Finite Group Databasebeta

← #28498293

Subgroups — #28498293

Order 15104.

532 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 91 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#21422434C211yesmaximal, characteristic, complemented
2#21422417C211yesmaximal, characteristic, complemented
3#21422026C212yesmaximal, complemented
4#21422024C212yesmaximal, complemented
5C59×#2534C211yesmaximal, characteristic
6#16837455—21no—
7#16837366—21no—
8#16837128—21no—
9#16837126—21no—
10#16837126C2212yes—
11#16837126C2212yes—
12C2×#123×C59C411yescharacteristic
13C2×#120×C59C411yescharacteristic
14C2×C4×D4×C59C2211yescharacteristic
15C22×C22⋊C4×C59C411yescharacteristic
16C59×#396—21no—
17C59×#394—21no—
18C59×#388—21no—
19C59×#386—21no—
20C59×#381C2211yescharacteristic
21C59×#380C411yescharacteristic
22C59×#379C2211yescharacteristic
23C2×#11745664—41no—
24C4⋊(C59⋊C8)—41no—
25C22⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—41no—
28C4×C59⋊C8—41no—
29C25×C59C2×C411yescharacteristic
30C22×D4×C59—21no—
31C22×D4×C59C2×C411yescharacteristic
32C23×C4×C59C2×C411yescharacteristic
33C23×C4×C59C2×C411yescharacteristic
34C23×C4×C59D411yescharacteristic
35#123×C59—42no—
36#120×C59—42no—
37C4×D4×C59—41no—
38C4×D4×C59—41no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
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×C59—41no—
49C2×C22⋊C4×C59—41no—
50C2×C22⋊C4×C59C2×C411yescharacteristic
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59C2311yescharacteristic
54C2×C42×C59D411yescharacteristic
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58C24×C59—21no—
59C24×C59—22no—
60C24×C59—41no—
61C24×C59—42no—
62C24×C59—41no—
63C24×C59—81no—
64C24×C59C22⋊C411yescharacteristic
65C24×C59C22×C411yescharacteristic
66C2×D4×C59—22no—
67C2×D4×C59—22no—
68C2×D4×C59—41no—
69C2×D4×C59—42no—
70C2×D4×C59—42no—
71C2×D4×C59—81no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—22no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59—81no—
83C22×C4×C59—81no—
84C22×C4×C59C22⋊C411yescharacteristic
85C22×C4×C59C22⋊C411yescharacteristic
86C22×C4×C59C22⋊C411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C4⋊C4×C59—41no—
90C4⋊C4×C59—42no—
91C4⋊C4×C59—81no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—42no—
95C22⋊C4×C59—42no—
96C22⋊C4×C59—42no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—42no—
100C22⋊C4×C59—41no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—81no—
103C42×C59—21no—
104C42×C59—21no—
105#11745664—82no—
106C2×C59⋊C8—42no—
107C2×C59⋊C8—42no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—22no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—42no—
122C23×C59—42no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—82no—
127C23×C59—84no—
128C23×C59—84no—
129C23×C59—82no—
130C23×C59C2×C22⋊C411yescharacteristic
131C23×C59#13112yes—
132D4×C59—44no—
133D4×C59—84no—
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—41no—
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—81no—
154C2×C4×C59—81no—
155C2×C4×C59—81no—
156C2×C4×C59—82no—
157C2×C4×C59#9911yescharacteristic
158C2×C4×C59#9911yescharacteristic
159C2×C4×C59#10012yes—
160C59⋊C8—82no—
161C59⋊C8—82no—
162#11559—591nomaximal, Sylow
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—22no—
166C22×C59—42no—
167C22×C59—42no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—41no—
174C22×C59—42no—
175C22×C59—41no—
176C22×C59—41no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59—81no—
180C22×C59—84no—
181C22×C59—84no—
182C22×C59—81no—
183C22×C59—81no—
184C22×C59#40811yescharacteristic
