Finite Group Databasebeta

← #28499240

Subgroups — #28499240

Order 15104.

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

normal subgroups only

This group has 786 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#21427452C211yesmaximal, characteristic
2#21422437C211yesmaximal, characteristic, complemented
3#21422092C214yesmaximal, complemented
4C4×#99×C59C211yesmaximal, characteristic, complemented
5C23×D236C411yescharacteristic, complemented
6C2×#14273701—21no—
7C2×#14273695—21no—
8#16837459—21no—
9#16837319C412yescomplemented
10#16837311C2212yes—
11#16837304—21no—
12#16837262C411yescharacteristic, complemented
13#16837257—21no—
14#16837126—21no—
15#16837126C2212yescomplemented
16C2×C4×D4×C59C2211yescharacteristic
17C2×#99×C59—21no—
18C4×C22⋊C4×C59—21no—
19C59×#342C2212yescomplemented
20C24×D59—21no—
21C22×D236—21no—
22C22×D236—44no—
23C22×D236—41no—
24C22×D236—41no—
25C22×D236C2×C412yes—
26C22×C22⋊D59—22no—
27#14273701—44no—
28#14273695—44no—
29C2×C4⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C2×(C2×D59)⋊C4—22no—
32C2×(C2×D59)⋊C4—22no—
33C2×(C2×D59)⋊C4—22no—
34C4⋊(C59⋊C8)—41no—
35C22⋊(C59⋊C8)—22no—
36#14273638—41no—
37C59⋊#95C2×C412yes—
38C22×D4×C59D411yescharacteristic
39C23×C4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C4×D4×C59—41no—
42C4×D4×C59—41no—
43C2×C4⋊C4×C59C2×C411yescharacteristic
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59D411yescharacteristic
46C2×C22⋊C4×C59C2311yescharacteristic
47C2×C42×C59—41no—
48C2×C42×C59D411yescharacteristic
49#99×C59—24no—
50#95×C59—41no—
51C24×C59C22⋊C411yescharacteristic
52C24×C59C2×D411yescharacteristic
53C23×D59—22no—
54C23×D59—44no—
55C23×D59—42no—
56C23×D59—42no—
57C23×D59—42no—
58C23×D59—82no—
59C2×D4×C59—22no—
60C2×D4×C59—22no—
61C2×D4×C59—41no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C2×C22⋊D59—44no—
75C2×C22⋊D59—44no—
76C2×C22⋊D59—82no—
77C22×Dic59—22no—
78C2×D236—24no—
79C2×D236—42no—
80C2×D236—42no—
81C2×D236—48no—
82C2×D236—44no—
83C2×D236—41no—
84C2×D236—41no—
85C2×D236—44no—
86C2×D236—44no—
87C4⋊C4×C59—41no—
88C22⋊C4×C59—24no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—42no—
92C42×C59—21no—
93C42×C59—21no—
94C42×C59—82no—
95C22⋊Dic59—42no—
96(C2×D59)⋊C4—44no—
97(C2×D59)⋊C4—44no—
98(C2×D59)⋊C4—82no—
99C4⋊Dic59—42no—
100C2×C59⋊C8—42no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59#10412yes—
110C23×C59C2×C22⋊C411yescharacteristic
111C23×C59AGL(1,ℤ/8)12yes—
112C22×D59—24no—
113C22×D59—42no—
114C22×D59—42no—
115C22×D59—42no—
116C22×D59—48no—
117C22×D59—42no—
118C22×D59—42no—
119C22×D59—42no—
120C22×D59—88no—
121C22×D59—88no—
122C22×D59—84no—
123C22×D59—84no—
124D4×C59—44no—
125C2×C4×C59—22no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—42no—
133C2×C4×C59—42no—
134C2×C4×C59—42no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—44no—
139C2×C4×C59—81no—
140C2×C4×C59—82no—
141C2×C4×C59—82no—
142C2×C4×C59#9911yescharacteristic
143C2×C4×C59#9911yescharacteristic
144C22⋊D59—88no—
145C2×Dic59—42no—
146C2×Dic59—42no—
147C2×Dic59—42no—
148D236—44no—
149D236—44no—
150D236—44no—
151D236—44no—
152C59⋊C8—82no—
153#11772—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—22no—
157C22×C59—22no—
158C22×C59—42no—
159C22×C59—42no—
160C22×C59—42no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—81no—
166C22×C59C2×#9911yescharacteristic
167C22×C59#41811yescharacteristic
168C22×C59C2×#10411yescharacteristic
169C2×D59—44no—
170C2×D59—44no—
171C2×D59—44no—
172C2×D59—82no—
173C2×D59—82no—
174C2×D59—816no—
175C2×D59—82no—
176C2×D59—82no—
177C4×C59—21no—
178C4×C59—41no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59—41no—
182C4×C59—84no—
183C4×C59AΣL(1,𝔽16)12yes—
184Dic59—82no—
185#2408—591no—
186C4×#99D5911yescharacteristic, complemented
187#1466—591no—
188#1343—594no—
189C2×C59—21no—
190C2×C59—21no—
