Finite Group Databasebeta

← #28559157

Subgroups — #28559157

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434105C211yesmaximal, characteristic
2#21433190C211yesmaximal, characteristic, complemented
3#21433136C211yesmaximal, characteristic
4#21423000C212yesmaximal, complemented
5#21422988C212yesmaximal, complemented
6#21422976C212yesmaximal, complemented
7#21422941C212yesmaximal, complemented
8#21422876C211yesmaximal, characteristic, complemented
9#21422540C211yesmaximal, characteristic
10#21422518C211yesmaximal, characteristic, complemented
11C59×#1453C211yesmaximal, characteristic, complemented
12#16837959C2211yescharacteristic
13C2×C4×D236C2211yescharacteristic
14#16837733C412yes—
15#16837693C412yes—
16#16837608C412yes—
17#16837555C412yes—
18#16837532C2212yescomplemented
19#16837531—22no—
20C8×Dic118C2212yescomplemented
21C4×#11745660C2212yes—
22C4×#11745660C2212yes—
23#16837481C2212yes—
24#16837481C2212yes—
25#16837479—22no—
26#16837460—22no—
27#16837459C2211yescharacteristic
28#16837455C2211yescharacteristic
29#16837364—22no—
30#16837155C2212yescomplemented
31#16837153C2212yescomplemented
32#16837134C2212yescomplemented
33#16837134C2212yescomplemented
34#16837131C2212yes—
35#16837131C2212yes—
36C59×#431C2211yescharacteristic
37C59×#431C2211yescharacteristic
38C4×M16×C59C2211yescharacteristic
39C59×#322C2211yescharacteristic, complemented
40C59×#322C2211yescharacteristic, complemented
41C59×#322C2211yescharacteristic, complemented
42C82×C59C2211yescharacteristic, complemented
43C22×D236C2×C411yescharacteristic
44C22×C4×D59—21no—
45C2×#11745660C2×C412yes—
46C2×#11745660C2×C412yes—
47C2×#11745659—22no—
48C2×#11745659—22no—
49#14273727C2×C411yescharacteristic
50#14273727C2×C411yescharacteristic
51#14273719—21no—
52#14273706—21no—
53#14273706C2×C412yes—
54C4×Dic118—21no—
55C4×Dic118C2312yes—
56C4×D236—21no—
57C4×D236—21no—
58C4×D236C2×C412yes—
59C4×D236C2312yes—
60#14273686—21no—
61C2×C4⋊Dic59C2×C411yescharacteristic
62C2×(C2×D59)⋊C4—21no—
63#14273659C2×C412yes—
64#14273659C2×C412yes—
65C4⋊(C59⋊C8)C2312yes—
66C4⋊(C59⋊C8)C2312yes—
67Dic59⋊C8—22no—
68Dic59⋊C8—22no—
69#14273652C2×C412yes—
70#14273652C2×C412yes—
71#14273649C2×C412yes—
72#14273649C2×C412yes—
73C22⋊(C59⋊C8)—21no—
74C22⋊(C59⋊C8)—21no—
75(C2×D59)⋊C8—22no—
76(C2×D59)⋊C8—22no—
77#14273638—21no—
78#14273637—22no—
79C4×C59⋊C8—21no—
80C8×Dic59—22no—
81C2×M16×C59—21no—
82C2×C42×C59C2311yescharacteristic
83C4⋊C8×C59C2×C412yes—
84C4⋊C8×C59D412yes—
85C22⋊C8×C59C2×C412yes—
86C22⋊C8×C59D412yes—
87#97×C59—21no—
88C4×C8×C59C2311yescharacteristic
89C4×C8×C59C2311yescharacteristic
90C4×C8×C59C2311yescharacteristic
91C4×C8×C59C2311yescharacteristic
92C4×C8×C59C2311yescharacteristic
93C4×C8×C59C2311yescharacteristic
94C23×D59—21no—
95C22×C4×C59C22×C411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59#4111yescharacteristic
98C22×Dic59—21no—
99C2×D236—21no—
100C2×D236—42no—
101C2×D236C22×C412yes—
102C2×C4×D59—22no—
103C2×C4×D59—22no—
104C2×C4×D59—82no—
105C2×Dic118—21no—
106C2×Dic118C22×C412yes—
107M16×C59—42no—
108C2×C8×C59—21no—
109C2×C8×C59—21no—
110C2×C8×C59C22×C412yes—
111C2×C8×C59C22×C412yes—
112C2×C8×C59C2×D412yes—
113C2×C8×C59C2×D412yes—
114C42×C59C22×C411yescharacteristic
115C42×C59C2×D411yescharacteristic
