Finite Group Databasebeta

← #28559360

Subgroups — #28559360

Order 15104.

600 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434655C211yesmaximal, characteristic
2#21433190C211yesmaximal, characteristic, complemented
3#21433171C211yesmaximal, characteristic
4#21423000C212yesmaximal, complemented
5#21422988C212yesmaximal, complemented
6#21422910C212yesmaximal, complemented
7C8×#11745660C212yesmaximal, complemented
8#21422859C211yesmaximal, characteristic, complemented
9#21422570C211yesmaximal, characteristic, complemented
10#21422566C211yesmaximal, characteristic
11C59×#1450C211yesmaximal, characteristic, complemented
12#16837959C2211yescharacteristic
13#16837952C2211yescharacteristic
14#16837800C411yescharacteristic
15#16837775C411yescharacteristic
16#16837753C411yescharacteristic
17#16837749C411yescharacteristic
18#16837631C412yes—
19#16837631C412yes—
20#16837532C2212yes—
21#16837531—22no—
22C8×Dic118C2212yes—
23C4×#11745660C2212yes—
24C4×#11745660C2212yes—
25#16837481C2212yes—
26#16837472—22no—
27C8×D236C2212yes—
28#16837460—22no—
29#16837459C2211yescharacteristic
30#16837459C2211yescharacteristic
31#16837367—22no—
32#16837155C2212yes—
33#16837153C2212yes—
34#16837134C2212yes—
35#16837134C2212yes—
36#16837131C2212yes—
37#16837131C2212yes—
38C59×#431C2211yescharacteristic
39C59×#431C2211yescharacteristic
40C2×C4×C8×C59C2211yescharacteristic
41C59×#322C2211yescharacteristic
42C59×#322C2211yescharacteristic
43C82×C59C2211yescharacteristic
44C82×C59C2211yescharacteristic
45C2×#11745685—21no—
46C2×#11745660C2×C412yes—
47C2×#11745660C2×C412yes—
48C2×#11745659—22no—
49C2×C8×D59—22no—
50#14273727C2×C411yescharacteristic
51#14273727C2×C411yescharacteristic
52#14273727C2×C411yescharacteristic
53#14273724C2×C411yescharacteristic
54#14273719—21no—
55#14273715—21no—
56#14273706—21no—
57#14273706C2×C412yes—
58#14273701—21no—
59#14273701C2×C412yes—
60C4×Dic118—21no—
61C4×Dic118C2312yes—
62C4×D236—21no—
63C4×D236C2312yes—
64#14273686—21no—
65#14273659C2×C412yes—
66#14273659C2×C412yes—
67C4⋊(C59⋊C8)C2312yes—
68C4⋊(C59⋊C8)C2312yes—
69Dic59⋊C8—22no—
70Dic59⋊C8—22no—
71#14273652C2×C412yes—
72#14273652C2×C412yes—
73#14273649C2×C412yes—
74#14273649C2×C412yes—
75C22⋊(C59⋊C8)—21no—
76C22⋊(C59⋊C8)—21no—
77(C2×D59)⋊C8—22no—
78(C2×D59)⋊C8—22no—
79#14273638—21no—
80#14273638—21no—
81#14273637—22no—
82C8×Dic59—22no—
83C22×C8×C59C2×C411yescharacteristic
84C22×C8×C59C2×C411yescharacteristic
85C2×C42×C59C2311yescharacteristic
86C4⋊C8×C59—21no—
87C4⋊C8×C59D412yes—
88C22⋊C8×C59—21no—
89C22⋊C8×C59D412yes—
90C4×C8×C59C2×C411yescharacteristic
91C4×C8×C59C2×C411yescharacteristic
92C4×C8×C59C2311yescharacteristic
93C4×C8×C59C2311yescharacteristic
94C4×C8×C59C2311yescharacteristic
95C4×C8×C59C2311yescharacteristic
96C4×C8×C59C2311yescharacteristic
97C4×C8×C59C2311yescharacteristic
98C22×C4×C59C22×C411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59#4111yescharacteristic
101C2×C22⋊D59—22no—
102#11745685—82no—
103C2×D236—21no—
104C2×D236—21no—
105C2×D236C22×C412yes—
106C2×C4×D59—22no—
107C2×Dic118—21no—
108C2×Dic118—21no—
109C2×Dic118C22×C412yes—
110C2×C8×C59—22no—
111C2×C8×C59—22no—
112C2×C8×C59—21no—
113C2×C8×C59—21no—
114C2×C8×C59C22×C411yescharacteristic
115C2×C8×C59C22×C411yescharacteristic
