Finite Group Databasebeta

← #28507363

Subgroups — #28507363

Order 15104.

500 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 107 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#21429269C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic
3#21422220C214yesmaximal
4C4×#99×C59C211yesmaximal, characteristic
5C2×#14273727C411yescharacteristic, complemented
6C2×#14273642C2212yes—
7#16837314—21no—
8#16837314—21no—
9#16837305C2212yes—
10#16837267C412yescomplemented
11#16837262—21no—
12#16837262—21no—
13#16837262—21no—
14C4×C4⋊Dic59C411yescharacteristic, complemented
15#16837169—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×#99×C59C2212yes—
18C4×C22⋊C4×C59—21no—
19C59×#342—21no—
20C22×Dic118—21no—
21#14273727—41no—
22#14273727—41no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—41no—
26C2×C4⋊Dic59C2×C412yes—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—22no—
30C2×C4×Dic59C2×C412yes—
31#14273642—24no—
32C59⋊#95—21no—
33C59⋊#95—22no—
34C59⋊#95—22no—
35C59⋊#95—41no—
36C22×D4×C59C2311yescharacteristic
37C23×C4×C59D411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C4×D4×C59—41no—
40C4×D4×C59—41no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C42×C59—41no—
46C2×C42×C59C2×C411yescharacteristic
47#99×C59—22no—
48#99×C59Q814yescomplemented
49#95×C59—41no—
50C24×C59C2×D411yescharacteristic
51C24×C59C2×D411yescharacteristic
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59C2×Q812yes—
55C2×D4×C59#4112yes—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—42no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22×C411yescharacteristic
68C22×Dic59—22no—
69C22×Dic59—22no—
70C22×Dic59—42no—
71C2×Dic118—42no—
72C2×Dic118—82no—
73C4⋊C4×C59—41no—
74C22⋊C4×C59—24no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—42no—
78C42×C59—21no—
79C42×C59—21no—
80C42×C59—82no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C22⋊Dic59—82no—
84C22⋊Dic59—82no—
85C4⋊Dic59—42no—
86C4⋊Dic59—41no—
87C4⋊Dic59—41no—
88C4⋊Dic59—42no—
89C4⋊Dic59—82no—
90C4×Dic59—42no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—21no—
94C23×C59—21no—
95C23×C59—41no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59C2×C22⋊C411yescharacteristic
100C23×C59C22⋊Q812yes—
101C23×C59#12411yescharacteristic
102C23×C59C4⋊Q811yescharacteristic
103D4×C59—42no—
104D4×C59—42no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—44no—
117C2×C4×C59—81no—
118C2×C4×C59—81no—
119C2×C4×C59—82no—
120C2×C4×C59—82no—
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#9911yescharacteristic
123C2×C4×C59C4×Q812yes—
124C2×Dic59—42no—
125C2×Dic59—42no—
126C2×Dic59—42no—
127C2×Dic59—42no—
128C2×Dic59—42no—
129C2×Dic59—42no—
130C2×Dic59—82no—
131C2×Dic59—82no—
132C2×Dic59—82no—
133Dic118—84no—
134#13299—591nomaximal, Sylow
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—22no—
139C22×C59—21no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—41no—
146C22×C59—41no—
147C22×C59—81no—
148C22×C59—81no—
149C22×C59#39111yescharacteristic
150C22×C59C2×#9911yescharacteristic
151C22×C59#41011yescharacteristic
152C4×C59—21no—
153C4×C59—22no—
154C4×C59—41no—
155C4×C59—41no—
156C4×C59—41no—
157C4×C59—84no—
158C4×C59#35412yes—
159Dic59—82no—
160Dic59—82no—
161Dic59—162no—
162#2563—591no—
163#1799—591no—
164C4×#99D5911yescharacteristic
165#1393—594no—
166C2×C59—21no—
167C2×C59—21no—
168C2×C59—41no—