185C22×C59C2×#9911yescharacteristic
186C22×C59C2×#10011yescharacteristic
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—22no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59—81no—
194C4×C59—81no—
195C4×C59—81no—
196#2534D5911yescharacteristic
197#1465—591no—
198#1460—591no—
199#1317—592no—
200#1316—592no—
201C2×C59—21no—
202C2×C59—21no—
203C2×C59—42no—
204C2×C59—81no—
205C2×C59—82no—
206C2×C59#147111yescharacteristic
207C2×C59#212011yescharacteristic
208C2×C59#212111yescharacteristic
209C2×#123Dic5911yescharacteristic
210C2×#120Dic5911yescharacteristic
211C2×C4×D4C2×D5911yescharacteristic
212C22×C22⋊C4Dic5911yescharacteristic
213#431—1181no—
214#407—1181no—
215#396—21no—
216#394—21no—
217#388—21no—
218#386—21no—
219#381C2×D5911yescharacteristic
220#380Dic5911yescharacteristic
221#379C2×D5911yescharacteristic
222#324—1181no—
223#323—592no—
224#323—592no—
225#323—1181no—
226C59#1155911yescharacteristic, complemented, Sylow
227C25C2×Dic5911yescharacteristic
228C22×D4—21no—
229C22×D4C2×Dic5911yescharacteristic
230C23×C4C2×Dic5911yescharacteristic
231C23×C4C2×Dic5911yescharacteristic
232C23×C4C22⋊D5911yescharacteristic
233C2×M16—2361no—
234#123—42no—
235#120—42no—
236C4×D4—41no—
237C4×D4—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4C22⋊D5911yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4C2×Dic5911yescharacteristic
250C2×C22⋊C4C2×Dic5911yescharacteristic
251C2×C22⋊C4C22⋊D5911yescharacteristic
252C2×C22⋊C4C22×D5911yescharacteristic
253C2×C42C22⋊D5911yescharacteristic
254C4⋊C8—2361no—
255C22⋊C8—1182no—
256C22⋊C8—1182no—
257C22⋊C8—2361no—
258C4×C8—2361no—
259#95—21no—
260#95—21no—
261#95—21no—
262C24—21no—
263C24—22no—
264C24—41no—
265C24—42no—
266C24—41no—
267C24—81no—
268C24C22⋊Dic5911yescharacteristic
269C24C22×Dic5911yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—81no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4C22⋊Dic5911yescharacteristic
289C22×C4C22⋊Dic5911yescharacteristic
290C22×C4C22⋊Dic5911yescharacteristic
291C22×C4C2×C22⋊D5911yescharacteristic
292C22×C4C2×C22⋊D5911yescharacteristic
293M16—4722no—
294C2×C8—2362no—
295C2×C8—2362no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C42—21no—
311C42—21no—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—42no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—82no—
331C23—84no—
332C23—84no—
333C23—82no—
334C23C2×C22⋊Dic5911yescharacteristic
335C23#1427373412yes—
336D4—44no—
337D4—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
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—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4#1427364211yescharacteristic
362C2×C4#1427364211yescharacteristic
363C2×C4#1427364412yes—
364C8—4722no—
365C8—4722no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—84no—
384C22—84no—
385C22—81no—
386C22—81no—
387C22#1683736911yescharacteristic
388C22C2×#1427364211yescharacteristic
389C22C2×#1427364411yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—81no—
403C2—82no—
404C2#2142245611yescharacteristic
405C2#2142573711yescharacteristic
406C2#2142574211yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2534C2
socleC22×C59C2×#99
prime radicalC2×#120×C59C4
cosocleC2×C22⋊C4×C59C23
hypercenter#2534D59
hypocenterC59#11559
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. #21422434C2semidirect
2. #21422417C2semidirect
3. #21422026C2semidirect
4. #21422024C2semidirect
226. C59#11559semidirect

Maximal subgroups

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

subgroupindex
#214224342
#214224172
#214220262
#214220242
C59×#25342
#1155959