Finite Group Databasebeta

← #28476897

Subgroups — #28476897

Order 15104.

506 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 101 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#21426634C211yesmaximal, characteristic
2#21424163C211yesmaximal, characteristic
3#21423857C211yesmaximal, characteristic
4#21421922C211yesmaximal, characteristic, complemented
5#21421920C211yesmaximal, characteristic, complemented
6#21421918C211yesmaximal, characteristic, complemented
7C59×#1272C211yesmaximal, characteristic, complemented
8C2×#14273703C411yescharacteristic
9C2×Q8×Dic59C411yescharacteristic
10C2×C42×D59C2211yescharacteristic
11C2×(C2×D59)⋊C8—21no—
12C2×(C2×D59)⋊C8—21no—
13C2×#14273637—21no—
14#16837300C2211yescharacteristic
15#16837289—21no—
16#16837276—21no—
17#16837259C411yescharacteristic
18C4⋊C4×Dic59C411yescharacteristic
19#16837253—21no—
20#16837249—21no—
21#16837157C2211yescharacteristic
22#16837157C2211yescharacteristic
23#16837156—21no—
24C59×#391C2211yescharacteristic
25C59×#336C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×C4×D59C2×C411yescharacteristic
28C22×C4×D59D411yescharacteristic
29C22×C4×D59D411yescharacteristic
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32#14273703—22no—
33#14273703—22no—
34Q8×Dic59—42no—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59C2×C411yescharacteristic
37C2×C4⋊Dic59C2×C411yescharacteristic
38C2×Dic59⋊C4—21no—
39C2×Dic59⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×C4×Dic59—21no—
43C2×C4×Dic59—21no—
44C2×C4×Dic59C2×C411yescharacteristic
45C2×C4×Dic59D411yescharacteristic
46C2×C4×Dic59D411yescharacteristic
47C42×D59—42no—
48(C2×D59)⋊C8—24no—
49(C2×D59)⋊C8—42no—
50#14273637—42no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C22×Q8×C59C2×C411yescharacteristic
55C22×C8×C59—21no—
56C22×C8×C59—21no—
57C2×C4⋊C4×C59C2×C411yescharacteristic
58C2×C42×C59C2311yescharacteristic
59#95×C59—21no—
60#95×C59—21no—
61C23×D59C22⋊C411yescharacteristic
62C2×Q8×C59—21no—
63C2×Q8×C59—41no—
64C2×Q8×C59—41no—
65C2×Q8×C59M1612yes—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59C22⋊C411yescharacteristic
74C22×Dic59C22⋊C411yescharacteristic
75C22×Dic59C22⋊C411yescharacteristic
76C2×C4×D59—22no—
77C2×C4×D59—21no—
78C2×C4×D59—21no—
79C2×C4×D59—21no—
80C2×C4×D59—21no—
81C2×C4×D59—21no—
82C2×C4×D59—21no—
83C2×C4×D59—22no—
84C2×C4×D59—41no—
85C2×C4×D59—41no—
86C2×C4×D59—41no—
87C2×C4×D59—41no—
88C2×C8×C59—22no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C2×C8×C59—41no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C4⋊C4×C59—41no—
95C42×C59—41no—
96(C2×D59)⋊C4—24no—
97(C2×D59)⋊C4—42no—
98C4⋊Dic59—21no—
99C4⋊Dic59—42no—
100C4⋊Dic59—41no—
101C4⋊Dic59M1612yes—
102Dic59⋊C4—24no—
103Dic59⋊C4—41no—
104Dic59⋊C4—41no—
105C4×Dic59—22no—
106C4×Dic59—22no—
107C4×Dic59—41no—
108C4×Dic59—42no—
109C4×Dic59—42no—
110C2×C59⋊C8—22no—
111C2×C59⋊C8—41no—
112C2×C59⋊C8—41no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C23×C59C2×C22⋊C411yescharacteristic
117C22×D59—21no—
