Finite Group Databasebeta

← #28499153

Subgroups — #28499153

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#21428370C211yesmaximal, characteristic
2#21422423C211yesmaximal, characteristic
3#21422119C212yesmaximal, complemented
4#21422095C212yesmaximal, complemented
5C2×C59×#342C211yesmaximal, characteristic
6C2×#14273686C411yescharacteristic, complemented
7#16837366—21no—
8#16837338C411yescharacteristic, complemented
9#16837338C2211yescharacteristic
10#16837327C411yescharacteristic, complemented
11#16837327C2211yescharacteristic
12#16837290—21no—
13#16837290—21no—
14#16837275—21no—
15#16837265C411yescharacteristic, complemented
16#16837258—21no—
17#16837126—21no—
18#16837126C2212yes—
19C22×C22⋊C4×C59—21no—
20C22×C22⋊C4×C59C2211yescharacteristic
21C59×#342—21no—
22C59×#342C2212yes—
23C2×#11745664—41no—
24#14273686—41no—
25#14273686—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic, complemented
27C2×C4⋊Dic59C2×C411yescharacteristic, complemented
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C4×Dic59—21no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)—41no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic, complemented
43C59⋊#95C2×C411yescharacteristic, complemented
44C25×C59D411yescharacteristic
45C23×C4×C59—41no—
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C2×C22⋊C4×C59—24no—
49C2×C22⋊C4×C59—22no—
50C2×C22⋊C4×C59—41no—
51C2×C22⋊C4×C59—41no—
52C2×C22⋊C4×C59—41no—
53C2×C22⋊C4×C59—42no—
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C22⋊C4×C59D411yescharacteristic
57C2×C22⋊C4×C59C2311yescharacteristic
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—44no—
74C22×C4×C59—42no—
75C22×C4×C59—81no—
76C22×C4×C59—82no—
77C22×C4×C59—82no—
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×Dic59—21no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22⋊C4×C59—24no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—44no—
92C22⋊C4×C59—42no—
93C22⋊Dic59—42no—
94C22⋊Dic59—41no—
95C22⋊Dic59—41no—
96C22⋊Dic59—42no—
97C22⋊Dic59—41no—
98C22⋊Dic59—41no—
99C22⋊Dic59—81no—
100C22⋊Dic59—81no—
101C4⋊Dic59—41no—
102C4⋊Dic59—41no—
103Dic59⋊C4—81no—
104Dic59⋊C4—81no—
105C4×Dic59—42no—
106#11745664—82no—
107C2×C59⋊C8—42no—
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—42no—
143C2×C4×C59—42no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—44no—
149C2×C4×C59—81no—
150C2×C4×C59—88no—
151C2×C4×C59—82no—
152C2×C4×C59—82no—
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—41no—
163C2×Dic59—41no—
164C2×Dic59—41no—
165C59⋊C8—82no—
166#11755—591nomaximal, Sylow
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—22no—
176C22×C59—21no—
177C22×C59—42no—
178C22×C59—42no—
179C22×C59—42no—
180C22×C59—41no—
181C22×C59—41no—
182C22×C59—44no—
183C22×C59—44no—
184C22×C59—41no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—41no—
188C22×C59—81no—
189C22×C59—81no—
190C22×C59—82no—
191C22×C59—82no—
192C22×C59#35212yes—
193C22×C59C2×#9911yescharacteristic
194C22×C59#41711yescharacteristic
195C22×C59C2×#10411yescharacteristic
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—41no—
200C4×C59—84no—
201Dic59—81no—
202Dic59—81no—
203Dic59—81no—
204Dic59—81no—
205#2482—591no—
206C2×#342D5911yescharacteristic
207#1462—591no—
208#1352—592no—
209#1344—592no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59—22no—
213C2×C59—41no—
214C2×C59—41no—
215C2×C59—42no—
216C2×C59—41no—
217C2×C59#150411yescharacteristic
218C2×C59C2×#35211yescharacteristic
219C2×C59#212711yescharacteristic
220C2×#117—591no—
221C22×C22⋊C4—21no—
222C22×C22⋊C4C2×D5911yescharacteristic
223#407—1181no—
