Finite Group Databasebeta

← #28499705

Subgroups — #28499705

Order 15104.

776 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 99 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#21424084C211yesmaximal, characteristic, complemented
3#21422094C214yesmaximal, complemented
4C59×#1460C211yesmaximal, characteristic, complemented
5C23×D236C411yescharacteristic, complemented
6C2×#14273701—21no—
7C2×#14273695—21no—
8C2×#14273642—21no—
9#16837319C412yescomplemented
10#16837311C2212yes—
11#16837304—21no—
12#16837262—21no—
13#16837262C411yescharacteristic, complemented
14#16837257—21no—
15#16837169C2212yescomplemented
16C2×C4×D4×C59C2211yescharacteristic
17C59×#431—21no—
18C59×#323—21no—
19C59×#323C2212yescomplemented
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—42no—
29#14273695—42no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59—41no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—22no—
34C2×(C2×D59)⋊C4—22no—
35C2×(C2×D59)⋊C4—22no—
36C2×(C2×D59)⋊C4—22no—
37#14273642—42no—
38C59⋊#95—41no—
39C59⋊#95C2×C412yes—
40C22×D4×C59D411yescharacteristic
41C23×C4×C59C2×C411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C4×D4×C59—41no—
44C4×D4×C59—41no—
45C2×C4⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59C2311yescharacteristic
48C2×C42×C59D411yescharacteristic
49C4⋊C8×C59—41no—
50C22⋊C8×C59—22no—
51C4×C8×C59—41no—
52C24×C59C22⋊C411yescharacteristic
53C24×C59C2×D411yescharacteristic
54C23×D59—22no—
55C23×D59—44no—
56C23×D59—42no—
57C23×D59—42no—
58C23×D59—42no—
59C23×D59—82no—
60C2×D4×C59—41no—
61C2×D4×C59—41no—
62C2×D4×C59D812yes—
63C2×D4×C59SD1612yes—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—42no—
69C22×C4×C59—42no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—44no—
76C2×C22⋊D59—42no—
77C2×C22⋊D59—42no—
78C2×C22⋊D59—82no—
79C22×Dic59—22no—
80C22×Dic59—42no—
81C2×D236—24no—
82C2×D236—42no—
83C2×D236—42no—
84C2×D236—48no—
85C2×D236—44no—
86C2×D236—41no—
87C2×D236—41no—
88C2×D236—44no—
89C2×D236—44no—
90C2×C8×C59—42no—
91C4⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C42×C59—21no—
95C42×C59—21no—
96C22⋊Dic59—42no—
97C22⋊Dic59—42no—
98C22⋊Dic59—42no—
99(C2×D59)⋊C4—44no—
100(C2×D59)⋊C4—44no—
101(C2×D59)⋊C4—82no—
102C4⋊Dic59—42no—
103C4⋊Dic59—82no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59#10212yes—
113C23×C59C2×C22⋊C411yescharacteristic
114C23×C59C2×D811yescharacteristic
115C23×C59C2×SD1611yescharacteristic
116C22×D59—24no—
117C22×D59—42no—
118C22×D59—42no—
119C22×D59—42no—
120C22×D59—48no—
121C22×D59—42no—
122C22×D59—42no—
123C22×D59—42no—
124C22×D59—88no—
125C22×D59—88no—
126C22×D59—84no—
127C22×D59—84no—
128D4×C59—42no—
129D4×C59—42no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—22no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—42no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—81no—
143C2×C4×C59#9911yescharacteristic
144C2×C4×C59#9911yescharacteristic
145C2×C4×C59#10212yes—
146C22⋊D59—84no—
147C22⋊D59—84no—
148C2×Dic59—42no—
149C2×Dic59—42no—
150C2×Dic59—42no—
151C2×Dic59—82no—
152C2×Dic59—82no—
153C2×Dic59—82no—
154D236—44no—
155D236—44no—
156D236—44no—
157D236—44no—
158C8×C59—82no—
159#11873—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—22no—
164C22×C59—21no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—42no—
168C22×C59—42no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—81no—
174C22×C59C2×#9911yescharacteristic
175C22×C59C2×#10211yescharacteristic
176C22×C59#42111yescharacteristic
177C2×D59—44no—
178C2×D59—44no—
179C2×D59—44no—
180C2×D59—82no—
181C2×D59—82no—
182C2×D59—816no—
183C2×D59—82no—
184C2×D59—82no—
185C4×C59—21no—
186C4×C59—22no—
187C4×C59—41no—
188C4×C59—41no—
189C4×C59—41no—
190C4×C59AΣL(1,𝔽16)12yes—
191Dic59—82no—
192Dic59—162no—
193#2408—591no—
194#1799—591no—
195#1460D5911yescharacteristic, complemented
196#1343—594no—
197C2×C59—21no—
198C2×C59—21no—
199C2×C59—41no—
200C2×C59—42no—
201C2×C59—41no—
202C2×C59#151911yescharacteristic
203C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
