Finite Group Databasebeta

← #28499754

Subgroups — #28499754

Order 15104.

422 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 83 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#21433675C211yesmaximal, characteristic
2C22⋊C16×D59C211yesmaximal, characteristic, complemented
3#21424120C211yesmaximal, characteristic, complemented
4#21422109C211yesmaximal, characteristic, complemented
5#21422107C211yesmaximal, characteristic, complemented
6#21422105C211yesmaximal, characteristic, complemented
7C59×#1347C211yesmaximal, characteristic, complemented
8C2×#14273746C411yescharacteristic, complemented
9C22×C8×D59C2211yescharacteristic
10#16837912—21no—
11C2×C16×D59—21no—
12#16837644C411yescharacteristic, complemented
13#16837643C2211yescharacteristic
14#16837635C411yescharacteristic, complemented
15#16837630—21no—
16#16837629—21no—
17#16837439C411yescharacteristic, complemented
18C2×#14273658—21no—
19#16837395—21no—
20#16837259—21no—
21C22⋊(C59⋊C16)C2211yescharacteristic
22(C2×D59)⋊C16—21no—
23#16837165C2211yescharacteristic
24C59×#466C2211yescharacteristic
25C59×C22⋊C16C2211yescharacteristic
26C59×#340C2211yescharacteristic
27C2×#11745687—21no—
28C22×C4×D59D411yescharacteristic
29#14273746—42no—
30C2×Dic236C2×C411yescharacteristic
31C2×#11745670—21no—
32C22×C59⋊C8D411yescharacteristic
33C2×C8×D59—42no—
34C2×C8×D59C2×C411yescharacteristic
35C2×C8×D59C2×C411yescharacteristic
36C2×C8×D59D411yescharacteristic
37C2×C8×D59D411yescharacteristic
38#14273709—41no—
39#14273707—21no—
40#14273706—21no—
41#14273698—21no—
42#14273695—41no—
43D4×Dic59—41no—
44(C22⋊D59)⋊C4—41no—
45#14273685—21no—
46C2×C4⋊Dic59—21no—
47C2×(C2×D59)⋊C4—41no—
48C2×C59⋊C16—21no—
49C16×D59—42no—
50#14273661C2×C411yescharacteristic
51#14273658—21no—
52#14273658—42no—
53#14273652—21no—
54#14273650—21no—
55#14273648—21no—
56C59⋊#95—41no—
57C2×D8×C59C2×C411yescharacteristic
58C22×C8×C59C2311yescharacteristic
59#121×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C16×C59—21no—
62#107×C59C2×C411yescharacteristic
63#102×C59—21no—
64C23×D59D811yescharacteristic
65C2×D4×C59—21no—
66C2×D4×C59—41no—
67C22×C4×C59—41no—
68C22×C4×C59C2×D411yescharacteristic
69C2×C22⋊D59—41no—
70C2×C22⋊D59—41no—
71C22×Dic59—41no—
72C22×Dic59—41no—
73C22×Dic59D811yescharacteristic
74#11745687—42no—
75#11745687—42no—
76C2×C4×D59—41no—
77C2×C4×D59—41no—
78C2×C4×D59C22⋊C411yescharacteristic
79C2×C4×D59C22⋊C411yescharacteristic
80C2×C4×D59SD1611yescharacteristic
81C2×C4×D59SD1611yescharacteristic
82C2×Dic118—21no—
83D8×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59C22×C411yescharacteristic
86C2×C8×C59C2×D411yescharacteristic
87C4⋊C4×C59—21no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊Dic59—41no—
92C22⋊Dic59—41no—
93#11745670—42no—
94(C2×D59)⋊C4—41no—
95(C2×D59)⋊C4—41no—
96(C2×D59)⋊C4—82no—
97C4⋊Dic59—21no—
98C4⋊Dic59—41no—
99C4⋊Dic59—41no—
100C4⋊Dic59—41no—
101Dic59⋊C4—41no—
102C4×Dic59—41no—
103C2×C59⋊C8—22no—
104C2×C59⋊C8C22⋊C411yescharacteristic
