Finite Group Databasebeta

← #28499043

Subgroups — #28499043

Order 15104.

436 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14273642C211yesmaximal, characteristic
2#21422417C211yesmaximal, characteristic
3#21422117C212yesmaximal, complemented
4#21422096C212yesmaximal, complemented
5C59×#2638C211yesmaximal, characteristic
6#16837455—21no—
7C2×#14273642—21no—
8C4×C22⋊Dic59—21no—
9#16837169C2212yes—
10#16837126—21no—
11#16837126C2212yes—
12C2×C22⋊Q8×C59C411yescharacteristic, complemented
13C2×C4×D4×C59C2211yescharacteristic
14C59×#398C411yescharacteristic, complemented
15C59×#397—21no—
16C59×#397C2211yescharacteristic
17C59×#395C411yescharacteristic, complemented
18C59×#394—21no—
19C59×#393C2211yescharacteristic
20C59×#387—21no—
21C59×#385—21no—
22C59×#381C411yescharacteristic, complemented
23C2×C22⋊Dic59—22no—
24C2×C4×Dic59—41no—
25C4⋊(C59⋊C8)—41no—
26#14273642—24no—
27C22⋊(C59⋊C8)—22no—
28C4×C59⋊C8—41no—
29C59⋊#95—41no—
30C22×Q8×C59C2×C411yescharacteristic
31C22×D4×C59D411yescharacteristic
32C23×C4×C59C2×C411yescharacteristic
33C23×C4×C59D411yescharacteristic
34C22⋊Q8×C59—41no—
35C22⋊Q8×C59—41no—
36C4×D4×C59—41no—
37C4×D4×C59—41no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59D411yescharacteristic
48C2×C22⋊C4×C59C2311yescharacteristic
49C2×C42×C59D411yescharacteristic
50#95×C59—21no—
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59C2×C411yescharacteristic
56#95×C59C2×C411yescharacteristic
57C24×C59C22⋊C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C2×Q8×C59—41no—
60C2×Q8×C59—81no—
61C2×D4×C59—22no—
62C2×D4×C59—22no—
63C2×D4×C59—41no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—81no—
73C22×C4×C59—81no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×Dic59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—42no—
85C4⋊C4×C59—81no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—42no—
89C22⋊C4×C59—42no—
90C22⋊C4×C59—81no—
91C22⋊C4×C59—81no—
92C42×C59—21no—
93C42×C59—21no—
94C22⋊Dic59—24no—
95C22⋊Dic59—42no—
96C4×Dic59—82no—
97C2×C59⋊C8—42no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59C2×C22⋊C411yescharacteristic
108C23×C59#13512yes—
109Q8×C59—82no—
110D4×C59—44no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—41no—
117C2×C4×C59—42no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—81no—
134C2×C4×C59—81no—
135C2×C4×C59—81no—
136C2×C4×C59#9911yescharacteristic
137C2×C4×C59#9911yescharacteristic
138C2×C4×C59#10412yes—
139C2×Dic59—44no—
140C2×Dic59—82no—
141C2×Dic59—82no—
142C59⋊C8—82no—
143#11731—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—22no—
147C22×C59—42no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59C2×#9911yescharacteristic
157C22×C59#41611yescharacteristic
158C22×C59C2×#10411yescharacteristic
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—22no—
162C4×C59—41no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59—81no—
166C4×C59—81no—
167C4×C59—81no—
168C4×C59—81no—
169Dic59—84no—
170#2638D5911yescharacteristic
171C4×#99—591no—
172#1460—591no—
173#1351—592no—
174#1344—592no—
175C2×C59—21no—
176C2×C59—21no—
177C2×C59—42no—
178C2×C59—81no—
179C2×C59#150111yescharacteristic
180C2×C59#212111yescharacteristic
181C2×C59#212911yescharacteristic
182C2×C22⋊Q8Dic5911yescharacteristic, complemented
183C2×C4×D4C2×D5911yescharacteristic
184#431—1181no—
185C2×#99—1181no—
186#398Dic5911yescharacteristic, complemented
187#397—21no—
188#397C2×D5911yescharacteristic
189#395Dic5911yescharacteristic, complemented
190#394—21no—
191#393C2×D5911yescharacteristic
192#387—21no—
193#385—21no—
194#381Dic5911yescharacteristic, complemented
195C4×C22⋊C4—1181no—
196#342—592no—
197#323—592no—
198#323—1181no—
199C59#1173111yescharacteristic, complemented, Sylow
200C22×Q8C2×Dic5911yescharacteristic
201C22×D4C22⋊D5911yescharacteristic
202C23×C4C2×Dic5911yescharacteristic
203C23×C4C22⋊D5911yescharacteristic
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×Dic5911yescharacteristic
212C2×C4⋊C4C2×Dic5911yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1182no—
218C2×C22⋊C4C22⋊D5911yescharacteristic
219C2×C22⋊C4C22×D5911yescharacteristic
220C2×C42—2361no—
221C2×C42C22⋊D5911yescharacteristic
222C4⋊C8—2361no—
223#99—1184no—
224C22⋊C8—1182no—
225C4×C8—2361no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—2361no—
232#95C2×Dic5911yescharacteristic
233#95C2×Dic5911yescharacteristic
234C24C22⋊Dic5911yescharacteristic
235C24C2×C22⋊D5911yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—2362no—
252C22×C4C22⋊Dic5911yescharacteristic
253C22×C4C22⋊Dic5911yescharacteristic
254C22×C4C22⋊Dic5911yescharacteristic
255C22×C4C22×Dic5911yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257C2×C8—2362no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—1184no—
271C22⋊C4—2362no—
272C42—21no—
273C42—21no—
274C42—4722no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊Dic5911yescharacteristic
285C23#1427374712yes—
286Q8—82no—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—4722no—
314C2×C4—4722no—
315C2×C4—2364no—
316C2×C4#1427364211yescharacteristic
317C2×C4#1427364211yescharacteristic
318C2×C4#1427365412yes—
319C8—4722no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#1427364211yescharacteristic
333C22#1683739811yescharacteristic
334C22C2×#1427365411yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—4724no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#2142258711yescharacteristic
351C2#2142574211yescharacteristic
352C2#2142577111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2638C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2638D59
hypocenterC59#11731
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #21422117C2semidirect
4. #21422096C2semidirect
12. C2×C22⋊Q8×C59C4semidirect
14. C59×#398C4semidirect
17. C59×#395C4semidirect
22. C59×#381C4semidirect
182. C2×C22⋊Q8Dic59semidirect
186. #398Dic59semidirect
189. #395Dic59semidirect
194. #381Dic59semidirect
199. C59#11731semidirect

Maximal subgroups

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

subgroupindex
C4×#142736422
#214224172
#214221172
#214220962
C59×#26382
#1173159