116C2×C8×C59C22×C411yescharacteristic
117C2×C8×C59C22×C411yescharacteristic
118C2×C8×C59C2×D412yes—
119C2×C8×C59C2×D412yes—
120C42×C59C22×C411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59#4111yescharacteristic
123C42×C59C2411yescharacteristic
124C22⋊Dic59—22no—
125(C2×D59)⋊C4—22no—
126(C2×D59)⋊C4—22no—
127C4⋊Dic59—21no—
128C4⋊Dic59—21no—
129C4⋊Dic59—21no—
130C4⋊Dic59—21no—
131C4⋊Dic59C22×C412yes—
132C4⋊Dic59C22×C412yes—
133Dic59⋊C4—22no—
134Dic59⋊C4—22no—
135C4×Dic59—22no—
136C2×C59⋊C8—22no—
137C2×C59⋊C8—22no—
138#11745660—24no—
139#11745660—24no—
140#11745659—82no—
141C8×D59—82no—
142C23×C59C4×D411yescharacteristic
143C22×D59—22no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—22no—
147C2×C4×C59—41no—
148C2×C4×C59C4×D411yescharacteristic
149C2×C4×C59C4×D411yescharacteristic
150C2×C4×C59C4×D411yescharacteristic
151C2×C4×C59C23×C411yescharacteristic
152C2×C4×C59C22×D411yescharacteristic
153C2×C4×C59C2×#4111yescharacteristic
154C22⋊D59—82no—
155C2×Dic59—22no—
156C2×Dic59—22no—
157C2×Dic59—22no—
158D236—22no—
159D236—22no—
160D236—81no—
161D236—81no—
162C4×D59—42no—
163C4×D59—42no—
164Dic118—22no—
165Dic118—22no—
166Dic118—81no—
167Dic118—81no—
168C8×C59—21no—
169C8×C59—21no—
170C8×C59—22no—
171C8×C59—22no—
172C8×C59—41no—
173C8×C59—41no—
174C8×C59#13112yes—
175C8×C59#13112yes—
176C59⋊C8—42no—
177C59⋊C8—42no—
178#17971—591nomaximal, Sylow
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—21no—
182C22×C59C2×C4×D411yescharacteristic
183C2×D59—42no—
184C2×D59—42no—
185C2×D59—42no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59C2×#13111yescharacteristic
191C4×C59C2×#13111yescharacteristic
192C4×C59#57712yes—
193Dic59—42no—
194Dic59—42no—
195Dic59—42no—
196#3167—591no—
197#2985—591no—
198#2981—591no—
199#1592—592no—
200#1590—592no—
201SD16⋊C8—592no—
202C8×SD16—592no—
203#1568—591no—
204#1497—591no—
205#1496—591no—
206#1450D5911yescharacteristic, complemented
207C2×C59—41no—
208C2×C59#293111yescharacteristic
209C2×C59#294911yescharacteristic
210C2×C59#295711yescharacteristic
211D59—82no—
212#533—591no—
213#532—591no—
214#498—591no—
215#492—591no—
216#487—591no—
217#486—591no—
218#465—592no—
219#465—592no—
220#446—592no—
221#446—1182no—
222C8×Q8—592no—
223C4×SD16—592no—
224C4×SD16—592no—
225#436—592no—
226#435—1182no—
227C8×D4—592no—
228#433—1182no—
229#432—591no—
230#432—591no—
231#431C2×D5911yescharacteristic
232#431C2×D5911yescharacteristic
233#408—1182no—
234C2×C4×C8C2×D5911yescharacteristic
235#335—592no—
236#334—592no—
237Q8⋊C8—592no—
238Q8⋊C8—592no—
239D4⋊C8—592no—
240D4⋊C8—592no—
241#322C2×D5911yescharacteristic
242#322C2×D5911yescharacteristic
243C82C2×D5911yescharacteristic
244C82C2×D5911yescharacteristic
245C59#1797111yescharacteristic, complemented, Sylow
246C2×#41—1181no—
247C2×SD16—592no—
248C2×SD16—592no—
249C2×M16—1182no—
250C22×C8—1182no—
251C22×C8C4×D5911yescharacteristic
252C22×C8C4×D5911yescharacteristic
253C4⋊Q8—591no—
254C4⋊Q8—591no—
255C4⋊Q8—591no—
256#127—591no—
257#125—1181no—
258#124—1181no—
259C22⋊Q8—592no—
260C22⋊Q8—1181no—
261#121—592no—
262#121—1181no—
263C4×Q8—592no—
264C4×Q8—1181no—
265C4×D4—592no—
266C4×D4—1181no—
267#117—1181no—
268C2×C42C22×D5911yescharacteristic
269#106—592no—
270#106—592no—
271C4⋊C8—21no—
272C4⋊C8—592no—
273C4⋊C8—592no—
274C4⋊C8—1182no—