116C42×C59#4111yescharacteristic
117C42×C59C2411yescharacteristic
118C22⋊Dic59—22no—
119(C2×D59)⋊C4—22no—
120(C2×D59)⋊C4—22no—
121C4⋊Dic59—21no—
122C4⋊Dic59—21no—
123C4⋊Dic59—21no—
124C4⋊Dic59—21no—
125C4⋊Dic59C22×C412yes—
126C4⋊Dic59C22×C412yes—
127Dic59⋊C4—22no—
128Dic59⋊C4—22no—
129C4×Dic59—22no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—22no—
132#11745660—42no—
133#11745660—42no—
134#11745659—82no—
135#11745659—82no—
136C23×C59C4×D411yescharacteristic
137C22×D59—22no—
138C22×D59—42no—
139C22×D59—81no—
140C22×D59—81no—
141C2×C4×C59—22no—
142C2×C4×C59—22no—
143C2×C4×C59—41no—
144C2×C4×C59C4×D411yescharacteristic
145C2×C4×C59C4×D411yescharacteristic
146C2×C4×C59C4×D411yescharacteristic
147C2×C4×C59C23×C411yescharacteristic
148C2×C4×C59C22×D411yescharacteristic
149C2×C4×C59C2×#4111yescharacteristic
150C2×Dic59—22no—
151C2×Dic59—22no—
152C2×Dic59—22no—
153C2×Dic59—81no—
154D236—22no—
155D236—22no—
156D236—42no—
157C4×D59—42no—
158C4×D59—42no—
159C4×D59—82no—
160Dic118—22no—
161Dic118—22no—
162Dic118—81no—
163C8×C59—22no—
164C8×C59—22no—
165C8×C59—22no—
166C8×C59—22no—
167C8×C59—22no—
168C8×C59—41no—
169C59⋊C8—42no—
170C59⋊C8—42no—
171#17952—591nomaximal, Sylow
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59C2×C4×D411yescharacteristic
176C2×D59—42no—
177C2×D59—42no—
178C2×D59—42no—
179C2×D59—84no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59#56812yes—
185C4×C59#57712yes—
186Dic59—42no—
187Dic59—42no—
188Dic59—42no—
189#3101—591no—
190#2985—591no—
191#2975—591no—
192#1592—592no—
193#1590—592no—
194#1588—592no—
195#1583—592no—
196#1572—591no—
197#1490—591no—
198#1485—591no—
199#1453D5911yescharacteristic, complemented
200C2×C59—41no—
201C2×C59#293011yescharacteristic
202C2×C59#294911yescharacteristic
203C2×C59#295611yescharacteristic
204D59—82no—
205#533—591no—
206C2×C4×D4—591no—
207#483—592no—
208#476—592no—
209#461—592no—
210#450—592no—
211#446—592no—
212#446—1182no—
213C8×Q8—592no—
214C4×SD16—592no—
215C4×SD16—592no—
216#436—592no—
217#436—592no—
218#436—1182no—
219#433—1182no—
220#432—591no—
221#431—591no—
222#431C2×D5911yescharacteristic
223#431C2×D5911yescharacteristic
224#407—1182no—
225C4×M16C2×D5911yescharacteristic
226#335—592no—
227#334—592no—
228Q8⋊C8—592no—
229Q8⋊C8—592no—
230D4⋊C8—592no—
231D4⋊C8—592no—
232#322C2×D5911yescharacteristic, complemented
233#322C2×D5911yescharacteristic, complemented
234#322C2×D5911yescharacteristic, complemented
235C82C2×D5911yescharacteristic, complemented
236C59#1795211yescharacteristic, complemented, Sylow
237C22×D4—591no—
238C23×C4—1181no—
239C2×SD16—592no—
240C2×SD16—592no—
241C2×M16—21no—
242C2×M16—1182no—
243C2×M16—1182no—
244C4⋊Q8—591no—
245C4⋊Q8—591no—
246#125—1181no—
247C22⋊Q8—592no—
248C22⋊Q8—1181no—
249C4×Q8—592no—
250C4×Q8—1181no—
251C4×D4—592no—
252C4×D4—592no—
253C4×D4—1181no—
254C4×D4—1181no—
255#117—1181no—
256C2×C4⋊C4—591no—
257C2×C22⋊C4—1181no—
258C2×C42C22×D5911yescharacteristic
259#106—592no—
260#106—592no—
261C4⋊C8—592no—
262C4⋊C8—592no—
263C4⋊C8—1182no—
264C4⋊C8—1182no—
265C4⋊C8C4×D5912yes—
266C4⋊C8D23612yes—
267#103—592no—
268#103—592no—
269#102—592no—
270#102—592no—
271C22⋊C8—1181no—
272C22⋊C8—1181no—
273C22⋊C8—1182no—
274C22⋊C8—1182no—
275C22⋊C8C4×D5912yes—
276C22⋊C8D23612yes—
