Finite Group Databasebeta

← #28499026

Subgroups — #28499026

Order 15104.

470 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 87 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#21427380C211yesmaximal, characteristic
2#21422414C211yesmaximal, characteristic
3#21422116C212yesmaximal, complemented
4#21422095C212yesmaximal, complemented
5C4×#99×C59C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic, complemented
7C2×C22⋊(C59⋊C8)—21no—
8#16837340—21no—
9#16837327C411yescharacteristic, complemented
10#16837327C2211yescharacteristic
11#16837316—21no—
12#16837314C411yescharacteristic, complemented
13#16837314C2211yescharacteristic
14C4×C22⋊Dic59—21no—
15C4×C22⋊Dic59—21no—
16#16837248C411yescharacteristic, complemented
17#16837128—21no—
18#16837126C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×#99×C59—21no—
21C4×C22⋊C4×C59—21no—
22C59×#342C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×C59⋊C8—41no—
25C4×Dic118—41no—
26C4×Dic118—41no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35C22⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—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×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C4×D4×C59—41no—
48C4×D4×C59—41no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59—22no—
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59—41no—
54C2×C42×C59C2×C411yescharacteristic
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—42no—
68C22×C4×C59—42no—
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—
78C2×Dic118—41no—
79C2×Dic118—42no—
80C4⋊C4×C59—41no—
81C22⋊C4×C59—24no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—42no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—82no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C22⋊Dic59—42no—
91C22⋊Dic59—42no—
92C4⋊Dic59—41no—
93Dic59⋊C4—42no—
94C4×Dic59—42no—
95C4×Dic59—41no—
96C4×Dic59—41no—
97C2×C59⋊C8—42no—
98C2×C59⋊C8—82no—
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×C59#10412yes—
108C23×C59C2×C22⋊C411yescharacteristic
109C23×C59#13512yes—
110D4×C59—44no—
111C2×C4×C59—22no—
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—42no—
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—82no—
127C2×C4×C59—82no—
128C2×C4×C59#9911yescharacteristic
129C2×C4×C59#9911yescharacteristic
130C2×Dic59—22no—
131C2×Dic59—41no—
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—
142Dic118—42no—
143Dic118—42no—
144C59⋊C8—82no—
145#11728—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—22no—
149C22×C59—22no—
150C22×C59—42no—
151C22×C59—42no—
152C22×C59—41no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—81no—
157C22×C59—81no—
158C22×C59C2×#9911yescharacteristic
159C22×C59#41611yescharacteristic
160C22×C59C2×#10411yescharacteristic
161C4×C59—21no—
162C4×C59—41no—
163C4×C59—41no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59—84no—
167C4×C59#35412yes—
168Dic59—42no—
169Dic59—81no—
170Dic59—81no—
171Dic59—81no—
172#2398—591no—
173C4×#99D5911yescharacteristic
174#1459—591no—
175#1351—592no—
176#1344—592no—
177C2×C59—21no—
178C2×C59—21no—
179C2×C59—42no—
180C2×C59—42no—
181C2×C59#150111yescharacteristic
182C2×C59#212011yescharacteristic
183C2×C59C2×#35411yescharacteristic
184C2×C4×Q8—591no—
185C2×C4×D4C2×D5911yescharacteristic
186C2×#99—21no—
187C2×C22⋊C8—1181no—
188#400—1181no—
189#397—591no—
190#397—591no—
191#394—1181no—
192#393—591no—
193#393—591no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—1181no—
196C4×C22⋊C4—1181no—
197#376—591no—
198#342C2×D5912yes—
199#324—1181no—
200#323—592no—
201C59#1172811yescharacteristic, complemented, Sylow
202C22×Q8—591no—
203C22×D4C22⋊D5911yescharacteristic
204C23×C4D23611yescharacteristic
205C23×C4C22⋊D5911yescharacteristic
206C22×C8—2361no—
207C4×Q8—2361no—
208C4×Q8—2361no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—591no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4D23611yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4—1181no—
219C2×C22⋊C4C4×D5911yescharacteristic
220C2×C22⋊C4C22×D5911yescharacteristic
221C2×C42—41no—
222C2×C42—1181no—
223C2×C42—1181no—
224C2×C42C4×D5911yescharacteristic
225#99—24no—
226C22⋊C8—1182no—
227C22⋊C8—2361no—
228#95—41no—
229#95—591no—
230#95—591no—
231#95—1181no—
232#95—1181no—
233#95—1181no—
234#95—1181no—
235#95—1181no—
236C24(C2×D59)⋊C411yescharacteristic
237C24C2×C22⋊D5911yescharacteristic
238C2×Q8—2361no—
239C2×Q8—2362no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—1181no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—1181no—
254C22×C4(C2×D59)⋊C411yescharacteristic
255C22×C4(C2×D59)⋊C411yescharacteristic
256C22×C4(C2×D59)⋊C411yescharacteristic
257C22×C4C2×C4×D5911yescharacteristic
258C22×C4C2×D23611yescharacteristic
259C2×C8—4722no—
260C2×C8—2362no—
261C4⋊C4—41no—
262C4⋊C4—2361no—
263C4⋊C4—2362no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—2362no—
269C22⋊C4—2362no—
270C22⋊C4—2362no—
271C22⋊C4—2362no—
272C42—21no—
273C42—21no—
274C42—82no—
275C42—2362no—
276C42—2361no—
277C42—2361no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1427365512yes—
287C23C2×(C2×D59)⋊C411yescharacteristic
288C23#1427374712yes—
289Q8—2362no—
290Q8—2362no—
291D4—44no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—1182no—
310C2×C4—2361no—
311C2×C4—2361no—
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#1427364111yescharacteristic
322C2×C4#1427364111yescharacteristic
323C8—4722no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#1427364111yescharacteristic
337C22#1683739611yescharacteristic
338C22C2×#1427365511yescharacteristic
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—84no—
345C4—4721no—
346C4—4721no—
347C4—4721no—
348C4—2362no—
349C4#1683719612yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2#2142258511yescharacteristic
355C2#2142573511yescharacteristic
356C2C2×#1683719611yescharacteristic
subgroupG/H
centerC2×C4#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#11728
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #21422116C2semidirect
4. #21422095C2semidirect
6. C2×C4×Dic118C4semidirect
9. #16837327C4semidirect
12. #16837314C4semidirect
16. #16837248C4semidirect
201. C59#11728semidirect

Maximal subgroups

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

subgroupindex
#214273802
#214224142
#214221162
#214220952
C4×#99×C592
#1172859