Finite Group Databasebeta

← #28499360

Subgroups — #28499360

Order 15104.

574 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429830C211yesmaximal, characteristic
2#21424170C211yesmaximal, characteristic, complemented
3#21422097C212yesmaximal, complemented
4#21422094C212yesmaximal, complemented
5C59×#1462C211yesmaximal, characteristic, complemented
6C2×#14273701C411yescharacteristic, complemented
7C2×#14273696—21no—
8#16837327C411yescharacteristic, complemented
9#16837327C2211yescharacteristic
10#16837318—21no—
11#16837318—21no—
12#16837311C411yescharacteristic, complemented
13#16837311C2211yescharacteristic
14#16837281—21no—
15#16837262C411yescharacteristic, complemented
16#16837258—21no—
17#16837169—21no—
18#16837169C2212yes—
19C22×C22⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C22×C22⋊D59—21no—
27#14273701—41no—
28#14273701—41no—
29#14273696—44no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—22no—
36C2×C22⋊Dic59—41no—
37C2×C22⋊Dic59—41no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95C2×C411yescharacteristic
44C59⋊#95C2×C411yescharacteristic
45C25×C59D411yescharacteristic
46C23×C4×C59C2×C411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C2×M16×C59—41no—
49C2×C22⋊C4×C59—41no—
50C2×C22⋊C4×C59—41no—
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2311yescharacteristic
55C22⋊C8×C59—22no—
56C22⋊C8×C59—41no—
57C24×C59—21no—
58C24×C59—21no—
59C24×C59—22no—
60C24×C59—41no—
61C24×C59—81no—
62C24×C59—81no—
63C24×C59C22⋊C411yescharacteristic
64C24×C59C22⋊C411yescharacteristic
65C24×C59C2×D411yescharacteristic
66C23×D59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—81no—
72C22×C4×C59—81no—
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C2×C22⋊D59—42no—
80C2×C22⋊D59—41no—
81C2×C22⋊D59—41no—
82C2×C22⋊D59—42no—
83C2×C22⋊D59—81no—
84C2×C22⋊D59—81no—
85C2×C22⋊D59—81no—
86C2×C22⋊D59—81no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C22×Dic59—21no—
90C22×Dic59—42no—
91C2×D236—41no—
92C2×D236—81no—
93M16×C59—82no—
94C2×C8×C59—42no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C22⋊Dic59—42no—
100C22⋊Dic59—42no—
101C22⋊Dic59—42no—
102C22⋊Dic59—42no—
103C22⋊Dic59—42no—
104C22⋊Dic59—82no—
105C22⋊Dic59—82no—
106(C2×D59)⋊C4—42no—
107(C2×D59)⋊C4—81no—
108(C2×D59)⋊C4—81no—
109C4⋊Dic59—41no—
110C4⋊Dic59—81no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—42no—
121C23×C59—42no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—42no—
127C23×C59—42no—
128C23×C59—81no—
129C23×C59—81no—
130C23×C59—84no—
131C23×C59—84no—
132C23×C59#9911yescharacteristic
133C23×C59#9911yescharacteristic
134C23×C59C2×C22⋊C411yescharacteristic
135C23×C59AGL(1,ℤ/8)12yes—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
139C22×D59—41no—
140C22×D59—41no—
141C22×D59—41no—
142C22×D59—41no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—42no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—81no—
153C2×C4×C59—81no—
154C2×C4×C59—81no—
155C22⋊D59—82no—
156C22⋊D59—82no—
157C22⋊D59—84no—
158C22⋊D59—84no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—82no—
169C2×Dic59—82no—
170C2×Dic59—82no—
171D236—82no—
172C8×C59—82no—
173#11799—591nomaximal, Sylow
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—21no—
180C22×C59—22no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—42no—
184C22×C59—42no—
185C22×C59—42no—
186C22×C59—42no—
187C22×C59—41no—
188C22×C59—41no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59—41no—
192C22×C59—81no—
193C22×C59—81no—
194C22×C59—84no—
195C22×C59—84no—
196C22×C59C2×#9911yescharacteristic
197C22×C59#41811yescharacteristic
198C22×C59#42111yescharacteristic
199C2×D59—81no—
200C2×D59—81no—
201C2×D59—81no—
202C2×D59—81no—
203C2×D59—81no—
204C2×D59—81no—
205C2×D59—81no—
206C4×C59—41no—
207C4×C59—41no—
208C4×C59—41no—
209C4×C59—41no—
210Dic59—81no—
211Dic59—81no—
212Dic59—81no—
213Dic59—162no—
214#2617—591no—
215#1819—591no—
216#1462D5911yescharacteristic, complemented
217#1344—592no—
218#1343—592no—
219C2×C59—21no—
220C2×C59—21no—
221C2×C59—42no—
222C2×C59—41no—
223C2×C59—41no—
224C2×C59—81no—