105C2×C59⋊C8C22⋊C411yescharacteristic
106Dic236—41no—
107C8×D59—22no—
108C8×D59—22no—
109C8×D59—41no—
110C8×D59—41no—
111C16×C59—41no—
112C59⋊C16—41no—
113C23×C59—41no—
114C23×C59C2×D811yescharacteristic
115C22×D59—21no—
116C22×D59—21no—
117C22×D59—22no—
118C22×D59—41no—
119C22×D59#10211yescharacteristic
120C22×D59#10211yescharacteristic
121D4×C59—41no—
122D4×C59—41no—
123D4×C59—81no—
124D4×C59—81no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—41no—
128C2×C4×C59—81no—
129C2×C4×C59—81no—
130C2×C4×C59—81no—
131C2×C4×C59C2×C22⋊C411yescharacteristic
132C2×C4×C59C2×SD1611yescharacteristic
133C22⋊D59—82no—
134C22⋊D59—82no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—82no—
139C2×Dic59—81no—
140C2×Dic59—81no—
141C2×Dic59—81no—
142C2×Dic59#10211yescharacteristic
143C2×Dic59#10211yescharacteristic
144C4×D59—21no—
145C4×D59—21no—
146C4×D59—21no—
147C4×D59—21no—
148C4×D59—41no—
149C4×D59—41no—
150Dic118—41no—
151Dic118—41no—
152C8×C59—21no—
153C8×C59—21no—
154C59⋊C8—21no—
155C59⋊C8#9912yes—
156#11885—591nomaximal, Sylow
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—81no—
160C22×C59—81no—
161C22×C59—81no—
162C22×C59C2×#10211yescharacteristic
163C22×C59#50811yescharacteristic
164C22×C59#50811yescharacteristic
165C2×D59—21no—
166C2×D59—21no—
167C2×D59—21no—
168C2×D59—21no—
169C2×D59—21no—
170C2×D59—42no—
171C2×D59—42no—
172C2×D59#36012yes—
173C4×C59—21no—
174C4×C59—81no—
175C4×C59—161no—
176C4×C59C2×#9911yescharacteristic
177C4×C59#42111yescharacteristic
178Dic59—21no—
179Dic59—21no—
180Dic59—81no—
181Dic59—81no—
182Dic59—161no—
183#3050—591no—
184C2×C22⋊C16—591no—
185#1807—591no—
186#1348—591no—
187#1348—591no—
188#1347—591no—
189#1347D5911yescharacteristic, complemented
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—161no—
193C2×C59#151911yescharacteristic
194C2×C59#213811yescharacteristic
195C2×C59C2×#36011yescharacteristic
196D59—22no—
197D59—41no—
198C2×#135—591no—
199C23×C8—591no—
200#525—1181no—
201C22×C16—1181no—
202#467—591no—
203#467—591no—
204#466—591no—
205#466C2×D5911yescharacteristic
206#465—1181no—
207#465—1181no—
208#427—591no—
209C2×#106—1181no—
210#416—1181no—
211#380—1181no—
212C22⋊C16—591no—
213C22⋊C16—1181no—
214C22⋊C16C2×D5911yescharacteristic
215#340—591no—
216#340C2×D5911yescharacteristic
217C59#1188511yescharacteristic, complemented, Sylow
218C2×#41—1181no—
219C23×C4—591no—
220#135—2362no—
221C2×Q16—591no—
222C2×SD16—1181no—
223C2×D8C4×D5911yescharacteristic
224C22×C8—591no—
225C22×C8—591no—
226C22×C8—591no—
227C22×C8—591no—
228C22×C8—591no—
229C22×C8—2362no—
230C22×C8C22×D5911yescharacteristic
231#123—2361no—
232C22⋊Q8—1181no—
233C22⋊Q8—1181no—
234#121—21no—
235#121—1181no—
236#120—2361no—
237C4×D4—2361no—
238C4×D4—2361no—
239#117—1181no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—1181no—
242C2×C22⋊C4—2361no—
243C2×C16—21no—
244C2×C16—1181no—
245C2×C16—2362no—
246#107—591no—
247#107C4×D5911yescharacteristic
248#106—1181no—
249#106—2362no—
250#103—1181no—
