Finite Group Databasebeta

← #28499767

Subgroups — #28499767

Order 15104.

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

normal subgroups only

This group has 464 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#466×D59C211yesmaximal, characteristic
2#21425705C211yesmaximal, characteristic, complemented
3#21424120C211yesmaximal, characteristic, complemented
4#21422106C211yesmaximal, characteristic, complemented
5#21422105C211yesmaximal, characteristic, complemented
6#21422104C211yesmaximal, characteristic, complemented
7C59×#1347C211yesmaximal, characteristic, complemented
8C2×D8×D59C411yescharacteristic, complemented
9C22×C8×D59C2211yescharacteristic
10#121×D59—21no—
11C2×#14273666—21no—
12#16837639C411yescharacteristic, complemented
13#16837638C2211yescharacteristic
14#16837637—21no—
15#16837636—21no—
16#16837635C411yescharacteristic, complemented
17#107×D59C411yescharacteristic, complemented
18C2×#14273658—21no—
19#102×D59—21no—
20#16837259—21no—
21C22⋊(C59⋊C16)C2211yescharacteristic, complemented
22(C2×D59)⋊C16—21no—
23#16837165C2211yescharacteristic, complemented
24C59×#466C2211yescharacteristic
25C59×C22⋊C16C2211yescharacteristic, complemented
26C59×#340C2211yescharacteristic, complemented
27C2×D4×D59—21no—
28C2×D4×D59—41no—
29C22×C4×D59D411yescharacteristic
30C2×D472C2×C411yescharacteristic
31C2×#11745669—21no—
32D8×D59—42no—
33C22×C59⋊C8D411yescharacteristic
34C2×C8×D59—42no—
35C2×C8×D59C2×C411yescharacteristic
36C2×C8×D59C2×C411yescharacteristic
37C2×C8×D59D411yescharacteristic
38C2×C8×D59D411yescharacteristic
39#14273702—21no—
40#14273701—21no—
41#14273700—41no—
42#14273699—41no—
43#14273698—21no—
44C2×C4⋊Dic59—21no—
45C4⋊C4×D59—21no—
46C2×(C2×D59)⋊C4—41no—
47C22⋊C4×D59—41no—
48#14273666—42no—
49C2×C59⋊C16—21no—
50#14273661C2×C411yescharacteristic
51#14273660—21no—
52#14273658—42no—
53#14273649—21no—
54#14273648—21no—
55#14273647—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×D59—41no—
65C23×D59D811yescharacteristic
66C2×D4×C59—21no—
67C2×D4×C59—41no—
68C22×C4×C59—41no—
69C22×C4×C59C2×D411yescharacteristic
70C2×C22⋊D59—41no—
71C2×C22⋊D59—41no—
72C2×C22⋊D59—41no—
73C22×Dic59—41no—
74C22×Dic59D811yescharacteristic
75D4×D59—42no—
76D4×D59—42no—
77D4×D59—82no—
78D4×D59—82no—
79C2×D236—21no—
80C2×D236—41no—
81C2×C4×D59—41no—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84C2×C4×D59C22⋊C411yescharacteristic
85C2×C4×D59C22⋊C411yescharacteristic
86C2×C4×D59SD1611yescharacteristic
87C2×C4×D59SD1611yescharacteristic
88D8×C59—41no—
89C2×C8×C59—41no—
90C2×C8×C59C22×C411yescharacteristic
91C2×C8×C59C2×D411yescharacteristic
92C4⋊C4×C59—21no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊Dic59—41no—
97#11745669—42no—
98(C2×D59)⋊C4—41no—
99(C2×D59)⋊C4—41no—
100(C2×D59)⋊C4—82no—
101C4⋊Dic59—21no—
102C4⋊Dic59—41no—
103C4⋊Dic59—41no—
104Dic59⋊C4—41no—
105C2×C59⋊C8—22no—
106C2×C59⋊C8C22⋊C411yescharacteristic
107C2×C59⋊C8C22⋊C411yescharacteristic
108D472—41no—
109C8×D59—22no—
110C8×D59—22no—
111C8×D59—41no—
112C8×D59—41no—
113C16×C59—41no—
114C59⋊C16—41no—
115C23×C59—41no—
