Finite Group Databasebeta

← #28499294

Subgroups — #28499294

Order 15104.

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

normal subgroups only

This group has 460 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#21422434C211yesmaximal, characteristic
3#21422120C212yesmaximal, complemented
4#21422117C212yesmaximal, complemented
5C59×#2580C211yesmaximal, characteristic
6C2×#14273642—21no—
7#16837366—21no—
8C4×C22⋊Dic59—21no—
9#16837169C2212yes—
10#16837128—21no—
11#16837126C2212yes—
12C2×C4×Q8×C59C411yescharacteristic, complemented
13C2×C4×D4×C59C2211yescharacteristic
14C59×#399C411yescharacteristic, complemented
15C59×#399C2211yescharacteristic
16C59×#393C411yescharacteristic, complemented
17C59×#393C2211yescharacteristic
18C59×#390—21no—
19C59×#389—21no—
20C59×#385—21no—
21C59×#385—21no—
22C59×#382C411yescharacteristic, complemented
23C2×#11745664—41no—
24C2×C22⋊Dic59—22no—
25C2×C4×Dic59—41no—
26#14273642—24no—
27C22⋊(C59⋊C8)—22no—
28C22⋊(C59⋊C8)—41no—
29C59⋊#95—41no—
30C22×Q8×C59C2×C411yescharacteristic
31C22×D4×C59D411yescharacteristic
32C23×C4×C59D411yescharacteristic
33C23×C4×C59D411yescharacteristic
34C4×Q8×C59—41no—
35C4×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×C59—21no—
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59C2×C411yescharacteristic
49C2×C22⋊C4×C59C2311yescharacteristic
50C2×C42×C59—21no—
51C2×C42×C59C2×C411yescharacteristic
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—42no—
60C2×Q8×C59—41no—
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—
86C4⋊C4×C59—81no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—42no—
90C22⋊C4×C59—42no—
91C22⋊C4×C59—81no—
92C22⋊C4×C59—81no—
93C42×C59—21no—
94C42×C59—21no—
95C42×C59—42no—
96C22⋊Dic59—24no—
97C22⋊Dic59—42no—
98C4×Dic59—82no—
99#11745664—82no—
100C2×C59⋊C8—42no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59C2×C22⋊C411yescharacteristic
111C23×C59#13712yes—
112Q8×C59—42no—
113Q8×C59—42no—
114D4×C59—44no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—22no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—42no—
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—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—81no—
139C2×C4×C59—81no—
140C2×C4×C59#9911yescharacteristic
141C2×C4×C59#9911yescharacteristic
142C2×C4×C59#10412yes—
143C2×Dic59—44no—
144C2×Dic59—82no—
145C2×Dic59—82no—
146C59⋊C8—82no—
147#11784—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—22no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59C2×#9911yescharacteristic
161C22×C59#41911yescharacteristic
162C22×C59C2×#10411yescharacteristic
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—22no—
166C4×C59—42no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59—81no—
171C4×C59—81no—
172C4×C59—81no—
173Dic59—84no—
174#2580D5911yescharacteristic
175C4×#99—591no—
176#1465—591no—
177#1352—592no—
178#1351—592no—
179C2×C59—21no—
180C2×C59—21no—
181C2×C59—42no—
182C2×C59—81no—
183C2×C59#150811yescharacteristic
184C2×C59#212011yescharacteristic
185C2×C59#212711yescharacteristic
186C2×C4×Q8Dic5911yescharacteristic, complemented
187C2×C4×D4C2×D5911yescharacteristic
188C2×#99—1181no—
189#407—1181no—
190#399Dic5911yescharacteristic, complemented
191#399C2×D5911yescharacteristic
192#393Dic5911yescharacteristic, complemented
193#393C2×D5911yescharacteristic
194#390—21no—
195#389—21no—
196#385—21no—
197#385—21no—
198#382Dic5911yescharacteristic, complemented
199C4×C22⋊C4—1181no—
200#342—592no—
201#324—1181no—
202#323—592no—
203C59#1178411yescharacteristic, complemented, Sylow
204C22×Q8C2×Dic5911yescharacteristic
205C22×D4C22⋊D5911yescharacteristic
206C23×C4C22⋊D5911yescharacteristic
207C23×C4C22⋊D5911yescharacteristic
208C2×M16—2361no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×D4—41no—
212C4×D4—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic5911yescharacteristic
218C2×C4⋊C4C22⋊D5911yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—1182no—
224C2×C22⋊C4C2×Dic5911yescharacteristic
225C2×C22⋊C4C22×D5911yescharacteristic
226C2×C42—21no—
227C2×C42—2361no—
228C2×C42C2×Dic5911yescharacteristic
229#99—1184no—
230C22⋊C8—1182no—
231C22⋊C8—2361no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—2361no—
236#95C2×Dic5911yescharacteristic
237#95C2×Dic5911yescharacteristic
238C24C22⋊Dic5911yescharacteristic
239C24C2×C22⋊D5911yescharacteristic
240C2×Q8—42no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—2362no—
256C22×C4C22⋊Dic5911yescharacteristic
257C22×C4C22⋊Dic5911yescharacteristic
258C22×C4C22⋊Dic5911yescharacteristic
259C22×C4C22×Dic5911yescharacteristic
260C22×C4C2×C22⋊D5911yescharacteristic
261M16—4722no—
262C2×C8—2362no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C22⋊C4—1184no—
277C22⋊C4—2362no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—4722no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊Dic5911yescharacteristic
292C23#1427375812yes—
293Q8—42no—
294Q8—42no—
295D4—44no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—4722no—
322C2×C4—4722no—
323C2×C4—2364no—
324C2×C4#1427364211yescharacteristic
325C2×C4#1427364211yescharacteristic
326C2×C4#1427365412yes—
327C8—4722no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22C2×#1427364211yescharacteristic
341C22#1683741111yescharacteristic
342C22C2×#1427365411yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—4724no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—81no—
358C2#2142261611yescharacteristic
359C2#2142573711yescharacteristic
360C2#2142576511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2580C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2580D59
hypocenterC59#11784
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #21422120C2semidirect
4. #21422117C2semidirect
12. C2×C4×Q8×C59C4semidirect
14. C59×#399C4semidirect
16. C59×#393C4semidirect
22. C59×#382C4semidirect
186. C2×C4×Q8Dic59semidirect
190. #399Dic59semidirect
192. #393Dic59semidirect
198. #382Dic59semidirect
203. C59#11784semidirect

Maximal subgroups

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

subgroupindex
C4×#142736422
#214224342
#214221202
#214221172
C59×#25802
#1178459