Finite Group Databasebeta

← #28499475

Subgroups — #28499475

Order 15104.

476 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 87 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#21422434C211yesmaximal, characteristic, complemented
3#21422117C212yesmaximal, complemented
4#21422093C212yesmaximal, complemented
5C59×#2631C211yesmaximal, characteristic
6C2×#14273642—21no—
7#16837366—21no—
8#16837262—21no—
9#16837169C2212yes—
10#16837128—21no—
11#16837126C2212yes—
12C2×#124×C59C411yescharacteristic, complemented
13C2×C4×D4×C59C2211yescharacteristic
14C59×#397—21no—
15C59×#393C411yescharacteristic, complemented
16C59×#393C2211yescharacteristic
17C59×#392—21no—
18C59×#392C411yescharacteristic, complemented
19C59×#392C2211yescharacteristic
20C59×#386—21no—
21C59×#386—21no—
22C59×#383C411yescharacteristic, complemented
23C2×#11745664—41no—
24C2×C4⋊Dic59—41no—
25C2×C22⋊Dic59—22no—
26#14273642—42no—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—41no—
29C59⋊#95—41no—
30C22×Q8×C59C2×C411yescharacteristic
31C22×D4×C59—21no—
32C22×D4×C59—21no—
33C22×D4×C59C2×C411yescharacteristic
34C22×D4×C59D411yescharacteristic
35C23×C4×C59D411yescharacteristic
36C23×C4×C59D411yescharacteristic
37#124×C59—41no—
38#124×C59—41no—
39C4×D4×C59—41no—
40C4×D4×C59—41no—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59C2×C411yescharacteristic
49C2×C22⋊C4×C59C2311yescharacteristic
50C2×C42×C59C2×C411yescharacteristic
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59C2×C411yescharacteristic
56#95×C59C2×C411yescharacteristic
57C24×C59—21no—
58C24×C59C22⋊C411yescharacteristic
59C24×C59C2×D411yescharacteristic
60C2×Q8×C59—41no—
61C2×Q8×C59—81no—
62C2×D4×C59—21no—
63C2×D4×C59—21no—
64C2×D4×C59—41no—
65C2×D4×C59—41no—
66C2×D4×C59—41no—
67C2×D4×C59—42no—
68C2×D4×C59—41no—
69C2×D4×C59—41no—
70C2×D4×C59—81no—
71C2×D4×C59—81no—
72C2×D4×C59—81no—
73C2×D4×C59—81no—
74C2×D4×C59—81no—
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—81no—
83C22×C4×C59—81no—
84C22×C4×C59C22⋊C411yescharacteristic
85C22×C4×C59C22⋊C411yescharacteristic
86C22×C4×C59C22⋊C411yescharacteristic
87C22×C4×C59C22×C411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×Dic59—42no—
90C4⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—42no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—81no—
97C22⋊C4×C59—81no—
98C22⋊C4×C59—81no—
99C22⋊C4×C59—81no—
100C42×C59—21no—
101C42×C59—21no—
102C22⋊Dic59—42no—
103C22⋊Dic59—42no—
104C4⋊Dic59—82no—
105#11745664—82no—
106C2×C59⋊C8—42no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
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×C59C2×C22⋊C411yescharacteristic
124C23×C59AGL(1,ℤ/8)11yescharacteristic
125C23×C59#13711yescharacteristic
126Q8×C59—82no—
127D4×C59—42no—
128D4×C59—42no—
129D4×C59—82no—
130D4×C59—82no—
131D4×C59—82no—
132D4×C59—84no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—41no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
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—81no—
155C2×C4×C59—81no—
156C2×C4×C59#9911yescharacteristic
157C2×C4×C59#9911yescharacteristic
158C2×Dic59—82no—
159C2×Dic59—82no—
160C2×Dic59—82no—
161C59⋊C8—82no—
162#11825—591nomaximal, Sylow
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—42no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—41no—
174C22×C59—41no—
175C22×C59—81no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59—81no—
180C22×C59—81no—
181C22×C59—81no—
182C22×C59—81no—
183C22×C59—81no—
184C22×C59C2×#9911yescharacteristic
185C22×C59#41711yescharacteristic
186C22×C59#42111yescharacteristic
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—41no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59—81no—
194C4×C59—81no—
195C4×C59—81no—