224#399—591no—
225#399—591no—
226#397—591no—
227#397—591no—
228#388—1181no—
229#388—1181no—
230#385—1181no—
231#381—591no—
232#379—1181no—
233#342—21no—
234#342C2×D5912yes—
235#323—592no—
236#323—1181no—
237C59#1175511yescharacteristic, complemented, Sylow
238C25C22⋊D5911yescharacteristic
239C23×C4—41no—
240C23×C4C4×D5911yescharacteristic
241C23×C4D23611yescharacteristic
242C2×M16—2361no—
243#117—2361no—
244#117—2361no—
245C2×C4⋊C4—591no—
246C2×C4⋊C4—591no—
247C2×C4⋊C4—1181no—
248C2×C4⋊C4—1181no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—1181no—
256C2×C22⋊C4—1181no—
257C2×C22⋊C4—1181no—
258C2×C22⋊C4—1181no—
259C2×C22⋊C4—1181no—
260C2×C22⋊C4—1181no—
261C2×C22⋊C4C4×D5911yescharacteristic
262C2×C22⋊C4D23611yescharacteristic
263C2×C22⋊C4C22⋊D5911yescharacteristic
264C2×C22⋊C4C22×D5911yescharacteristic
265C2×C42—1181no—
266C22⋊C8—1182no—
267C22⋊C8—2361no—
268#95—591no—
269#95—591no—
270#95—1181no—
271#95—1181no—
272#95—1181no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—21no—
277C24—21no—
278C24—42no—
279C24—81no—
280C24(C2×D59)⋊C411yescharacteristic
281C24(C2×D59)⋊C411yescharacteristic
282C24C2×C22⋊D5911yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—1181no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4—1181no—
297C22×C4(C2×D59)⋊C411yescharacteristic
298C22×C4(C2×D59)⋊C411yescharacteristic
299C22×C4C2×C4×D5911yescharacteristic
300C22×C4C2×D23611yescharacteristic
301M16—4722no—
302C2×C8—2362no—
303C4⋊C4—4721no—
304C4⋊C4—4721no—
305C4⋊C4—2361no—
306C4⋊C4—2361no—
307C22⋊C4—24no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—44no—
313C22⋊C4—42no—
314C22⋊C4—4721no—
315C22⋊C4—4721no—
316C22⋊C4—2362no—
317C22⋊C4—2361no—
318C22⋊C4—2361no—
319C22⋊C4—2362no—
320C22⋊C4—2361no—
321C22⋊C4—2361no—
322C42—2362no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—84no—
344C23—82no—
345C23—82no—
346C23#1427364111yescharacteristic
347C23#1427364111yescharacteristic
348C23#1427365512yes—
349C23C2×(C2×D59)⋊C411yescharacteristic
350C23#1427375311yescharacteristic
351C23#1427375811yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—81no—
365C2×C4—88no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—2361no—
369C2×C4—2361no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4—2361no—
377C2×C4—2361no—
378C2×C4—2361no—
379C2×C4—2361no—
380C8—4722no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—41no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22—81no—
404C22—82no—
405C22—82no—
406C22#1683719112yes—
407C22C2×#1427364111yescharacteristic
408C22#1683740111yescharacteristic
409C22C2×#1427365511yescharacteristic
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—84no—
415C4—4721no—
416C4—4721no—
417C4—4721no—
418C4—4721no—
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—41no—
424C2—42no—
425C2—41no—
426C2#2142259811yescharacteristic
427C2C2×#1683719111yescharacteristic
428C2#2142576311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC2×C59×#342C2
socleC22×C59C2×#99
prime radicalC25×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenterC2×#342D59
hypocenterC59#11755
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. #21422119C2semidirect
4. #21422095C2semidirect
6. C2×#14273686C4semidirect
8. #16837338C4semidirect
10. #16837327C4semidirect
15. #16837265C4semidirect
26. C2×C4⋊Dic59C2×C4semidirect
27. C2×C4⋊Dic59C2×C4semidirect
42. C59⋊#95C2×C4semidirect
43. C59⋊#95C2×C4semidirect
237. C59#11755semidirect

Maximal subgroups

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

subgroupindex
#214283702
#214224232
#214221192
#214220952
C2×C59×#3422
#1175559