Finite Group Databasebeta

← #28499711

Subgroups — #28499711

Order 15104.

536 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 95 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#21422414C211yesmaximal, characteristic, complemented
3#21422095C212yesmaximal, complemented
4#21422092C212yesmaximal, complemented
5C59×#1799C211yesmaximal, characteristic, complemented
6C2×#14273698—21no—
7C2×C4×D236C411yescharacteristic, complemented
8C2×C22⋊(C59⋊C8)—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—
21C59×#380—21no—
22C59×#342C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C22×C4×D59—21no—
27C22×C59⋊C8—41no—
28#14273698—42no—
29#14273698—42no—
30C4×D236—41no—
31C4×D236—41no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59C2×C411yescharacteristic
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C22⋊(C59⋊C8)—22no—
40C22⋊(C59⋊C8)—41no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C22×D4×C59D411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C4×D4×C59—41no—
50C4×D4×C59—41no—
51C2×C4⋊C4×C59—41no—
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59—22no—
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59C2311yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57#99×C59—42no—
58#95×C59—41no—
59C24×C59C22⋊C411yescharacteristic
60C24×C59C2×D411yescharacteristic
61C23×D59—21no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59D812yes—
65C2×D4×C59SD1612yes—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—42no—
71C22×C4×C59—81no—
72C22×C4×C59—81no—
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C2×C22⋊D59—42no—
79C2×C22⋊D59—42no—
80C2×C22⋊D59—42no—
81C2×C22⋊D59—41no—
82C2×C22⋊D59—41no—
83C2×C22⋊D59—81no—
84C2×C22⋊D59—81no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C2×D236—41no—
89C2×D236—42no—
90C2×C4×D59—42no—
91C2×C4×D59—42no—
92C2×C4×D59—42no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—82no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—42no—
98C22⋊C4×C59—42no—
99C42×C59—21no—
100C42×C59—21no—
101C22⋊Dic59—42no—
102C22⋊Dic59—41no—
103C22⋊Dic59—41no—
104(C2×D59)⋊C4—41no—
105(C2×D59)⋊C4—41no—
106(C2×D59)⋊C4—81no—
107(C2×D59)⋊C4—81no—
108C4⋊Dic59—42no—
109C4⋊Dic59—41no—
110C2×C59⋊C8—42no—
111C2×C59⋊C8—82no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59#10212yes—
121C23×C59C2×C22⋊C411yescharacteristic
122C23×C59C2×D811yescharacteristic
123C23×C59C2×SD1611yescharacteristic
124C22×D59—22no—
125C22×D59—41no—
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—41no—
129C22×D59—41no—
130C22×D59—41no—
131D4×C59—42no—
132D4×C59—42no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—81no—
145C2×C4×C59—81no—
146C2×C4×C59—82no—
147C2×C4×C59—82no—
148C2×C4×C59—82no—
149C2×C4×C59#9911yescharacteristic
150C2×C4×C59#9911yescharacteristic
151C2×C4×C59#10212yes—
152C22⋊D59—84no—
153C22⋊D59—82no—
154C22⋊D59—82no—
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—
164D236—42no—
165D236—42no—
166C4×D59—84no—
167C59⋊C8—82no—
168#11875—591nomaximal, Sylow
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—22no—
173C22×C59—21no—
174C22×C59—42no—
175C22×C59—41no—
176C22×C59—41no—
177C22×C59—41no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59—81no—
182C22×C59—81no—
183C22×C59C2×#9911yescharacteristic
184C22×C59C2×#10211yescharacteristic
185C22×C59#42111yescharacteristic
186C2×D59—42no—
187C2×D59—42no—
188C2×D59—42no—
189C2×D59—81no—
190C2×D59—81no—
191C2×D59—81no—
192C2×D59—81no—
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—22no—
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—162no—
200Dic59—81no—
201Dic59—81no—
202Dic59—81no—
203#2535—591no—
204#1799D5911yescharacteristic, complemented
205#1459—591no—
206#1344—592no—
207#1343—592no—
208C2×C59—21no—
209C2×C59—21no—
210C2×C59—41no—
211C2×C59—42no—
212C2×C59—41no—
213C2×C59#151911yescharacteristic
214C2×C59#212011yescharacteristic
215C2×C59#212711yescharacteristic
