Finite Group Databasebeta

← #28499315

Subgroups — #28499315

Order 15104.

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

normal subgroups only

This group has 462 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#21422437C211yesmaximal, characteristic
3#21422120C212yesmaximal, complemented
4#21422117C212yesmaximal, complemented
5C59×#2564C211yesmaximal, characteristic
6#16837459—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×#400C411yescharacteristic, complemented
15C59×#399C2211yescharacteristic
16C59×#395C411yescharacteristic, complemented
17C59×#393C2211yescharacteristic
18C59×#387—21no—
19C59×#386—21no—
20C59×#385—21no—
21C59×#380—21no—
22C2×#95×C59C411yescharacteristic, complemented
23C2×C22⋊Dic59—22no—
24C2×C4×Dic59—41no—
25C4⋊(C59⋊C8)—41no—
26#14273642—24no—
27C22⋊(C59⋊C8)—22no—
28#14273638—41no—
29C59⋊#95—41no—
30C22×Q8×C59C2×C411yescharacteristic
31C22×D4×C59D411yescharacteristic
32C23×C4×C59—21no—
33C23×C4×C59C2×C411yescharacteristic
34C23×C4×C59D411yescharacteristic
35C22⋊Q8×C59—41no—
36C22⋊Q8×C59—41no—
37C4×D4×C59—41no—
38C4×D4×C59—41no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59C2×C411yescharacteristic
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—42no—
73C22×C4×C59—42no—
74C22×C4×C59—81no—
75C22×C4×C59—81no—
76C22×C4×C59—81no—
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×Dic59—42no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—42no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—81no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—42no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—81no—
94C42×C59—21no—
95C42×C59—21no—
96C22⋊Dic59—24no—
97C22⋊Dic59—42no—
98C4×Dic59—82no—
99C2×C59⋊C8—42no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59#10412yes—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59#13712yes—
111Q8×C59—82no—
112D4×C59—44no—
113C2×C4×C59—22no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
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—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59—84no—
140C2×C4×C59#9911yescharacteristic
141C2×C4×C59#9911yescharacteristic
142C2×Dic59—44no—
143C2×Dic59—82no—
144C2×Dic59—82no—
145C59⋊C8—82no—
146#11789—591nomaximal, Sylow
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—22no—
150C22×C59—22no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59—42no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—81no—
159C22×C59C2×#9911yescharacteristic
160C22×C59#41911yescharacteristic
161C22×C59C2×#10411yescharacteristic
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—42no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—81no—
170C4×C59—81no—
171C4×C59—81no—
172Dic59—84no—
173#2564D5911yescharacteristic
174C4×#99—591no—
175#1466—591no—
176#1352—592no—
177#1351—592no—
178C2×C59—21no—
179C2×C59—21no—
180C2×C59—42no—
181C2×C59—42no—
182C2×C59#150811yescharacteristic
183C2×C59#212111yescharacteristic
184C2×C59#212611yescharacteristic
185C2×C22⋊Q8Dic5911yescharacteristic, complemented
186C2×C4×D4C2×D5911yescharacteristic
187#432—1181no—
188C2×#99—1181no—
189#400Dic5911yescharacteristic, complemented
190#399C2×D5911yescharacteristic
191#395Dic5911yescharacteristic, complemented
192#393C2×D5911yescharacteristic
193#387—21no—
194#386—21no—
195#385—21no—
196#380—21no—
197C4×C22⋊C4—1181no—
198C2×#95Dic5911yescharacteristic, complemented
199#342—592no—
200#323—592no—
201#323—1181no—
202C59#1178911yescharacteristic, complemented, Sylow
203C22×Q8C2×Dic5911yescharacteristic
204C22×D4C22⋊D5911yescharacteristic
205C23×C4—21no—
206C23×C4C2×Dic5911yescharacteristic
207C23×C4C22⋊D5911yescharacteristic
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C2×Dic5911yescharacteristic
216C2×C4⋊C4C2×Dic5911yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—1182no—
221C2×C22⋊C4C22⋊D5911yescharacteristic
222C2×C22⋊C4C22×D5911yescharacteristic
223C2×C42—2361no—
224C2×C42C22⋊D5911yescharacteristic
225C4⋊C8—2361no—
226#99—1184no—
227C22⋊C8—1182no—
228#97—2361no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—2361no—
235#95C2×Dic5911yescharacteristic
236#95C2×Dic5911yescharacteristic
237C24C22⋊Dic5911yescharacteristic
238C24C2×C22⋊D5911yescharacteristic
239C2×Q8—41no—
240C2×Q8—81no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—2362no—
258C22×C4C22⋊Dic5911yescharacteristic
259C22×C4C22⋊Dic5911yescharacteristic
260C22×C4C22⋊Dic5911yescharacteristic
261C22×C4C22×Dic5911yescharacteristic
262C22×C4C2×C22⋊D5911yescharacteristic
263C2×C8—2362no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4—1184no—
276C22⋊C4—2362no—
277C42—21no—
278C42—21no—
279C42—4722no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#1427365412yes—
289C23C2×C22⋊Dic5911yescharacteristic
290C23#1427375812yes—
291Q8—82no—
292D4—44no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—42no—
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—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—84no—
320C2×C4—4722no—
321C2×C4—4722no—
322C2×C4—2364no—
323C2×C4#1427364211yescharacteristic
324C2×C4#1427364211yescharacteristic
325C8—4722no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22C2×#1427364211yescharacteristic
339C22#1683741111yescharacteristic
340C22C2×#1427365411yescharacteristic
341C4—21no—
342C4—21no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—4724no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—42no—
356C2#2142261611yescharacteristic
357C2#2142574211yescharacteristic
358C2#2142576111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2564C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2564D59
hypocenterC59#11789
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #21422120C2semidirect
4. #21422117C2semidirect
12. C2×C22⋊Q8×C59C4semidirect
14. C59×#400C4semidirect
16. C59×#395C4semidirect
22. C2×#95×C59C4semidirect
185. C2×C22⋊Q8Dic59semidirect
189. #400Dic59semidirect
191. #395Dic59semidirect
198. C2×#95Dic59semidirect
202. C59#11789semidirect

Maximal subgroups

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

subgroupindex
C4×#142736422
#214224372
#214221202
#214221172
C59×#25642
#1178959