196Dic59—162no—
197#2631D5911yescharacteristic
198#1799—591no—
199#1465—591no—
200#1351—592no—
201#1343—592no—
202C2×C59—21no—
203C2×C59—21no—
204C2×C59—41no—
205C2×C59—41no—
206C2×C59—81no—
207C2×C59—161no—
208C2×C59#151311yescharacteristic
209C2×C59#212211yescharacteristic
210C2×C59#212711yescharacteristic
211C2×#124Dic5911yescharacteristic, complemented
212C2×C4×D4C2×D5911yescharacteristic
213C2×#99—1181no—
214#407—1181no—
215#397—21no—
216#393Dic5911yescharacteristic, complemented
217#393C2×D5911yescharacteristic
218#392—21no—
219#392Dic5911yescharacteristic, complemented
220#392C2×D5911yescharacteristic
221#386—21no—
222#386—21no—
223#383Dic5911yescharacteristic, complemented
224#380—1181no—
225#342—592no—
226#324—1181no—
227#323—592no—
228C59#1182511yescharacteristic, complemented, Sylow
229C22×Q8C2×Dic5911yescharacteristic
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2×Dic5911yescharacteristic
233C22×D4C22⋊D5911yescharacteristic
234C23×C4C22⋊D5911yescharacteristic
235C23×C4C22⋊D5911yescharacteristic
236C2×M16—2361no—
237#124—41no—
238#124—41no—
239C4×D4—41no—
240C4×D4—41no—
241C2×C4⋊C4—2361no—
242C2×C4⋊C4C22⋊D5911yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—1182no—
250C2×C22⋊C4C2×Dic5911yescharacteristic
251C2×C22⋊C4C22×D5911yescharacteristic
252C2×C42C2×Dic5911yescharacteristic
253#99—2362no—
254C22⋊C8—1182no—
255C22⋊C8—2361no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—2361no—
261#95C2×Dic5911yescharacteristic
262#95C2×Dic5911yescharacteristic
263C24—21no—
264C24C22⋊Dic5911yescharacteristic
265C24C2×C22⋊D5911yescharacteristic
266C2×Q8—41no—
267C2×Q8—81no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—81no—
280C2×D4—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—2362no—
291C22×C4C22⋊Dic5911yescharacteristic
292C22×C4C22⋊Dic5911yescharacteristic
293C22×C4C22⋊Dic5911yescharacteristic
294C22×C4C22×Dic5911yescharacteristic
295C22×C4C2×C22⋊D5911yescharacteristic
296M16—4722no—
297C2×C8—2362no—
298C4⋊C4—41no—
299C4⋊C4—4722no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—2362no—
310C22⋊C4—2362no—
311C42—21no—
312C42—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23C2×C22⋊Dic5911yescharacteristic
330C23#1427375211yescharacteristic
331C23#1427375711yescharacteristic
332Q8—82no—
333D4—42no—
334D4—42no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—84no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—4722no—
363C2×C4—4722no—
364C2×C4—4722no—
365C2×C4#1427364211yescharacteristic
366C2×C4#1427364211yescharacteristic
367C8—4722no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C2×#1427364211yescharacteristic
390C22#1683740311yescharacteristic
391C22#1683741911yescharacteristic
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—9442no—
402C2—21no—
403C2—21no—
404C2—41no—
405C2—41no—
406C2—81no—
407C2—161no—
408C2#2142263911yescharacteristic
409C2#2142574611yescharacteristic
410C2#2142576511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
Fitting, prime radicalC59×#2631C2
socleC22×C59C2×#99
cosocleC2×C22⋊C4×C59C23
hypercenter#2631D59
hypocenterC59#11825
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2631

Complements

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

NH ≅ G/Ntype
1. #21424084C2semidirect
2. #21422434C2semidirect
3. #21422117C2semidirect
4. #21422093C2semidirect
12. C2×#124×C59C4semidirect
15. C59×#393C4semidirect
18. C59×#392C4semidirect
22. C59×#383C4semidirect
211. C2×#124Dic59semidirect
216. #393Dic59semidirect
219. #392Dic59semidirect
223. #383Dic59semidirect
228. C59#11825semidirect

Maximal subgroups

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

subgroupindex
#214240842
#214224342
#214221172
#214220932
C59×#26312
#1182559