Finite Group Databasebeta

← #28499248

Subgroups — #28499248

Order 15104.

544 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 83 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#21428993C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic, complemented
3#21422095C212yesmaximal, complemented
4#21422092C212yesmaximal, complemented
5C4×#99×C59C211yesmaximal, characteristic, complemented
6C2×#14273698—21no—
7C2×C4×D236C411yescharacteristic, complemented
8#16837366—21no—
9#16837327C411yescharacteristic, complemented
10#16837327C2211yescharacteristic
11#16837311C411yescharacteristic, complemented
12#16837311C2211yescharacteristic
13#16837281—21no—
14#16837281—21no—
15#16837268C411yescharacteristic, complemented
16#16837259—21no—
17#16837128—21no—
18#16837126C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×#99×C59—21no—
21C4×C22⋊C4×C59—21no—
22C59×#342C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C22×C4×D59—21no—
27C2×#11745664—41no—
28#14273698—44no—
29C4×D236—41no—
30C4×D236—41no—
31C2×C4⋊Dic59—21no—
32C2×C4⋊Dic59C2×C411yescharacteristic
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C22⋊(C59⋊C8)—22no—
39C22⋊(C59⋊C8)—41no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C22×D4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C4×D4×C59—41no—
49C4×D4×C59—41no—
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59—22no—
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59C2311yescharacteristic
54C2×C42×C59—41no—
55C2×C42×C59C2×C411yescharacteristic
56#99×C59—24no—
57#95×C59—41no—
58C24×C59C22⋊C411yescharacteristic
59C24×C59C2×D411yescharacteristic
60C23×D59—21no—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59—41no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—42no—
69C22×C4×C59—81no—
70C22×C4×C59—81no—
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C2×C22⋊D59—42no—
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C2×C22⋊D59—42no—
80C2×C22⋊D59—81no—
81C2×C22⋊D59—81no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C2×D236—41no—
86C2×D236—42no—
87C2×C4×D59—42no—
88C2×C4×D59—42no—
89C2×C4×D59—42no—
90C4⋊C4×C59—41no—
91C22⋊C4×C59—24no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—42no—
95C42×C59—21no—
96C42×C59—21no—
97C42×C59—82no—
98C22⋊Dic59—42no—
99C22⋊Dic59—42no—
100(C2×D59)⋊C4—41no—
101(C2×D59)⋊C4—41no—
102(C2×D59)⋊C4—81no—
103(C2×D59)⋊C4—81no—
104C4⋊Dic59—42no—
105C4⋊Dic59—41no—
106#11745664—82no—
107C2×C59⋊C8—42no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59C2×C22⋊C411yescharacteristic
118C23×C59AGL(1,ℤ/8)12yes—
119C22×D59—22no—
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—41no—
124C22×D59—41no—
125C22×D59—41no—
126D4×C59—44no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—41no—
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—81no—
141C2×C4×C59—82no—
142C2×C4×C59—82no—
143C2×C4×C59#9911yescharacteristic
144C2×C4×C59#9911yescharacteristic
145C2×C4×C59#10412yes—
146C22⋊D59—84no—
147C22⋊D59—84no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157D236—42no—
158D236—42no—
159C4×D59—84no—
160C59⋊C8—82no—
161#11774—591nomaximal, Sylow
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—22no—
165C22×C59—42no—
166C22×C59—42no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59C2×#9911yescharacteristic
175C22×C59#41811yescharacteristic
176C22×C59C2×#10411yescharacteristic
177C2×D59—42no—
178C2×D59—42no—
179C2×D59—42no—
180C2×D59—81no—
181C2×D59—81no—
182C2×D59—81no—
183C2×D59—81no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—22no—
187C4×C59—41no—
188C4×C59—41no—
189C4×C59—41no—
190C4×C59—84no—
191Dic59—81no—
192Dic59—81no—
193Dic59—81no—
194#2535—591no—
195C4×#99D5911yescharacteristic, complemented
196#1465—591no—
197#1344—592no—
198#1343—592no—
199C2×C59—21no—
200C2×C59—21no—
201C2×C59—42no—
202C2×C59—81no—
203C2×C59#150711yescharacteristic
204C2×C59#212011yescharacteristic