251#103—1181no—
252#102—21no—
253#102—1181no—
254#95—2361no—
255C24—591no—
256#41—2362no—
257#41—2362no—
258C2×Q8—1181no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—2361no—
262C2×D4—2361no—
263C22×C4—41no—
264C22×C4—591no—
265C22×C4—591no—
266C22×C4—591no—
267C22×C4—591no—
268C22×C4—591no—
269C22×C4—2361no—
270C22×C4—2361no—
271C22×C4—2361no—
272C22×C4—2361no—
273C22×C4D4×D5911yescharacteristic
274Q16—2361no—
275SD16—2362no—
276D8—41no—
277C2×C8—41no—
278C2×C8—591no—
279C2×C8—591no—
280C2×C8—1182no—
281C2×C8—1182no—
282C2×C8—1182no—
283C2×C8—2361no—
284C2×C8—2361no—
285C2×C8C2×C4×D5911yescharacteristic
286C2×C8D4×D5911yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—1181no—
291C4⋊C4—2361no—
292C4⋊C4—2361no—
293C4⋊C4—2361no—
294C4⋊C4—2361no—
295C22⋊C4—41no—
296C22⋊C4—4722no—
297C22⋊C4—2361no—
298C22⋊C4—2361no—
299C22⋊C4—2361no—
300C22⋊C4—2361no—
301C42—2361no—
302C16—41no—
303C16—2361no—
304C23—41no—
305C23—591no—
306C23—591no—
307C23—1182no—
308C23—1181no—
309C23—1181no—
310C23—2361no—
311C23D8×D5911yescharacteristic
312Q8—2361no—
313Q8—2361no—
314D4—41no—
315D4—41no—
316D4—81no—
317D4—81no—
318D4—4722no—
319D4—4722no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—591no—
327C2×C4—591no—
328C2×C4—1181no—
329C2×C4—1181no—
330C2×C4—1181no—
331C2×C4—1181no—
332C2×C4—4722no—
333C2×C4—4721no—
334C2×C4—4721no—
335C2×C4—4721no—
336C2×C4—2361no—
337C2×C4—2361no—
338C2×C4—2361no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4C22⋊C4×D5911yescharacteristic
342C2×C4SD16×D5911yescharacteristic
343C8—21no—
344C8—21no—
345C8—592no—
346C8—1181no—
347C22—21no—
348C22—21no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—592no—
353C22—1181no—
354C22—1181no—
355C22—1181no—
356C22—1181no—
357C22—1181no—
358C22—2362no—
359C22—2362no—
360C22#102×D5911yescharacteristic
361C22#1683783111yescharacteristic
362C22#1683783311yescharacteristic
363C4—21no—
364C4—81no—
365C4—161no—
366C4—1181no—
367C4—1181no—
368C4—9441no—
369C4—4721no—
370C4—4721no—
371C4#99×D5911yescharacteristic
372C4#1683741711yescharacteristic
373C2—21no—
374C2—21no—
375C2—161no—
376C2—1182no—
377C2—2361no—
378C2#2142266211yescharacteristic
379C2#2142580011yescharacteristic
380C2#360×D5911yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1347C2
socleC22×C59C2×#102
prime radical#21433675C2
cosocleC22×C8×C59C23
hypercenter#1347D59
hypocenterC59#11885
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1347

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D59C2semidirect
3. #21424120C2semidirect
4. #21422109C2semidirect
5. #21422107C2semidirect
6. #21422105C2semidirect
7. C59×#1347C2semidirect
8. C2×#14273746C4semidirect
12. #16837644C4semidirect
14. #16837635C4semidirect
17. #16837439C4semidirect
189. #1347D59semidirect
217. C59#11885semidirect

Maximal subgroups

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

subgroupindex
#214336752
C22⋊C16×D592
#214241202
#214221092
#214221072
#214221052
C59×#13472
#1188559