C2×#121×C59—21no—
13C2×#121×C59C411yescharacteristic, complemented
14C2×C4×D4×C59C2211yescharacteristic
15C59×#398C411yescharacteristic, complemented
16C59×#397C2211yescharacteristic
17C59×#394C411yescharacteristic, complemented
18C59×#392C2211yescharacteristic
19C59×#388—21no—
20C59×#386—21no—
21C59×#386—21no—
22C2×#95×C59C411yescharacteristic, complemented
23C2×C4⋊Dic59—41no—
24C2×C22⋊Dic59—22no—
25C4⋊(C59⋊C8)—41no—
26#14273642—42no—
27C22⋊(C59⋊C8)—22no—
28C4×C59⋊C8—41no—
29C59⋊#95—41no—
30C22×D4×C59—21no—
31C22×D4×C59—21no—
32C22×D4×C59C2×C411yescharacteristic
33C22×D4×C59D411yescharacteristic
34C23×C4×C59—21no—
35C23×C4×C59C2×C411yescharacteristic
36C23×C4×C59D411yescharacteristic
37#121×C59—42no—
38#121×C59—42no—
39#121×C59—41no—
40#121×C59—41no—
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59D411yescharacteristic
54#95×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59C2×C411yescharacteristic
58#95×C59C2×C411yescharacteristic
59C24×C59—21no—
60C24×C59C22⋊C411yescharacteristic
61C24×C59C2×D411yescharacteristic
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59—42no—
65C2×D4×C59—41no—
66C2×D4×C59—42no—
67C2×D4×C59—42no—
68C2×D4×C59—42no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59—81no—
72C2×D4×C59—81no—
73C2×D4×C59D812yes—
74C2×D4×C59SD1612yes—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—42no—
83C22×C4×C59—42no—
84C22×C4×C59—81no—
85C22×C4×C59—81no—
86C22×C4×C59—81no—
87C22×C4×C59C22⋊C411yescharacteristic
88C22×C4×C59C22⋊C411yescharacteristic
89C22×C4×C59C22⋊C411yescharacteristic
90C22×C4×C59C22×C411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×Dic59—42no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C4⋊C4×C59—42no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—42no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—42no—
104C22⋊C4×C59—81no—
105C42×C59—21no—
106C42×C59—21no—
107C22⋊Dic59—42no—
108C22⋊Dic59—42no—
109C4⋊Dic59—82no—
110C2×C59⋊C8—42no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59#10212yes—
127C23×C59C2×C22⋊C411yescharacteristic
128C23×C59C2×D811yescharacteristic
129C23×C59C2×SD1611yescharacteristic
130D4×C59—42no—
131D4×C59—42no—
132D4×C59—82no—
133D4×C59—82no—
134D4×C59—82no—
135D4×C59—84no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—22no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—41no—
143C2×C4×C59—42no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—81no—
156C2×C4×C59—81no—
157C2×C4×C59—81no—
158C2×C4×C59—84no—
159C2×C4×C59#9911yescharacteristic
160C2×C4×C59#9911yescharacteristic
161C2×C4×C59#10212yes—
162C2×Dic59—82no—
163C2×Dic59—82no—
164C2×Dic59—82no—
165C59⋊C8—82no—
166#11876—591nomaximal, Sylow
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—22no—
171C22×C59—21no—
172C22×C59—42no—
173C22×C59—41no—
174C22×C59—41no—
175C22×C59—42no—
176C22×C59—41no—
177C22×C59—41no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—81no—
181C22×C59—81no—
182C22×C59—81no—
183C22×C59—81no—
184C22×C59—81no—
185C22×C59—81no—
186C22×C59—81no—
187C22×C59—81no—
188C22×C59C2×#9911yescharacteristic
189C22×C59C2×#10211yescharacteristic
190C22×C59#42111yescharacteristic
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—22no—
194C4×C59—42no—
195C4×C59—41no—
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—81no—
199C4×C59—81no—
200Dic59—162no—
201#2543D5911yescharacteristic
202#1799—591no—
203#1460—591no—
204#1344—592no—
205#1343—592no—
206C2×C59—21no—
207C2×C59—21no—
208C2×C59—41no—
209C2×C59—42no—
210C2×C59—41no—
211C2×C59—161no—
212C2×C59#151911yescharacteristic