118C22×D59—21no—
119C22×D59—21no—
120C22×D59—21no—
121C22×D59—21no—
122C22×D59—21no—
123C22×D59#10412yes—
124Q8×C59—42no—
125Q8×C59—42no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—22no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59C2×M1611yescharacteristic
139C2×C4×C59C2×M1611yescharacteristic
140C2×C4×C59#13511yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C2×C4×C59#13711yescharacteristic
143C2×C4×C59#13711yescharacteristic
144C2×Dic59—21no—
145C2×Dic59—21no—
146C2×Dic59—21no—
147C2×Dic59—22no—
148C2×Dic59—21no—
149C2×Dic59—21no—
150C2×Dic59—21no—
151C2×Dic59—21no—
152C2×Dic59—21no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59#9912yes—
159C4×D59—42no—
160C4×D59—42no—
161C4×D59—42no—
162C4×D59—42no—
163C4×D59—42no—
164C4×D59—42no—
165C8×C59—42no—
166C8×C59—81no—
167C59⋊C8—42no—
168C59⋊C8—81no—
169#7623—591nomaximal, Sylow
170C22×C59#40711yescharacteristic
171C22×C59C2×#9911yescharacteristic
172C22×C59#41311yescharacteristic
173C22×C59#41611yescharacteristic
174C22×C59#41911yescharacteristic
175C22×C59C2×#10411yescharacteristic
176C22×C59#42111yescharacteristic
177C2×D59—22no—
178C2×D59—22no—
179C2×D59—22no—
180C2×D59—41no—
181C2×D59—41no—
182C2×D59—41no—
183C2×D59—41no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—21no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—42no—
191C4×C59—81no—
192Dic59—22no—
193Dic59—41no—
194Dic59—42no—
195Dic59—41no—
196Dic59—81no—
197C4×C22⋊Q8—591no—
198#1817—591no—
199#1749—591no—
200#1272—591no—
201#1272D5911yescharacteristic, complemented
202#1271—591no—
203#1271—591no—
204C2×C59#146511yescharacteristic
205C2×C59#148811yescharacteristic
206C2×C59#149111yescharacteristic
207C2×C59#150111yescharacteristic
208C2×C59#151111yescharacteristic
209C2×C59#153211yescharacteristic
210C2×C59#155411yescharacteristic
211D59—42no—
212C2×C22⋊Q8—591no—
213C2×C4×Q8—591no—
214C22×C42—591no—
215C2×C22⋊C8—1181no—
216C2×C22⋊C8—1181no—
217C2×#97—1181no—
218#391C2×D5911yescharacteristic
219#389—591no—
220#387—1181no—
221#385—1181no—
222#380—591no—
223C4×C4⋊C4—591no—
224C4×C4⋊C4—1181no—
225C4×C22⋊C4—1181no—
226#336—591no—
227#336—591no—
228#336—1181no—
229#336C2×D5911yescharacteristic
230#336C2×D5911yescharacteristic
231C59#762311yescharacteristic, complemented, Sylow
232C22×Q8C4×D5911yescharacteristic
233C23×C4—591no—
234C23×C4—591no—
235C23×C4—591no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—1181no—
239C22×C8—1181no—
240C22⋊Q8—1182no—
241C22⋊Q8—1182no—
242C4×Q8—2362no—
243C2×C4⋊C4—591no—
244C2×C4⋊C4—591no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4—1181no—
247C2×C4⋊C4—1181no—
248C2×C4⋊C4C4×D5911yescharacteristic
249C2×C22⋊C4—1181no—
250C2×C22⋊C4—1181no—
251C2×C42—591no—
252C2×C42—591no—
253C2×C42—591no—
254C2×C42—1181no—
255C2×C42—1181no—
256C2×C42—2362no—
257C2×C42C22×D5911yescharacteristic
258C22⋊C8—1184no—
259C22⋊C8—2362no—
260#97—2362no—
261#95—21no—
262#95—21no—
263#95—1181no—
264#95—1181no—
265#95—1181no—