191C2×C59—42no—
192C2×C59—42no—
193C2×C59#150711yescharacteristic
194C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
195C2×C59#212811yescharacteristic
196D59—84no—
197C23×D4—591no—
198C2×#121—1181no—
199C2×#120—1181no—
200C2×C4×D4C2×D5911yescharacteristic
201#432—1181no—
202C2×#99—21no—
203#394—592no—
204#392—592no—
205#390—1181no—
206#380—591no—
207#379—1181no—
208C4×C22⋊C4—21no—
209#342C2×D5912yescomplemented
210#323—592no—
211#323—1181no—
212C59#1177211yescharacteristic, complemented, Sylow
213C25—1181no—
214C22×D4—592no—
215C22×D4—1182no—
216C22×D4—1181no—
217C22×D4—2361no—
218C22×D4—2361no—
219C22×D4—2364no—
220C22×D4C22⋊D5911yescharacteristic
221C23×C4C4×D5911yescharacteristic
222C23×C4C22⋊D5911yescharacteristic
223#121—2364no—
224#120—2364no—
225C4×D4—41no—
226C4×D4—41no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4C4×D5911yescharacteristic
229C2×C22⋊C4—22no—
230C2×C22⋊C4—1182no—
231C2×C22⋊C4—1181no—
232C2×C22⋊C4—1182no—
233C2×C22⋊C4—1182no—
234C2×C22⋊C4D23611yescharacteristic
235C2×C22⋊C4C22×D5911yescharacteristic
236C2×C42—41no—
237C2×C42D23611yescharacteristic
238C4⋊C8—2361no—
239#99—24no—
240C22⋊C8—1182no—
241#97—2361no—
242#95—41no—
243#95—592no—
244C24—1182no—
245C24—4722no—
246C24—2362no—
247C24—2362no—
248C24—2364no—
249C24—2362no—
250C24(C2×D59)⋊C411yescharacteristic
251C24C2×C22⋊D5911yescharacteristic
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—1184no—
256C2×D4—4722no—
257C2×D4—2362no—
258C2×D4—2362no—
259C2×D4—2368no—
260C2×D4—2364no—
261C2×D4—2364no—
262C2×D4—2364no—
263C2×D4—2364no—
264C2×D4—2364no—
265C2×D4—2361no—
266C2×D4—2361no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—1182no—
275C22×C4(C2×D59)⋊C411yescharacteristic
276C22×C4(C2×D59)⋊C411yescharacteristic
277C22×C4(C2×D59)⋊C411yescharacteristic
278C22×C4C2×C4×D5911yescharacteristic
279C22×C4C2×D23611yescharacteristic
280C2×C8—2362no—
281C4⋊C4—41no—
282C4⋊C4—2362no—
283C22⋊C4—24no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—4722no—
288C22⋊C4—2364no—
289C22⋊C4—2364no—
290C22⋊C4—2362no—
291C42—21no—
292C42—21no—
293C42—82no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—1184no—
303C23—4724no—
304C23—4724no—
305C23—4728no—
306C23—4728no—
307C23—2362no—
308C23—2362no—
309C23—2362no—
310C23—2368no—
311C23—2362no—
312C23—2362no—
313C23—2362no—
314C23#1427365512yes—
315C23C2×(C2×D59)⋊C411yescharacteristic
316C23#1427375212yes—
317D4—44no—
318D4—4728no—
319D4—2364no—
320D4—2364no—
321D4—2364no—
322D4—2364no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—2362no—
341C2×C4—2362no—
342C2×C4—2362no—
343C2×C4#1427364111yescharacteristic
344C2×C4#1427364111yescharacteristic
345C8—4722no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—4722no—
359C22—4722no—
360C22—47216no—
361C22—4722no—
362C22—4722no—
363C22—2364no—
364C22—2364no—
365C22—2364no—
366C22C2×#1427364111yescharacteristic
367C22#1683740611yescharacteristic
368C22C2×#1427365511yescharacteristic
369C4—21no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—84no—
375C4—4722no—
376C4#1683718812yes—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—42no—
381C2—4724no—
382C2#2142261111yescharacteristic
383C2C2×#1683718811yescharacteristic
384C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radical#21427452C2
cosocleC2×C22⋊C4×C59C23
hypercenterC4×#99D59
hypocenterC59#11772
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. #21422437C2semidirect
3. #21422092C2semidirect
4. C4×#99×C59C2semidirect
5. C23×D236C4semidirect
9. #16837319C4semidirect
12. #16837262C4semidirect
15. #16837126C22semidirect
19. C59×#342C22semidirect
186. C4×#99D59semidirect
209. #342C2×D59semidirect
212. C59#11772semidirect

Maximal subgroups

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

subgroupindex
#214274522
#214224372
#214220922
C4×#99×C592
#1177259