Finite Group Databasebeta

← #28499176

Subgroups — #28499176

Order 15104.

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

normal subgroups only

This group has 586 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#21429254C211yesmaximal, characteristic
2C2×#16837169C211yesmaximal, characteristic
3#21422118C212yesmaximal, complemented
4#21422097C212yesmaximal, complemented
5C59×#1462C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic, complemented
7C22×C22⋊Dic59—21no—
8#16837327C411yescharacteristic, complemented
9#16837327C2211yescharacteristic
10#16837314C411yescharacteristic, complemented
11#16837314C2211yescharacteristic
12#16837296—21no—
13#16837296—21no—
14#16837265—21no—
15#16837258—21no—
16C4×C22⋊Dic59C411yescharacteristic, complemented
17#16837169—21no—
18#16837169C2212yes—
19C22×C22⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×Dic118C2×C411yescharacteristic, complemented
24C23×Dic59—41no—
25#14273706—41no—
26#14273706—41no—
27C2×C4⋊Dic59C2×C411yescharacteristic, complemented
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—24no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—41no—
38C2×C22⋊Dic59—42no—
39C2×C4×Dic59—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic, complemented
45C59⋊#95C2×C411yescharacteristic, complemented
46C25×C59D411yescharacteristic
47C23×C4×C59C2×C411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C2×M16×C59—41no—
50C2×C22⋊C4×C59—41no—
51C2×C22⋊C4×C59—41no—
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59C2311yescharacteristic
56C22⋊C8×C59—22no—
57C22⋊C8×C59—41no—
58C24×C59—21no—
59C24×C59—21no—
60C24×C59—22no—
61C24×C59—21no—
62C24×C59—21no—
63C24×C59—42no—
64C24×C59—81no—
65C24×C59C22⋊C411yescharacteristic
66C24×C59C22⋊C411yescharacteristic
67C24×C59C2×D411yescharacteristic
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—42no—
73C22×C4×C59—81no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×Dic59—21no—
79C22×Dic59—21no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—44no—
83C22×Dic59—42no—
84C22×Dic59—82no—
85C22×Dic59—82no—
86C2×Dic118—41no—
87C2×Dic118—81no—
88M16×C59—82no—
89C2×C8×C59—42no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊Dic59—24no—
95C22⋊Dic59—42no—
96C22⋊Dic59—44no—
97C22⋊Dic59—41no—
98C22⋊Dic59—41no—
99C22⋊Dic59—42no—
100C22⋊Dic59—41no—
101C22⋊Dic59—42no—
102C22⋊Dic59—41no—
103C22⋊Dic59—81no—
104C22⋊Dic59—81no—
105C4⋊Dic59—41no—
106Dic59⋊C4—42no—
107C4×Dic59—81no—
108C23×C59—22no—
109C23×C59—22no—
110C23×C59—22no—
111C23×C59—22no—
112C23×C59—22no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—41no—
120C23×C59—42no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—42no—
126C23×C59—81no—
127C23×C59—81no—
128C23×C59—84no—
129C23×C59—82no—
130C23×C59—82no—
131C23×C59#9911yescharacteristic
132C23×C59#9911yescharacteristic
133C23×C59#10412yes—
134C23×C59C2×C22⋊C411yescharacteristic
135C23×C59AGL(1,ℤ/8)11yescharacteristic
136C23×C59#13711yescharacteristic
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—42no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—81no—
147C2×C4×C59—81no—
148C2×C4×C59—81no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—44no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—88no—
163C2×Dic59—82no—
164C2×Dic59—82no—
165Dic118—82no—
166C8×C59—82no—
167#11759—591nomaximal, Sylow
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—22no—
177C22×C59—21no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—42no—
181C22×C59—41no—
182C22×C59—41no—
183C22×C59—44no—
184C22×C59—44no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—41no—
188C22×C59—41no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59—82no—
192C22×C59—82no—
193C22×C59#35412yes—
194C22×C59C2×#9911yescharacteristic
195C22×C59#41711yescharacteristic
196C22×C59C2×#10411yescharacteristic
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—41no—
200C4×C59—41no—
201Dic59—81no—
202Dic59—81no—
203Dic59—84no—
204Dic59—81no—
205Dic59—81no—
206#2561—591no—
207C2×#342—591no—
208#1462D5911yescharacteristic
209#1351—592no—
210#1344—592no—
211C2×C59—21no—
212C2×C59—21no—