277#97—21no—
278#97—1182no—
279#97—1181no—
280C4×C8—1182no—
281C4×C8—1181no—
282C4×C8C22×D5911yescharacteristic
283C4×C8C22×D5911yescharacteristic
284C4×C8C22×D5911yescharacteristic
285C4×C8C22×D5911yescharacteristic
286C4×C8C22×D5911yescharacteristic
287C4×C8C22×D5911yescharacteristic
288C24—1181no—
289C2×Q8—592no—
290C2×Q8—1181no—
291C2×D4—592no—
292C2×D4—1181no—
293C2×D4—2362no—
294C22×C4—1181no—
295C22×C4—1182no—
296C22×C4—1182no—
297C22×C4—4722no—
298C22×C4C2×C4×D5911yescharacteristic
299C22×C4C2×D23611yescharacteristic
300C22×C4#1174568511yescharacteristic
301SD16—2362no—
302SD16—2362no—
303M16—42no—
304M16—4722no—
305M16—4722no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—1182no—
309C2×C8—1182no—
310C2×C8C2×C4×D5912yes—
311C2×C8C2×C4×D5912yes—
312C2×C8C2×D23612yes—
313C2×C8C2×D23612yes—
314C4⋊C4—592no—
315C4⋊C4—592no—
316C4⋊C4—1181no—
317C4⋊C4—1181no—
318C4⋊C4—1181no—
319C4⋊C4—1182no—
320C4⋊C4—1181no—
321C4⋊C4—1182no—
322C22⋊C4—1182no—
323C22⋊C4—1182no—
324C22⋊C4—1182no—
325C42—1182no—
326C42C2×C4×D5911yescharacteristic
327C42C2×D23611yescharacteristic
328C42#1174568511yescharacteristic
329C42C23×D5911yescharacteristic
330C23—1182no—
331C23—4721no—
332C23—4721no—
333C23—2362no—
334C23C4×D23611yescharacteristic
335Q8—1182no—
336Q8—1182no—
337Q8—4721no—
338D4—1182no—
339D4—1182no—
340D4—2362no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—1182no—
345C2×C4—1182no—
346C2×C4—1182no—
347C2×C4—4722no—
348C2×C4—4721no—
349C2×C4—2362no—
350C2×C4—2362no—
351C2×C4C4×D23611yescharacteristic
352C2×C4C4×D23611yescharacteristic
353C2×C4C4×D23611yescharacteristic
354C2×C4C22×C4×D5911yescharacteristic
355C2×C4C22×D23611yescharacteristic
356C2×C4C2×#1174568511yescharacteristic
357C8—22no—
358C8—22no—
359C8—22no—
360C8—22no—
361C8—22no—
362C8—41no—
363C8—2362no—
364C8—2362no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—4724no—
369C22—2362no—
370C22—2362no—
371C22—2362no—
372C22C2×C4×D23611yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—2362no—
378C4—2362no—
379C4—2362no—
380C4#1683816712yes—
381C4#1683821412yes—
382C2—41no—
383C2—4722no—
384C2#2143280111yescharacteristic
385C2#2143295711yescharacteristic
386C2#2143300711yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1453C2
socleC22×C59C2×C4×D4
prime radicalC22×D236C2×C4
cosocleC42×C59C24
hypercenter#1453D59
hypocenterC59#17952
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #21433190C2semidirect
4. #21423000C2semidirect
5. #21422988C2semidirect
6. #21422976C2semidirect
7. #21422941C2semidirect
8. #21422876C2semidirect
10. #21422518C2semidirect
11. C59×#1453C2semidirect
18. #16837532C22semidirect
20. C8×Dic118C22semidirect
30. #16837155C22semidirect
31. #16837153C22semidirect
32. #16837134C22semidirect
33. #16837134C22semidirect
39. C59×#322C22semidirect
40. C59×#322C22semidirect
41. C59×#322C22semidirect
42. C82×C59C22semidirect
199. #1453D59semidirect
232. #322C2×D59semidirect
233. #322C2×D59semidirect
234. #322C2×D59semidirect
235. C82C2×D59semidirect
236. C59#17952semidirect

Maximal subgroups

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

subgroupindex
#214341052
#214331902
#214331362
#214230002
#214229882
#214229762
#214229412
#214228762
#214225402
#214225182
C59×#14532
#1795259