275C4⋊C8—1182no—
276C4⋊C8D23612yes—
277#103—592no—
278#103—592no—
279#102—592no—
280#102—592no—
281C22⋊C8—21no—
282C22⋊C8—1181no—
283C22⋊C8—1181no—
284C22⋊C8—1182no—
285C22⋊C8—1182no—
286C22⋊C8D23612yes—
287#97—1182no—
288#97—1181no—
289#97—1181no—
290C4×C8—1182no—
291C4×C8C4×D5911yescharacteristic
292C4×C8C4×D5911yescharacteristic
293C4×C8C22×D5911yescharacteristic
294C4×C8C22×D5911yescharacteristic
295C4×C8C22×D5911yescharacteristic
296C4×C8C22×D5911yescharacteristic
297C4×C8C22×D5911yescharacteristic
298C4×C8C22×D5911yescharacteristic
299#41—4722no—
300C2×Q8—592no—
301C2×Q8—1181no—
302C2×Q8—1181no—
303C2×D4—592no—
304C2×D4—1182no—
305C2×D4—1181no—
306C2×D4—1181no—
307C22×C4—1182no—
308C22×C4C2×C4×D5911yescharacteristic
309C22×C4C2×D23611yescharacteristic
310C22×C4#1174568511yescharacteristic
311SD16—1184no—
312SD16—1184no—
313M16—4722no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—1182no—
319C2×C8—1182no—
320C2×C8—4722no—
321C2×C8C2×C4×D5911yescharacteristic
322C2×C8C2×C4×D5911yescharacteristic
323C2×C8C2×C4×D5911yescharacteristic
324C2×C8C2×C4×D5911yescharacteristic
325C2×C8C2×D23612yes—
326C2×C8C2×D23612yes—
327C4⋊C4—592no—
328C4⋊C4—592no—
329C4⋊C4—1181no—
330C4⋊C4—1181no—
331C4⋊C4—1181no—
332C4⋊C4—1182no—
333C4⋊C4—1181no—
334C4⋊C4—1182no—
335C22⋊C4—1182no—
336C22⋊C4—1182no—
337C22⋊C4—1182no—
338C42—1182no—
339C42C2×C4×D5911yescharacteristic
340C42C2×D23611yescharacteristic
341C42#1174568511yescharacteristic
342C42C23×D5911yescharacteristic
343C23—1182no—
344C23C4×D23611yescharacteristic
345Q8—1182no—
346Q8—1182no—
347Q8—4721no—
348Q8—4721no—
349D4—1182no—
350D4—1182no—
351D4—4721no—
352D4—4721no—
353D4—4722no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—1182no—
359C2×C4—1182no—
360C2×C4—1182no—
361C2×C4—2362no—
362C2×C4—2362no—
363C2×C4C4×D23611yescharacteristic
364C2×C4C4×D23611yescharacteristic
365C2×C4C4×D23611yescharacteristic
366C2×C4C22×C4×D5911yescharacteristic
367C2×C4C22×D23611yescharacteristic
368C2×C4C2×#1174568511yescharacteristic
369C8—21no—
370C8—21no—
371C8—22no—
372C8—22no—
373C8—41no—
374C8—41no—
375C8—2362no—
376C8—2362no—
377C8#1427373312yes—
378C8#1427373312yes—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—2362no—
383C22—2362no—
384C22—2362no—
385C22C2×C4×D23611yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—2362no—
391C4—2362no—
392C4—2362no—
393C4C2×#1427373311yescharacteristic
394C4C2×#1427373311yescharacteristic
395C4#1683821412yes—
396C2—41no—
397C2—4722no—
398C2#2143280911yescharacteristic
399C2#2143295711yescharacteristic
400C2#2143301411yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1450C2
socleC22×C59C2×C4×D4
prime radical#16837952C22
cosocleC42×C59C24
hypercenter#1450D59
hypocenterC59#17971
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #21433190C2semidirect
4. #21423000C2semidirect
5. #21422988C2semidirect
6. #21422910C2semidirect
7. C8×#11745660C2semidirect
8. #21422859C2semidirect
9. #21422570C2semidirect
11. C59×#1450C2semidirect
206. #1450D59semidirect
245. C59#17971semidirect

Maximal subgroups

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

subgroupindex
#214346552
#214331902
#214331712
#214230002
#214229882
#214229102
C8×#117456602
#214228592
#214225702
#214225662
C59×#14502
#1797159