216D59—82no—
217C2×#121—1181no—
218C2×C4×D4—591no—
219C2×C4×D4C2×D5911yescharacteristic
220C2×#99—21no—
221C2×C22⋊C8—1181no—
222#397—591no—
223#397—591no—
224#392—591no—
225#392—591no—
226#386—1181no—
227#386—1181no—
228#382—591no—
229#380—21no—
230#380—1181no—
231#342C2×D5912yes—
232#324—1181no—
233#323—592no—
234C59#1187511yescharacteristic, complemented, Sylow
235C22×D4—591no—
236C22×D4—1181no—
237C22×D4—1181no—
238C22×D4C22⋊D5911yescharacteristic
239C23×C4—1181no—
240C23×C4D23611yescharacteristic
241C23×C4C22⋊D5911yescharacteristic
242C22×C8—2361no—
243#121—2362no—
244#121—2362no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—2361no—
248C4×D4—2361no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—591no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4D23611yescharacteristic
253C2×C22⋊C4—22no—
254C2×C22⋊C4—1181no—
255C2×C22⋊C4—1181no—
256C2×C22⋊C4—1181no—
257C2×C22⋊C4—1181no—
258C2×C22⋊C4—1181no—
259C2×C22⋊C4C4×D5911yescharacteristic
260C2×C22⋊C4C22×D5911yescharacteristic
261C2×C42C4×D5911yescharacteristic
262#99—42no—
263C22⋊C8—1182no—
264C22⋊C8—2361no—
265#95—41no—
266#95—591no—
267#95—591no—
268#95—1181no—
269#95—1181no—
270#95—1181no—
271C24—1181no—
272C24(C2×D59)⋊C411yescharacteristic
273C24C2×C22⋊D5911yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—4721no—
277C2×D4—4721no—
278C2×D4—2361no—
279C2×D4—2362no—
280C2×D4—2362no—
281C2×D4—2362no—
282C2×D4—2362no—
283C2×D4—2361no—
284C2×D4—2361no—
285C2×D4#1174566912yes—
286C2×D4#1174567112yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4—1181no—
297C22×C4—2362no—
298C22×C4—2362no—
299C22×C4—2362no—
300C22×C4(C2×D59)⋊C411yescharacteristic
301C22×C4(C2×D59)⋊C411yescharacteristic
302C22×C4(C2×D59)⋊C411yescharacteristic
303C22×C4C2×C4×D5911yescharacteristic
304C22×C4C2×D23611yescharacteristic
305C2×C8—4722no—
306C2×C8—2362no—
307C4⋊C4—41no—
308C4⋊C4—82no—
309C4⋊C4—2361no—
310C4⋊C4—2362no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—4721no—
316C22⋊C4—4721no—
317C22⋊C4—2362no—
318C22⋊C4—2361no—
319C22⋊C4—2361no—
320C22⋊C4—2361no—
321C22⋊C4—2361no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—1182no—
333C23—2361no—
334C23—2361no—
335C23—2361no—
336C23—2361no—
337C23—2361no—
338C23—2361no—
339C23#1427364712yes—
340C23C2×(C2×D59)⋊C411yescharacteristic
341C23C2×#1174566911yescharacteristic
342C23C2×#1174567111yescharacteristic
343D4—42no—
344D4—42no—
345D4—4724no—
346D4—4722no—
347D4—4722no—
348D4—2362no—
349D4—2362no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—4724no—
367C2×C4—2361no—
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#1427364111yescharacteristic
377C2×C4#1427364111yescharacteristic
378C2×C4#1427364712yes—
379C8—4722no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—81no—
394C22—4721no—
395C22—4721no—
396C22—4721no—
397C22—4721no—
398C22—2362no—
399C22—2362no—
400C22—2362no—
401C22C2×#1427364111yescharacteristic
402C22C2×#1427364711yescharacteristic
403C22#1683742011yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—162no—
411C4—4721no—
412C4—4721no—
413C4—4721no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—42no—
418C2—41no—
419C2—4722no—
420C2#2142266311yescharacteristic
421C2#2142573511yescharacteristic
422C2#2142576311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1799C2
socleC22×C59C2×#99
prime radical#21428993C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1799D59
hypocenterC59#11875
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #21422414C2semidirect
3. #21422095C2semidirect
4. #21422092C2semidirect
5. C59×#1799C2semidirect
7. C2×C4×D236C4semidirect
9. #16837327C4semidirect
11. #16837311C4semidirect
15. #16837268C4semidirect
204. #1799D59semidirect
234. C59#11875semidirect

Maximal subgroups

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

subgroupindex
#214289932
#214224142
#214220952
#214220922
C59×#17992
#1187559