116C23×C59C2×D811yescharacteristic
117C22×D59—21no—
118C22×D59—21no—
119C22×D59—22no—
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—82no—
123C22×D59—82no—
124C22×D59—82no—
125C22×D59#10211yescharacteristic
126C22×D59#10211yescharacteristic
127D4×C59—41no—
128D4×C59—41no—
129D4×C59—81no—
130D4×C59—81no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—41no—
134C2×C4×C59—81no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59C2×C22⋊C411yescharacteristic
138C2×C4×C59C2×SD1611yescharacteristic
139C22⋊D59—82no—
140C22⋊D59—82no—
141C22⋊D59—81no—
142C22⋊D59—81no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—81no—
146C2×Dic59—81no—
147C2×Dic59—81no—
148C2×Dic59#10211yescharacteristic
149C2×Dic59#10211yescharacteristic
150D236—41no—
151D236—41no—
152D236—82no—
153C4×D59—21no—
154C4×D59—21no—
155C4×D59—21no—
156C4×D59—21no—
157C4×D59—41no—
158C4×D59—41no—
159C4×D59—82no—
160C8×C59—21no—
161C8×C59—21no—
162C59⋊C8—21no—
163C59⋊C8#9912yes—
164#11888—591nomaximal, Sylow
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59C2×#10211yescharacteristic
171C22×C59#50911yescharacteristic
172C22×C59#50911yescharacteristic
173C2×D59—21no—
174C2×D59—21no—
175C2×D59—21no—
176C2×D59—21no—
177C2×D59—21no—
178C2×D59—42no—
179C2×D59—42no—
180C2×D59—81no—
181C2×D59—81no—
182C2×D59—81no—
183C2×D59—164no—
184C2×D59#36012yes—
185C4×C59—21no—
186C4×C59—81no—
187C4×C59—161no—
188C4×C59C2×#9911yescharacteristic
189C4×C59#42111yescharacteristic
190Dic59—21no—
191Dic59—21no—
192Dic59—81no—
193Dic59—161no—
194C2×#466—591no—
195#2112—591no—
196#1807—591no—
197#1347—591no—
198#1347—591no—
199#1347—591no—
200#1347D5911yescharacteristic, complemented
201C2×C59—21no—
202C2×C59—21no—
203C2×C59—161no—
204C2×C59#151911yescharacteristic
205C2×C59#214011yescharacteristic
206C2×C59C2×#36011yescharacteristic
207D59—22no—
208D59—41no—
209D59—161no—
210C22×D8—591no—
211C23×C8—591no—
212C2×#121—1181no—
213C2×M32—1181no—
214#466—591no—
215#466—591no—
216#466—591no—
217#466—1181no—
218#466—1181no—
219#466C2×D5911yescharacteristic
220C2×#107—591no—
221C2×#106—1181no—
222C2×#102—1181no—
223#380—1181no—
224C22⋊C16—591no—
225C22⋊C16—1181no—
226C22⋊C16C2×D5911yescharacteristic, complemented
227#340—591no—
228#340C2×D5911yescharacteristic, complemented
229C59#1188811yescharacteristic, complemented, Sylow
230C22×D4—1181no—
231C22×D4—2361no—
232C23×C4—591no—
233C2×D8—591no—
234C2×D8—1181no—
235C2×D8—2362no—
236C2×D8C4×D5911yescharacteristic
237C22×C8—591no—
238C22×C8—591no—
239C22×C8—591no—
240C22×C8—591no—
241C22×C8—591no—
242C22×C8—2362no—
243C22×C8C22×D5911yescharacteristic
244#121—21no—
245#121—1181no—
246#121—1181no—
247#121—1181no—
248#121—2361no—
249#121—2361no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—1181no—
253C2×C22⋊C4—2361no—
254C2×C22⋊C4—2361no—
255M32—2362no—
256C2×C16—21no—
257C2×C16—1181no—
258#107—591no—
259#107—1181no—
260#107C4×D5911yescharacteristic
261#106—2362no—
262#102—21no—
263#102—1181no—
264#102—1181no—
265#102—1181no—
266#95—2361no—
267C24—591no—
268C24—2361no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—1181no—