225C2×C59#151011yescharacteristic
226C2×C59#212111yescharacteristic
227C2×C59#212711yescharacteristic
228D59—161no—
229C2×#121—591no—
230C2×#120—1181no—
231C22×C22⋊C4C2×D5911yescharacteristic
232#407—21no—
233#397—591no—
234#397—591no—
235#394—1181no—
236#394—1181no—
237#392—591no—
238#392—591no—
239#386—1181no—
240#380—591no—
241#379—1181no—
242#342—592no—
243#342—1181no—
244#323—21no—
245#323C2×D5912yes—
246C59#1179911yescharacteristic, complemented, Sylow
247C25D23611yescharacteristic
248C22×D4—591no—
249C22×D4—1181no—
250C22×D4—1181no—
251C22×D4—1181no—
252C23×C4C4×D5911yescharacteristic
253C23×C4C22⋊D5911yescharacteristic
254C2×M16—41no—
255#121—2361no—
256#121—2361no—
257#120—2364no—
258C2×C4⋊C4—591no—
259C2×C4⋊C4—1181no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—1181no—
263C2×C22⋊C4—1181no—
264C2×C22⋊C4—1181no—
265C2×C22⋊C4—1181no—
266C2×C22⋊C4—1181no—
267C2×C22⋊C4—1181no—
268C2×C22⋊C4—1182no—
269C2×C22⋊C4—2361no—
270C2×C22⋊C4—2361no—
271C2×C22⋊C4C4×D5911yescharacteristic
272C2×C22⋊C4D23611yescharacteristic
273C2×C22⋊C4C22⋊D5911yescharacteristic
274C2×C22⋊C4C22×D5911yescharacteristic
275C22⋊C8—22no—
276C22⋊C8—41no—
277#95—591no—
278#95—591no—
279#95—1181no—
280#95—1181no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—41no—
285C24—81no—
286C24—81no—
287C24—1181no—
288C24(C2×D59)⋊C411yescharacteristic
289C24(C2×D59)⋊C411yescharacteristic
290C24C2×D23611yescharacteristic
291C2×D4—4721no—
292C2×D4—4721no—
293C2×D4—4721no—
294C2×D4—4721no—
295C2×D4—4721no—
296C2×D4—2361no—
297C2×D4—2362no—
298C2×D4—2362no—
299C2×D4—2362no—
300C2×D4—2361no—
301C2×D4—2361no—
302C2×D4—2362no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—1181no—
310C22×C4—1181no—
311C22×C4—1181no—
312C22×C4—2362no—
313C22×C4(C2×D59)⋊C411yescharacteristic
314C22×C4(C2×D59)⋊C411yescharacteristic
315C22×C4C2×C4×D5911yescharacteristic
316C22×C4C2×C22⋊D5911yescharacteristic
317M16—82no—
318C2×C8—42no—
319C4⋊C4—4721no—
320C4⋊C4—2361no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—4721no—
326C22⋊C4—4721no—
327C22⋊C4—4722no—
328C22⋊C4—4722no—
329C22⋊C4—2362no—
330C22⋊C4—2362no—
331C22⋊C4—2362no—
332C22⋊C4—2362no—
333C22⋊C4—2362no—
334C22⋊C4—2362no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—81no—
353C23—81no—
354C23—84no—
355C23—84no—
356C23—2361no—
357C23—2361no—
358C23—2361no—
359C23—2361no—
360C23—2361no—
361C23—2361no—
362C23—2361no—
363C23#1427364111yescharacteristic
364C23#1427364111yescharacteristic
365C23C2×(C2×D59)⋊C411yescharacteristic
366C23#1427375512yes—
367D4—4722no—
368D4—4722no—
369D4—4722no—
370D4—4724no—
371D4—4724no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—4722no—
385C2×C4—4722no—
386C2×C4—4722no—
387C2×C4—2361no—
388C2×C4—2361no—
389C2×C4—2361no—
390C2×C4—2361no—
391C2×C4—2361no—
392C2×C4—2361no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2361no—
396C8—82no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—81no—
417C22—84no—
418C22—84no—
419C22—4721no—
420C22—4721no—
421C22—4721no—
422C22—4721no—
423C22—4721no—
424C22—4721no—
425C22—4721no—
426C22C2×#1427364111yescharacteristic
427C22#1683740811yescharacteristic
428C22#1683741811yescharacteristic
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—9442no—
434C4—4721no—
435C4—4721no—
436C4—4721no—
437C2—21no—
438C2—21no—
439C2—42no—
440C2—41no—
441C2—41no—
442C2—81no—
443C2—9441no—
444C2#2142262611yescharacteristic
445C2#2142574311yescharacteristic
446C2#2142576311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1462C2
socleC22×C59C2×#99
prime radical#21429830C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1462D59
hypocenterC59#11799
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
2. #21424170C2semidirect
3. #21422097C2semidirect
4. #21422094C2semidirect
5. C59×#1462C2semidirect
6. C2×#14273701C4semidirect
8. #16837327C4semidirect
12. #16837311C4semidirect
15. #16837262C4semidirect
216. #1462D59semidirect
246. C59#11799semidirect

Maximal subgroups

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

subgroupindex
#214298302
#214241702
#214220972
#214220942
C59×#14622
#1179959