Finite Group Databasebeta

← #28499397

Subgroups — #28499397

Order 15104.

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

normal subgroups only

This group has 680 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#21424170C211yesmaximal, characteristic
2#21422423C211yesmaximal, characteristic
3#21422117C214yesmaximal, complemented
4C2×C59×#393C211yesmaximal, characteristic
5#16837366—21no—
6#16837258—21no—
7#16837169—21no—
8#16837169C2212yes—
9#16837126—21no—
10#16837126C2212yes—
11C23×Q8×C59C411yescharacteristic, complemented
12C22×C22⋊C4×C59—21no—
13C22×C22⋊C4×C59C2211yescharacteristic
14C59×#393—21no—
15C59×#393—21no—
16C59×#393C412yescomplemented
17C59×#393C2212yes—
18C2×#95×C59—21no—
19C2×#95×C59C411yescharacteristic, complemented
20C2×#11745664—41no—
21C2×C22⋊Dic59—22no—
22C2×C22⋊Dic59—41no—
23C2×C22⋊Dic59—41no—
24C22⋊(C59⋊C8)—22no—
25C22⋊(C59⋊C8)—41no—
26C25×C59D411yescharacteristic
27C22×Q8×C59—21no—
28C22×Q8×C59—44no—
29C22×Q8×C59—41no—
30C22×Q8×C59—41no—
31C22×Q8×C59C2×C412yes—
32C23×C4×C59—21no—
33C23×C4×C59—21no—
34C23×C4×C59C2×C411yescharacteristic
35C23×C4×C59D411yescharacteristic
36C2×C22⋊C4×C59—22no—
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59—41no—
39C2×C22⋊C4×C59—41no—
40C2×C22⋊C4×C59—44no—
41C2×C22⋊C4×C59C2×C411yescharacteristic
42C2×C22⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59C2311yescharacteristic
45#95×C59—22no—
46#95×C59—21no—
47#95×C59—22no—
48#95×C59C2×C412yes—
49C24×C59—22no—
50C24×C59—21no—
51C24×C59—21no—
52C24×C59—41no—
53C24×C59—42no—
54C24×C59—42no—
55C24×C59C22⋊C411yescharacteristic
56C24×C59C22⋊C411yescharacteristic
57C24×C59C2×D411yescharacteristic
58C2×Q8×C59—44no—
59C2×Q8×C59—42no—
60C2×Q8×C59—44no—
61C2×Q8×C59—42no—
62C2×Q8×C59—48no—
63C2×Q8×C59—44no—
64C2×Q8×C59—44no—
65C22×C4×C59—24no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—22no—
70C22×C4×C59—21no—
71C22×C4×C59—22no—
72C22×C4×C59—42no—
73C22×C4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59—42no—
76C22×C4×C59—42no—
77C22×C4×C59—42no—
78C22×C4×C59—42no—
79C22×C4×C59—42no—
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×Dic59—42no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—44no—
90C22⋊C4×C59—44no—
91C22⋊Dic59—42no—
92C22⋊Dic59—42no—
93C22⋊Dic59—82no—
94C22⋊Dic59—82no—
95#11745664—82no—
96C2×C59⋊C8—42no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—42no—
107C23×C59—42no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—48no—
113C23×C59—42no—
114C23×C59—42no—
115C23×C59—81no—
116C23×C59—81no—
117C23×C59—84no—
118C23×C59#9911yescharacteristic
119C23×C59#9911yescharacteristic
120C23×C59C2×C22⋊C411yescharacteristic
121C23×C59#13712yes—
122Q8×C59—44no—
123Q8×C59—48no—
124Q8×C59—44no—
125C2×C4×C59—24no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—42no—
131C2×C4×C59—44no—
132C2×C4×C59—42no—
133C2×C4×C59—44no—
134C2×C4×C59—42no—
135C2×C4×C59—42no—
136C2×C4×C59—42no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—42no—
144C2×C4×C59—81no—
145C2×C4×C59—84no—
146C2×Dic59—82no—
147C2×Dic59—82no—
148C2×Dic59—82no—
149C59⋊C8—82no—
150#11808—591nomaximal, Sylow
151C22×C59—24no—
152C22×C59—24no—
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—22no—
160C22×C59—42no—
161C22×C59—42no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—81no—
168C22×C59—84no—
169C22×C59—84no—
170C22×C59—81no—
171C22×C59AΣL(1,𝔽16)12yes—
172C22×C59#35412yes—
173C22×C59C2×#9911yescharacteristic
174C22×C59#41911yescharacteristic
175C22×C59#42111yescharacteristic
176C4×C59—44no—
177C4×C59—41no—
178C4×C59—41no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59—82no—
182Dic59—162no—
183C2×#393D5911yescharacteristic