272C2×D4—4722no—
273C2×D4—4722no—
274C2×D4—2361no—
275C2×D4—2361no—
276C2×D4—2362no—
277C2×D4—2362no—
278C2×D4—2361no—
279C2×D4—2361no—
280C22×C4—41no—
281C22×C4—591no—
282C22×C4—591no—
283C22×C4—591no—
284C22×C4—591no—
285C22×C4—591no—
286C22×C4—2361no—
287C22×C4—2361no—
288C22×C4—2361no—
289C22×C4—2361no—
290C22×C4D4×D5911yescharacteristic
291D8—41no—
292D8—2361no—
293D8—2362no—
294C2×C8—41no—
295C2×C8—591no—
296C2×C8—591no—
297C2×C8—1182no—
298C2×C8—1182no—
299C2×C8—1182no—
300C2×C8—2361no—
301C2×C8—2361no—
302C2×C8C2×C4×D5911yescharacteristic
303C2×C8D4×D5911yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1181no—
308C4⋊C4—2361no—
309C4⋊C4—2361no—
310C4⋊C4—2361no—
311C22⋊C4—41no—
312C22⋊C4—4722no—
313C22⋊C4—2361no—
314C22⋊C4—2361no—
315C22⋊C4—2361no—
316C16—41no—
317C16—2361no—
318C23—41no—
319C23—591no—
320C23—591no—
321C23—1182no—
322C23—1181no—
323C23—1181no—
324C23—4722no—
325C23—4722no—
326C23—4722no—
327C23—2361no—
328C23—2361no—
329C23D8×D5911yescharacteristic
330D4—41no—
331D4—41no—
332D4—81no—
333D4—81no—
334D4—4722no—
335D4—4722no—
336D4—4722no—
337D4—4721no—
338D4—4721no—
339D4—2361no—
340D4—2361no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—591no—
348C2×C4—591no—
349C2×C4—1181no—
350C2×C4—1181no—
351C2×C4—1181no—
352C2×C4—1181no—
353C2×C4—4722no—
354C2×C4—4721no—
355C2×C4—4721no—
356C2×C4—4721no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4C22⋊C4×D5911yescharacteristic
362C2×C4SD16×D5911yescharacteristic
363C8—21no—
364C8—21no—
365C8—592no—
366C8—1181no—
367C22—21no—
368C22—21no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—592no—
373C22—1181no—
374C22—1181no—
375C22—1181no—
376C22—1181no—
377C22—1181no—
378C22—9444no—
379C22—4721no—
380C22—4721no—
381C22—4721no—
382C22—2362no—
383C22—2362no—
384C22#102×D5911yescharacteristic
385C22#1683783611yescharacteristic
386C22#1683783911yescharacteristic
387C4—21no—
388C4—81no—
389C4—161no—
390C4—1181no—
391C4—1181no—
392C4—9441no—
393C4—4721no—
394C4#99×D5911yescharacteristic
395C4#1683741711yescharacteristic
396C2—21no—
397C2—21no—
398C2—161no—
399C2—1182no—
400C2—9441no—
401C2—2361no—
402C2#2142266211yescharacteristic
403C2#2142581011yescharacteristic
404C2#360×D5911yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1347C2
socleC22×C59C2×#102
prime radical#466×D59C2
cosocleC22×C8×C59C23
hypercenter#1347D59
hypocenterC59#11888
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. #21425705C2semidirect
3. #21424120C2semidirect
4. #21422106C2semidirect
5. #21422105C2semidirect
6. #21422104C2semidirect
7. C59×#1347C2semidirect
8. C2×D8×D59C4semidirect
12. #16837639C4semidirect
16. #16837635C4semidirect
17. #107×D59C4semidirect
21. C22⋊(C59⋊C16)C22semidirect
23. #16837165C22semidirect
25. C59×C22⋊C16C22semidirect
26. C59×#340C22semidirect
200. #1347D59semidirect
226. C22⋊C16C2×D59semidirect
228. #340C2×D59semidirect
229. C59#11888semidirect

Maximal subgroups

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

subgroupindex
#466×D592
#214257052
#214241202
#214221062
#214221052
#214221042
C59×#13472
#1188859