169C2×C59—42no—
170C2×C59—41no—
171C2×C59#199611yescharacteristic
172C2×C59C2×#35411yescharacteristic
173C2×C59#212811yescharacteristic
174C2×C4⋊Q8—591no—
175C2×C4×D4C2×D5911yescharacteristic
176C2×#99—592no—
177C2×#99C2×D5912yes—
178#393—1181no—
179#393—1181no—
180#390—592no—
181#382—592no—
182#380—1181no—
183#380—1181no—
184#380—1181no—
185C4×C4⋊C4—591no—
186C4×C22⋊C4—21no—
187#342—21no—
188#342—1181no—
189C59#1329911yescharacteristic, complemented, Sylow
190C22×Q8—1181no—
191C22×D4C22×D5911yescharacteristic
192C23×C4D23611yescharacteristic
193C23×C4C22⋊D5911yescharacteristic
194C4⋊Q8—2361no—
195C4⋊Q8—2361no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—592no—
199C2×C4⋊C4—1181no—
200C2×C4⋊C4—1181no—
201C2×C4⋊C4—2361no—
202C2×C4⋊C4C4×D5911yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1182no—
205C2×C22⋊C4—1182no—
206C2×C22⋊C4—1182no—
207C2×C22⋊C4D23611yescharacteristic
208C2×C22⋊C4C22⋊D5911yescharacteristic
209C2×C42—41no—
210C2×C42—592no—
211C2×C42C4×D5911yescharacteristic
212#99—22no—
213#99—1184no—
214#99Dic11814yescomplemented
215#95—41no—
216#95—1181no—
217#95—1182no—
218#95—1182no—
219#95—2361no—
220C24C2×D23611yescharacteristic
221C24C2×C22⋊D5911yescharacteristic
222C2×Q8—4722no—
223C2×Q8—2362no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4C2×Dic11812yes—
227C2×D4#1174568512yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1182no—
236C22×C4—1182no—
237C22×C4—2362no—
238C22×C4(C2×D59)⋊C411yescharacteristic
239C22×C4(C2×D59)⋊C411yescharacteristic
240C22×C4(C2×D59)⋊C411yescharacteristic
241C22×C4(C2×D59)⋊C411yescharacteristic
242C22×C4C2×C4×D5911yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—4722no—
245C4⋊C4—2362no—
246C4⋊C4—2362no—
247C4⋊C4—2361no—
248C4⋊C4—2361no—
249C22⋊C4—24no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—42no—
253C22⋊C4—4722no—
254C22⋊C4—4722no—
255C22⋊C4—2362no—
256C22⋊C4—2362no—
257C42—21no—
258C42—21no—
259C42—82no—
260C42—2362no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23C2×(C2×D59)⋊C411yescharacteristic
270C23#1427370612yes—
271C23#1427371511yescharacteristic
272C23#1427372711yescharacteristic
273Q8—4724no—
274D4—42no—
275D4—42no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—4722no—
293C2×C4—4722no—
294C2×C4—4722no—
295C2×C4—2362no—
296C2×C4—2362no—
297C2×C4—2362no—
298C2×C4—2362no—
299C2×C4—2362no—
300C2×C4—2362no—
301C2×C4#1427364111yescharacteristic
302C2×C4#1427364111yescharacteristic
303C2×C4C4×Dic11812yes—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—81no—
318C22#1683730811yescharacteristic
319C22C2×#1427364111yescharacteristic
320C22#1683737511yescharacteristic
321C4—21no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—84no—
327C4—9442no—
328C4—4722no—
329C4—4722no—
330C4#1683719612yes—
331C2—21no—
332C2—21no—
333C2—42no—
334C2—41no—
335C2—41no—
336C2#2142508111yescharacteristic
337C2C2×#1683719611yescharacteristic
338C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenterC4×#99D59
hypocenterC59#13299
LCSG ▷ C22×C4×C59 ▷ C22×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
5. C2×#14273727C4semidirect
10. #16837267C4semidirect
14. C4×C4⋊Dic59C4semidirect
48. #99×C59Q8semidirect
189. C59#13299semidirect
214. #99Dic118semidirect

Maximal subgroups

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

subgroupindex
#214292692
#214240842
#214222202
C4×#99×C592
#1329959