205C2×C59#212711yescharacteristic
206D59—82no—
207C2×#121—1181no—
208C2×C4×D4—591no—
209C2×C4×D4C2×D5911yescharacteristic
210C2×#99—21no—
211#407—1181no—
212#397—591no—
213#397—591no—
214#392—591no—
215#392—591no—
216#386—1181no—
217#386—1181no—
218#382—591no—
219#380—1181no—
220C4×C22⋊C4—21no—
221#342C2×D5912yes—
222#324—1181no—
223#323—592no—
224C59#1177411yescharacteristic, complemented, Sylow
225C22×D4—591no—
226C22×D4—1181no—
227C22×D4—1181no—
228C22×D4C22⋊D5911yescharacteristic
229C23×C4—1181no—
230C23×C4D23611yescharacteristic
231C23×C4C22⋊D5911yescharacteristic
232C2×M16—2361no—
233#121—2364no—
234C4×D4—41no—
235C4×D4—41no—
236C4×D4—2361no—
237C4×D4—2361no—
238C2×C4⋊C4—591no—
239C2×C4⋊C4—1181no—
240C2×C4⋊C4D23611yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—1181no—
243C2×C22⋊C4—1181no—
244C2×C22⋊C4—1181no—
245C2×C22⋊C4—1181no—
246C2×C22⋊C4—1181no—
247C2×C22⋊C4C4×D5911yescharacteristic
248C2×C22⋊C4C22×D5911yescharacteristic
249C2×C42—41no—
250C2×C42C4×D5911yescharacteristic
251#99—24no—
252C22⋊C8—1182no—
253C22⋊C8—2361no—
254#95—41no—
255#95—591no—
256#95—591no—
257#95—1181no—
258#95—1181no—
259#95—1181no—
260C24—1181no—
261C24(C2×D59)⋊C411yescharacteristic
262C24C2×C22⋊D5911yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—4721no—
267C2×D4—4721no—
268C2×D4—2361no—
269C2×D4—2362no—
270C2×D4—2362no—
271C2×D4—2362no—
272C2×D4—2362no—
273C2×D4—2362no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—1181no—
282C22×C4—1181no—
283C22×C4—1181no—
284C22×C4—2362no—
285C22×C4—2362no—
286C22×C4—2362no—
287C22×C4(C2×D59)⋊C411yescharacteristic
288C22×C4(C2×D59)⋊C411yescharacteristic
289C22×C4(C2×D59)⋊C411yescharacteristic
290C22×C4C2×C4×D5911yescharacteristic
291C22×C4C2×D23611yescharacteristic
292M16—4722no—
293C2×C8—2362no—
294C4⋊C4—41no—
295C4⋊C4—2361no—
296C4⋊C4—2362no—
297C22⋊C4—24no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22⋊C4—4721no—
302C22⋊C4—4721no—
303C22⋊C4—2362no—
304C22⋊C4—2362no—
305C22⋊C4—2361no—
306C22⋊C4—2361no—
307C42—21no—
308C42—21no—
309C42—82no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—1182no—
320C23—2361no—
321C23—2361no—
322C23—2361no—
323C23—2361no—
324C23—2361no—
325C23—2361no—
326C23C2×(C2×D59)⋊C411yescharacteristic
327C23#1427375212yes—
328D4—44no—
329D4—4724no—
330D4—4724no—
331D4—2362no—
332D4—2362no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—44no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—4724no—
350C2×C4—2361no—
351C2×C4—2361no—
352C2×C4—2361no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4—2361no—
356C2×C4—2361no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4#1427364111yescharacteristic
360C2×C4#1427364111yescharacteristic
361C2×C4#1427365512yes—
362C8—4722no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—42no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—81no—
375C22—4721no—
376C22—4721no—
377C22—4721no—
378C22—4721no—
379C22—2362no—
380C22—2362no—
381C22—2362no—
382C22C2×#1427364111yescharacteristic
383C22#1683740611yescharacteristic
384C22C2×#1427365511yescharacteristic
385C4—22no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—84no—
392C4—4721no—
393C4—4721no—
394C4—4721no—
395C2—21no—
396C2—21no—
397C2—42no—
398C2—81no—
399C2—4722no—
400C2#2142261111yescharacteristic
401C2#2142573511yescharacteristic
402C2#2142576311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radical#21428993C2
cosocleC2×C22⋊C4×C59C23
hypercenterC4×#99D59
hypocenterC59#11774
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. #21422434C2semidirect
3. #21422095C2semidirect
4. #21422092C2semidirect
5. C4×#99×C59C2semidirect
7. C2×C4×D236C4semidirect
9. #16837327C4semidirect
11. #16837311C4semidirect
15. #16837268C4semidirect
195. C4×#99D59semidirect
224. C59#11774semidirect

Maximal subgroups

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

subgroupindex
#214289932
#214224342
#214220952
#214220922
C4×#99×C592
#1177459