184#1819—591no—
185#1462—591no—
186#1351—594no—
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—22no—
190C2×C59—22no—
191C2×C59—42no—
192C2×C59—81no—
193C2×C59#151111yescharacteristic
194C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
195C2×C59C2×#35411yescharacteristic
196C23×Q8Dic5911yescharacteristic, complemented
197C22×C22⋊C4—21no—
198C22×C22⋊C4C2×D5911yescharacteristic
199#407—1181no—
200#393—21no—
201#393—21no—
202#393Dic5912yescomplemented
203#393C2×D5912yes—
204#379—1181no—
205C2×#95—21no—
206C2×#95Dic5911yescharacteristic, complemented
207#342—592no—
208#342—1181no—
209#323—592no—
210#323—1181no—
211C59#1180811yescharacteristic, complemented, Sylow
212C25C22⋊D5911yescharacteristic
213C22×Q8—21no—
214C22×Q8—44no—
215C22×Q8—41no—
216C22×Q8—41no—
217C22×Q8C2×Dic5912yes—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C2×Dic5911yescharacteristic
221C23×C4C22⋊D5911yescharacteristic
222C2×M16—2361no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—41no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—1182no—
229C2×C22⋊C4—2361no—
230C2×C22⋊C4—2361no—
231C2×C22⋊C4C2×Dic5911yescharacteristic
232C2×C22⋊C4C22⋊D5911yescharacteristic
233C2×C22⋊C4C22⋊D5911yescharacteristic
234C2×C22⋊C4C22×D5911yescharacteristic
235C22⋊C8—1182no—
236C22⋊C8—2361no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95C2×Dic5912yes—
241C24—22no—
242C24—21no—
243C24—21no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24C22⋊Dic5911yescharacteristic
248C24C22⋊Dic5911yescharacteristic
249C24C2×C22⋊D5911yescharacteristic
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—48no—
255C2×Q8—44no—
256C2×Q8—44no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—2362no—
273C22×C4C22⋊Dic5911yescharacteristic
274C22×C4C22⋊Dic5911yescharacteristic
275C22×C4C22×Dic5911yescharacteristic
276C22×C4C2×C22⋊D5911yescharacteristic
277M16—4722no—
278C2×C8—2362no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—4722no—
286C22⋊C4—4722no—
287C22⋊C4—2362no—
288C22⋊C4—2362no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—42no—
299C23—42no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—48no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—81no—
309C23—84no—
310C23#1427364211yescharacteristic
311C23#1427364211yescharacteristic
312C23C2×C22⋊Dic5911yescharacteristic
313C23#1427375812yes—
314Q8—44no—
315Q8—44no—
316Q8—48no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—81no—
337C2×C4—84no—
338C2×C4—4722no—
339C2×C4—4722no—
340C2×C4—4722no—
341C8—4722no—
342C22—24no—
343C22—24no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—84no—
360C22—84no—
361C22—81no—
362C22#1683718912yes—
363C22#1683719712yes—
364C22C2×#1427364211yescharacteristic
365C22#1683741111yescharacteristic
366C22#1683741911yescharacteristic
367C4—44no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—82no—
373C4—9442no—
374C2—21no—
375C2—21no—
376C2—22no—
377C2—22no—
378C2—42no—
379C2—81no—
380C2#2142262911yescharacteristic
381C2C2×#1683718911yescharacteristic
382C2C2×#1683719711yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC2×C59×#393C2
socleC22×C59C2×#99
prime radicalC25×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenterC2×#393D59
hypocenterC59#11808
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
3. #21422117C2semidirect
11. C23×Q8×C59C4semidirect
16. C59×#393C4semidirect
19. C2×#95×C59C4semidirect
196. C23×Q8Dic59semidirect
202. #393Dic59semidirect
206. C2×#95Dic59semidirect
211. C59#11808semidirect

Maximal subgroups

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

subgroupindex
#214241702
#214224232
#214221172
C2×C59×#3932
#1180859