Finite Group Databasebeta

← #28499313

Subgroups — #28499313

Order 15104.

462 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 83 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#21429272C211yesmaximal, characteristic
2#21422437C211yesmaximal, characteristic
3#21422119C212yesmaximal, complemented
4#21422116C212yesmaximal, complemented
5C4×#99×C59C211yesmaximal, characteristic
6C2×#14273706C411yescharacteristic, complemented
7#16837459—21no—
8#16837341C411yescharacteristic, complemented
9#16837338C2211yescharacteristic
10#16837322C411yescharacteristic, complemented
11#16837314C2211yescharacteristic
12#16837286—21no—
13#16837282—21no—
14#16837275—21no—
15#16837260—21no—
16C2×C59⋊#95C411yescharacteristic, complemented
17#16837126—21no—
18#16837126C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×#99×C59—21no—
21C4×C22⋊C4×C59—21no—
22C59×#342C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C23×Dic59—21no—
25#14273706—41no—
26#14273706—41no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C4⋊(C59⋊C8)—41no—
35C22⋊(C59⋊C8)—22no—
36#14273638—41no—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C22×D4×C59D411yescharacteristic
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C4×D4×C59—41no—
48C4×D4×C59—41no—
49C2×C4⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59—22no—
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59—41no—
54C2×C42×C59D411yescharacteristic
55#99×C59—24no—
56#95×C59—41no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C2×D4×C59—22no—
60C2×D4×C59—22no—
61C2×D4×C59—41no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—42no—
67C22×C4×C59—81no—
68C22×C4×C59—81no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—42no—
79C22×Dic59—42no—
80C22×Dic59—81no—
81C2×Dic118—41no—
82C2×Dic118—81no—
83C4⋊C4×C59—41no—
84C22⋊C4×C59—24no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—42no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—82no—
91C22⋊Dic59—42no—
92C22⋊Dic59—42no—
93C22⋊Dic59—81no—
94C4⋊Dic59—41no—
95Dic59⋊C4—42no—
96Dic59⋊C4—41no—
97Dic59⋊C4—41no—
98Dic59⋊C4—81no—
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—
111D4×C59—44no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—42no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—44no—
124C2×C4×C59—81no—
125C2×C4×C59—81no—
126C2×C4×C59—81no—
127C2×C4×C59—82no—
128C2×C4×C59—82no—
129C2×C4×C59#9911yescharacteristic
130C2×C4×C59#9911yescharacteristic
131C2×Dic59—22no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—84no—
144Dic118—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—41no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—84no—
169Dic59—42no—
170Dic59—81no—
171Dic59—81no—
172Dic59—81no—
173#2564—591no—
174C4×#99D5911yescharacteristic
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⋊Q8—591no—
186C2×C4×D4C2×D5911yescharacteristic
187#432—1181no—
188C2×#99—21no—
189#400—591no—
190#399—591no—
191#395—591no—
192#393—591no—
193#387—1181no—
194#386—1181no—
195#385—1181no—
196#380—1181no—
197C4×C22⋊C4—21no—
198C2×#95—591no—
199#342C2×D5912yes—
200#323—592no—
201#323—1181no—
202C59#1178911yescharacteristic, complemented, Sylow
203C22×Q8—591no—
204C22×D4C22⋊D5911yescharacteristic
205C23×C4—1181no—
206C23×C4C4×D5911yescharacteristic
207C23×C4C22⋊D5911yescharacteristic
208C22⋊Q8—2361no—
209C22⋊Q8—2361no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—591no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4—1181no—
215C2×C4⋊C4—1181no—
216C2×C4⋊C4C4×D5911yescharacteristic
217C2×C22⋊C4—22no—
218C2×C22⋊C4—1181no—
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1181no—
221C2×C22⋊C4D23611yescharacteristic
222C2×C22⋊C4C22×D5911yescharacteristic
223C2×C42—41no—
224C2×C42D23611yescharacteristic
225C4⋊C8—2361no—
226#99—24no—
227C22⋊C8—1182no—
228#97—2361no—
229#95—41no—
230#95—591no—
231#95—591no—
232#95—1181no—
233#95—1181no—
234#95—1181no—
235#95—1181no—
236#95—1181no—
237C24(C2×D59)⋊C411yescharacteristic
238C24C2×C22⋊D5911yescharacteristic
239C2×Q8—4721no—
240C2×Q8—2361no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4—1181no—
255C22×C4—4721no—
256C22×C4—2362no—
257C22×C4—2362no—
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4(C2×D59)⋊C411yescharacteristic
260C22×C4(C2×D59)⋊C411yescharacteristic
261C22×C4C2×C4×D5911yescharacteristic
262C22×C4C2×D23611yescharacteristic
263C2×C8—2362no—
264C4⋊C4—41no—
265C4⋊C4—4721no—
266C4⋊C4—2361no—
267C4⋊C4—2362no—
268C4⋊C4—2361no—
269C4⋊C4—2361no—
270C22⋊C4—24no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—4721no—
275C22⋊C4—2362no—
276C22⋊C4—2362no—
277C42—21no—
278C42—21no—
279C42—82no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#1427365512yes—
289C23C2×(C2×D59)⋊C411yescharacteristic
290C23#1427375712yes—
291Q8—4722no—
292D4—44no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—1182no—
311C2×C4—4724no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4#1427364111yescharacteristic
324C2×C4#1427364111yescharacteristic
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×#1427364111yescharacteristic
339C22#1683741011yescharacteristic
340C22C2×#1427365511yescharacteristic
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—84no—
348C4—4721no—
349C4—4721no—
350C4—4721no—
351C4—2362no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—42no—
356C2#2142261511yescharacteristic
357C2#2142574011yescharacteristic
358C2#2142575911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radicalC22×D4×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenterC4×#99D59
hypocenterC59#11789
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #21422119C2semidirect
4. #21422116C2semidirect
6. C2×#14273706C4semidirect
8. #16837341C4semidirect
10. #16837322C4semidirect
16. C2×C59⋊#95C4semidirect
202. C59#11789semidirect

Maximal subgroups

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

subgroupindex
#214292722
#214224372
#214221192
#214221162
C4×#99×C592
#1178959