266C24—591no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8#1174565912yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—591no—
274C22×C4—591no—
275C22×C4—591no—
276C22×C4—1182no—
277C22×C4—1182no—
278C22×C4—1181no—
279C22×C4—1181no—
280C22×C4—1181no—
281C22×C4—1181no—
282C22×C4—1181no—
283C22×C4—1181no—
284C22×C4—1181no—
285C22×C4—1181no—
286C22×C4—2361no—
287C22×C4—2361no—
288C22×C4—2361no—
289C22×C4—2361no—
290C22×C4C2×C4×D5911yescharacteristic
291C22×C4D4×D5911yescharacteristic
292C22×C4D4×D5911yescharacteristic
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—1182no—
300C2×C8—2361no—
301C2×C8—2361no—
302C2×C8—2361no—
303C2×C8—2361no—
304C2×C8—2361no—
305C4⋊C4—41no—
306C4⋊C4—592no—
307C4⋊C4—1184no—
308C4⋊C4—1181no—
309C4⋊C4—2361no—
310C4⋊C4—2362no—
311C4⋊C4—2361no—
312C4⋊C4—2361no—
313C22⋊C4—1184no—
314C22⋊C4—2362no—
315C42—41no—
316C42—1182no—
317C42—1182no—
318C42—2361no—
319C42—2362no—
320C42—2362no—
321C23—592no—
322C23—1181no—
323C23—1181no—
324C23—1181no—
325C23—1181no—
326C23—1181no—
327C23—1181no—
328C23C22⋊C4×D5911yescharacteristic
329Q8—42no—
330Q8—42no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—592no—
344C2×C4—1181no—
345C2×C4—1181no—
346C2×C4—1181no—
347C2×C4—1181no—
348C2×C4—1181no—
349C2×C4—1182no—
350C2×C4—1181no—
351C2×C4—1181no—
352C2×C4—1181no—
353C2×C4—2362no—
354C2×C4—2362no—
355C2×C4—2362no—
356C2×C4—2362no—
357C2×C4—2362no—
358C2×C4—2362no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2361no—
362C2×C4—2361no—
363C2×C4—2361no—
364C2×C4M16×D5911yescharacteristic
365C2×C4C2×#1174565911yescharacteristic
366C2×C4#1427374811yescharacteristic
367C2×C4#1427375011yescharacteristic
368C2×C4#1427375611yescharacteristic
369C2×C4#1427375911yescharacteristic
370C8—42no—
371C8—81no—
372C8—4721no—
373C8—2362no—
374C22—1182no—
375C22—1182no—
376C22—1182no—
377C22—2361no—
378C22—2361no—
379C22—2361no—
380C22—2361no—
381C22#1683736411yescharacteristic
382C22#99×D5911yescharacteristic
383C22#1683738311yescharacteristic
384C22#1683739711yescharacteristic
385C22#1683740911yescharacteristic
386C22#104×D5911yescharacteristic
387C22#1683741711yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—1182no—
397C4—4721no—
398C4—2361no—
399C4—2362no—
400C4—2361no—
401C2—2362no—
402C2#2142243111yescharacteristic
403C2#2142252911yescharacteristic
404C2#2142254111yescharacteristic
405C2#2142258611yescharacteristic
406C2#2142262711yescharacteristic
407C2#2142271311yescharacteristic
408C2#2142280511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socleC23×C59C2×C22⋊C4
prime radicalC23×D59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1272D59
hypocenterC59#7623
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #21421922C2semidirect
5. #21421920C2semidirect
6. #21421918C2semidirect
7. C59×#1272C2semidirect
201. #1272D59semidirect
231. C59#7623semidirect

Maximal subgroups

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

subgroupindex
#214266342
#214241632
#214238572
#214219222
#214219202
#214219182
C59×#12722
#762359