213C2×C59—22no—
214C2×C59—41no—
215C2×C59—41no—
216C2×C59—42no—
217C2×C59—41no—
218C2×C59#150411yescharacteristic
219C2×C59#212111yescharacteristic
220C2×C59C2×#35411yescharacteristic
221C2×C22⋊Q8—591no—
222C22×C22⋊C4—1181no—
223C22×C22⋊C4C2×D5911yescharacteristic
224#407—21no—
225#397—591no—
226#397—591no—
227#393—591no—
228#393—591no—
229#388—1181no—
230#388—1181no—
231#381—1181no—
232#379—1181no—
233C4×C22⋊C4—591no—
234#342—592no—
235#342—1181no—
236#323—21no—
237#323C2×D5912yes—
238C59#1175911yescharacteristic, complemented, Sylow
239C25D23611yescharacteristic
240C22×Q8—591no—
241C23×C4—2361no—
242C23×C4C4×D5911yescharacteristic
243C23×C4C22⋊D5911yescharacteristic
244C2×M16—41no—
245C22⋊Q8—2361no—
246C22⋊Q8—2361no—
247C2×C4⋊C4—591no—
248C2×C4⋊C4—1181no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—1184no—
252C2×C22⋊C4—1182no—
253C2×C22⋊C4—1181no—
254C2×C22⋊C4—1181no—
255C2×C22⋊C4—1181no—
256C2×C22⋊C4—1181no—
257C2×C22⋊C4—1181no—
258C2×C22⋊C4—1181no—
259C2×C22⋊C4—2361no—
260C2×C22⋊C4—2362no—
261C2×C22⋊C4C4×D5911yescharacteristic
262C2×C22⋊C4D23611yescharacteristic
263C2×C22⋊C4C22⋊D5911yescharacteristic
264C2×C22⋊C4C22×D5911yescharacteristic
265C2×C42—1181no—
266C22⋊C8—22no—
267C22⋊C8—41no—
268#95—591no—
269#95—591no—
270#95—1181no—
271#95—1181no—
272#95—1181no—
273#95—1181no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—42no—
280C24—81no—
281C24(C2×D59)⋊C411yescharacteristic
282C24(C2×D59)⋊C411yescharacteristic
283C24C2×D23611yescharacteristic
284C2×Q8—4721no—
285C2×Q8—2361no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—81no—
292C22×C4—1181no—
293C22×C4—1181no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4—4722no—
297C22×C4—4722no—
298C22×C4—2364no—
299C22×C4—2362no—
300C22×C4(C2×D59)⋊C411yescharacteristic
301C22×C4(C2×D59)⋊C411yescharacteristic
302C22×C4C2×C4×D5911yescharacteristic
303C22×C4C2×C22⋊D5911yescharacteristic
304M16—82no—
305C2×C8—42no—
306C4⋊C4—2361no—
307C4⋊C4—2362no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—1184no—
313C22⋊C4—4721no—
314C22⋊C4—4721no—
315C22⋊C4—2364no—
316C22⋊C4—2362no—
317C22⋊C4—2361no—
318C22⋊C4—2361no—
319C22⋊C4—2362no—
320C22⋊C4—2362no—
321C22⋊C4—2361no—
322C22⋊C4—2361no—
323C42—4721no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23—84no—
345C23—82no—
346C23—82no—
347C23#1427364111yescharacteristic
348C23#1427364111yescharacteristic
349C23#1427365312yes—
350C23C2×(C2×D59)⋊C411yescharacteristic
351C23#1427375511yescharacteristic
352C23#1427376011yescharacteristic
353Q8—4722no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—4728no—
367C2×C4—4722no—
368C2×C4—4722no—
369C2×C4—2361no—
370C2×C4—2361no—
371C2×C4—2364no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4—2361no—
377C2×C4—2361no—
378C2×C4—2361no—
379C2×C4—2361no—
380C2×C4—2361no—
381C2×C4—2361no—
382C8—82no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—44no—
399C22—44no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22—82no—
407C22—82no—
408C22#1683719612yes—
409C22C2×#1427364111yescharacteristic
410C22#1683740411yescharacteristic
411C22C2×#1427365311yescharacteristic
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—4721no—
417C4—4721no—
418C4—4724no—
419C4—4721no—
420C4—4721no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—41no—
425C2—41no—
426C2—42no—
427C2—41no—
428C2#2142260111yescharacteristic
429C2#2142574311yescharacteristic
430C2C2×#1683719611yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1462C2
socleC22×C59C2×#99
prime radicalC25×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1462D59
hypocenterC59#11759
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #21422118C2semidirect
4. #21422097C2semidirect
6. C2×#14273706C4semidirect
8. #16837327C4semidirect
10. #16837314C4semidirect
16. C4×C22⋊Dic59C4semidirect
23. C22×Dic118C2×C4semidirect
27. C2×C4⋊Dic59C2×C4semidirect
44. C59⋊#95C2×C4semidirect
45. C59⋊#95C2×C4semidirect
238. C59#11759semidirect

Maximal subgroups

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

subgroupindex
#214292542
C2×#168371692
#214221182
#214220972
C59×#14622
#1175959