213C2×C59#212111yescharacteristic
214C2×C59#212611yescharacteristic
215C2×#121—21no—
216C2×#121Dic5911yescharacteristic, complemented
217C2×C4×D4C2×D5911yescharacteristic
218#431—1181no—
219C2×#99—1181no—
220#398Dic5911yescharacteristic, complemented
221#397C2×D5911yescharacteristic
222#394Dic5911yescharacteristic, complemented
223#392C2×D5911yescharacteristic
224#388—21no—
225#386—21no—
226#386—21no—
227#380—1181no—
228C2×#95Dic5911yescharacteristic, complemented
229#342—592no—
230#323—592no—
231#323—1181no—
232C59#1187611yescharacteristic, complemented, Sylow
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2×Dic5911yescharacteristic
236C22×D4C22⋊D5911yescharacteristic
237C23×C4—21no—
238C23×C4C2×Dic5911yescharacteristic
239C23×C4C22⋊D5911yescharacteristic
240#121—42no—
241#121—42no—
242#121—41no—
243#121—41no—
244C4×D4—41no—
245C4×D4—41no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—2361no—
248C2×C4⋊C4C2×Dic5911yescharacteristic
249C2×C4⋊C4C2×Dic5911yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—1182no—
256C2×C22⋊C4C22⋊D5911yescharacteristic
257C2×C22⋊C4C22×D5911yescharacteristic
258C2×C42C22⋊D5911yescharacteristic
259C4⋊C8—2361no—
260#99—2362no—
261C22⋊C8—1182no—
262C4×C8—2361no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—2361no—
267#95C2×Dic5911yescharacteristic
268#95C2×Dic5911yescharacteristic
269C24—21no—
270C24C22⋊Dic5911yescharacteristic
271C24C2×C22⋊D5911yescharacteristic
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4#1174566912yes—
284C2×D4#1174567012yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—2362no—
298C22×C4C22⋊Dic5911yescharacteristic
299C22×C4C22⋊Dic5911yescharacteristic
300C22×C4C22⋊Dic5911yescharacteristic
301C22×C4C22×Dic5911yescharacteristic
302C22×C4C2×C22⋊D5911yescharacteristic
303C2×C8—2362no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—4722no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—81no—
317C22⋊C4—2362no—
318C22⋊C4—2362no—
319C42—21no—
320C42—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1427364812yes—
337C23C2×C22⋊Dic5911yescharacteristic
338C23C2×#1174566911yescharacteristic
339C23C2×#1174567011yescharacteristic
340D4—42no—
341D4—42no—
342D4—82no—
343D4—82no—
344D4—82no—
345D4—84no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—84no—
369C2×C4—4722no—
370C2×C4—4722no—
371C2×C4—4722no—
372C2×C4#1427364211yescharacteristic
373C2×C4#1427364211yescharacteristic
374C2×C4#1427364812yes—
375C8—4722no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C2×#1427364211yescharacteristic
398C22C2×#1427364811yescharacteristic
399C22#1683741911yescharacteristic
400C4—22no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4—81no—
409C4—9442no—
410C2—21no—
411C2—21no—
412C2—41no—
413C2—42no—
414C2—41no—
415C2—161no—
416C2#2142266411yescharacteristic
417C2#2142574211yescharacteristic
418C2#2142576111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
Fitting, prime radicalC59×#2543C2
socleC22×C59C2×#99
cosocleC2×C22⋊C4×C59C23
hypercenter#2543D59
hypocenterC59#11876
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. #21424084C2semidirect
2. #21422417C2semidirect
3. #21422096C2semidirect
4. #21422093C2semidirect
13. C2×#121×C59C4semidirect
15. C59×#398C4semidirect
17. C59×#394C4semidirect
22. C2×#95×C59C4semidirect
216. C2×#121Dic59semidirect
220. #398Dic59semidirect
222. #394Dic59semidirect
228. C2×#95Dic59semidirect
232. C59#11876semidirect

Maximal subgroups

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

subgroupindex
#214240842
#214224172
#214220962
#214220932
C59×#25432
#1187659