204C2×C59#212811yescharacteristic
205D59—84no—
206C23×D4—591no—
207C2×#121—1181no—
208C2×#120—1181no—
209C2×C4×D4C2×D5911yescharacteristic
210#431—21no—
211C2×#99—1181no—
212#394—592no—
213#392—592no—
214#390—1181no—
215#380—591no—
216#380—1181no—
217#379—1181no—
218#342—592no—
219#323—21no—
220#323C2×D5912yescomplemented
221C59#1187311yescharacteristic, complemented, Sylow
222C25—1181no—
223C22×D4—592no—
224C22×D4—1182no—
225C22×D4—1181no—
226C22×D4—2361no—
227C22×D4—2361no—
228C22×D4—2364no—
229C22×D4D23611yescharacteristic
230C23×C4C4×D5911yescharacteristic
231C23×C4D23611yescharacteristic
232#121—2364no—
233#120—2362no—
234#120—2362no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—1181no—
238C2×C4⋊C4—2361no—
239C2×C4⋊C4C4×D5911yescharacteristic
240C2×C22⋊C4—1182no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4—1182no—
243C2×C22⋊C4—1182no—
244C2×C22⋊C4—1182no—
245C2×C22⋊C4C22⋊D5911yescharacteristic
246C2×C22⋊C4C22×D5911yescharacteristic
247C2×C42C22⋊D5911yescharacteristic
248C4⋊C8—41no—
249#99—2362no—
250C22⋊C8—22no—
251C4×C8—41no—
252#95—592no—
253#95—2361no—
254C24—1182no—
255C24—4722no—
256C24—2362no—
257C24—2362no—
258C24—2364no—
259C24—2362no—
260C24(C2×D59)⋊C411yescharacteristic
261C24C2×D23611yescharacteristic
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—1184no—
265C2×D4—4722no—
266C2×D4—2362no—
267C2×D4—2362no—
268C2×D4—2368no—
269C2×D4—2364no—
270C2×D4—2364no—
271C2×D4—2364no—
272C2×D4—2364no—
273C2×D4—2362no—
274C2×D4—2362no—
275C2×D4—2361no—
276C2×D4—2361no—
277C2×D4#1174566012yes—
278C2×D4D47212yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—1182no—
286C22×C4—2362no—
287C22×C4(C2×D59)⋊C411yescharacteristic
288C22×C4(C2×D59)⋊C411yescharacteristic
289C22×C4(C2×D59)⋊C411yescharacteristic
290C22×C4C2×C4×D5911yescharacteristic
291C22×C4C2×C22⋊D5911yescharacteristic
292C2×C8—42no—
293C4⋊C4—41no—
294C4⋊C4—4722no—
295C4⋊C4—2362no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—4722no—
299C22⋊C4—2364no—
300C22⋊C4—2362no—
301C22⋊C4—2362no—
302C22⋊C4—2362no—
303C22⋊C4—2364no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—1184no—
315C23—4724no—
316C23—4724no—
317C23—4728no—
318C23—4728no—
319C23—2362no—
320C23—2362no—
321C23—2362no—
322C23—2368no—
323C23—2362no—
324C23—2362no—
325C23—2362no—
326C23#1427364912yes—
327C23C2×(C2×D59)⋊C411yescharacteristic
328C23C2×D47211yescharacteristic
329C23C2×#1174566011yescharacteristic
330D4—42no—
331D4—42no—
332D4—4724no—
333D4—4724no—
334D4—2364no—
335D4—2364no—
336D4—2364no—
337D4—2364no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—4722no—
352C2×C4—4722no—
353C2×C4—4722no—
354C2×C4—2362no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4#1427364111yescharacteristic
358C2×C4#1427364111yescharacteristic
359C2×C4#1427364912yes—
360C8—82no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—4722no—
376C22—4722no—
377C22—47216no—
378C22—4722no—
379C22—4722no—
380C22—2364no—
381C22—2364no—
382C22—2364no—
383C22C2×#1427364111yescharacteristic
384C22C2×#1427364911yescharacteristic
385C22#1683741811yescharacteristic
386C4—21no—
387C4—22no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—9442no—
392C4—4722no—
393C4#1683719012yes—
394C2—21no—
395C2—21no—
396C2—41no—
397C2—42no—
398C2—41no—
399C2—4724no—
400C2#2142266511yescharacteristic
401C2C2×#1683719011yescharacteristic
402C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1460C2
socleC22×C59C2×#99
prime radical#21427452C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1460D59
hypocenterC59#11873
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #21424084C2semidirect
3. #21422094C2semidirect
4. C59×#1460C2semidirect
5. C23×D236C4semidirect
9. #16837319C4semidirect
13. #16837262C4semidirect
15. #16837169C22semidirect
19. C59×#323C22semidirect
195. #1460D59semidirect
220. #323C2×D59semidirect
221. C59#11873semidirect

Maximal subgroups

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

subgroupindex
#214274522
#214240842
#214220942
C59×#14602
#1187359