Finite Group Databasebeta

← #28503858

Subgroups — #28503858

Order 15104.

564 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 113 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#21426806C211yesmaximal, characteristic
2C2×#16837196C212yesmaximal
3#99×Dic59C212yesmaximal
4#21422221C211yesmaximal, characteristic
5C59×#1393C211yesmaximal, characteristic
6C2×#14273714C2211yescharacteristic
7C2×#14273692C411yescharacteristic, complemented
8C2×D4×Dic59C2211yescharacteristic
9C2×#14273642C2212yes—
10#16837308C411yescharacteristic, complemented
11#16837296—22no—
12#16837274—22no—
13#16837271C411yescharacteristic, complemented
14C22⋊C4×Dic59—22no—
15#16837248C411yescharacteristic, complemented
16#16837196C418yescomplemented
17#16837168—22no—
18C2×#99×C59C2212yes—
19C59×#390C2211yescharacteristic
20C22×Dic118C2×C411yescharacteristic
21C23×Dic59D412yes—
22#14273714—21no—
23#14273714—21no—
24#14273714C2×C414yes—
25#14273692—42no—
26D4×Dic59—42no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59D412yes—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—42no—
35C2×C4×Dic59C2×C411yescharacteristic
36C2×C4×Dic59C2×C411yescharacteristic
37#14273642—22no—
38#14273642C2×C414yes—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—22no—
42C59⋊#95—42no—
43C22×D4×C59C2311yescharacteristic
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C42×C59C2×C411yescharacteristic
48#99×C59—22no—
49#99×C59C2×C414yes—
50C24×C59C2×D412yes—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59C22×C412yes—
54C2×D4×C59#4111yescharacteristic
55C2×D4×C59#4111yescharacteristic
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—42no—
59C22×C4×C59C22×C411yescharacteristic
60C22×Dic59—21no—
61C22×Dic59—21no—
62C22×Dic59—22no—
63C22×Dic59—22no—
64C22×Dic59—42no—
65C22×Dic59—82no—
66C22×Dic59C22⋊C411yescharacteristic
67C22×Dic59C22⋊C412yes—
68C22×Dic59C22⋊C411yescharacteristic
69C2×Dic118—21no—
70C2×Dic118—41no—
71C2×Dic118—41no—
72C2×Dic118C4212yescomplemented
73C4⋊C4×C59—41no—
74C22⋊C4×C59—24no—
75C22⋊C4×C59—42no—
76C22⋊C4×C59—82no—
77C42×C59—41no—
78C22⋊Dic59—24no—
79C22⋊Dic59—24no—
80C22⋊Dic59—24no—
81C22⋊Dic59—42no—
82C22⋊Dic59—42no—
83C4⋊Dic59—41no—
84Dic59⋊C4—42no—
85Dic59⋊C4—82no—
86C4×Dic59—21no—
87C4×Dic59—22no—
88C4×Dic59—42no—
89C4×Dic59—84no—
90C4×Dic59C4212yescomplemented
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—42no—
94C23×C59—42no—
95C23×C59C2×C22⋊C411yescharacteristic
96C23×C59C4×D412yes—
97C23×C59C4×D412yes—
98D4×C59—42no—
99D4×C59—41no—
100D4×C59—41no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—42no—
104C2×C4×C59—44no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—82no—
110C2×C4×C59—82no—
111C2×C4×C59C2×C4211yescharacteristic
112C2×C4×C59#11711yescharacteristic
113C2×Dic59—22no—
114C2×Dic59—22no—
115C2×Dic59—21no—
116C2×Dic59—22no—
117C2×Dic59—41no—
118C2×Dic59—44no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59—41no—
122C2×Dic59—41no—
123C2×Dic59—42no—
124C2×Dic59—41no—
125C2×Dic59—44no—
126C2×Dic59—82no—
127C2×Dic59—82no—
128C2×Dic59—82no—
129C2×Dic59#9912yes—
130Dic118—42no—
131Dic118—42no—
132#12627—591nomaximal, Sylow
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—22no—
136C22×C59—22no—
137C22×C59—42no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—42no—
141C22×C59—82no—
142C22×C59C4×C22⋊C411yescharacteristic
143C22×C59C2×#9911yescharacteristic
144C22×C59#41011yescharacteristic
145C4×C59—21no—
146C4×C59—21no—
147C4×C59—81no—
148C4×C59—84no—
149C4×C59—81no—
150Dic59—22no—
151Dic59—42no—
152Dic59—42no—
153Dic59—41no—
154Dic59—42no—
155Dic59—84no—
156#2326—591no—
157C2×#354—592no—
158C4×#99—592no—
159#1393—591no—
160#1393D5911yescharacteristic
161C2×C59—21no—
162C2×C59—21no—
163C2×C59—42no—
164C2×C59—42no—
165C2×C59C4×#9911yescharacteristic
166C2×C59#212711yescharacteristic
167C2×C59#212911yescharacteristic
168C2×#124—591no—
169C2×C4×Q8—591no—
170C2×C4×D4—591no—
171C2×#99—592no—
172C2×#99C2×D5912yes—
173#391—591no—
174#390C2×D5911yescharacteristic
175#388—1182no—
176#385—1182no—
177#384—591no—
178C4×C22⋊C4—1182no—
179#376—591no—
180#354—598no—
181#342—1182no—
182C59#1262711yescharacteristic, complemented, Sylow
183C22×Q8—591no—
184C22×D4C22×D5911yescharacteristic
185C23×C4—592no—
186#124—594no—
187#124—1181no—
188#124—1181no—
189C4×Q8—2362no—
190C4×D4—2362no—
191C2×C4⋊C4—591no—
192C2×C4⋊C4—1182no—
193C2×C4⋊C4—1181no—
194C2×C4⋊C4C4×D5911yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—592no—
198C2×C22⋊C4—1182no—
199C2×C22⋊C4—1182no—
200C2×C42—591no—
201C2×C42—591no—
202C2×C42—1181no—
203C2×C42—2362no—
204C2×C42C4×D5911yescharacteristic
205#99—22no—
206#99—594no—
207#99—1182no—
208#99C4×D5914yes—
209#95—1181no—
210#95—1181no—
211#95—1182no—
212#95—2362no—
213C24D4×D5912yes—
214C2×Q8—592no—
215C2×Q8—1181no—
216C2×Q8—2361no—
217C2×Q8—2361no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4C2×C4×D5912yes—
221C2×D4#1174568711yescharacteristic
222C2×D4#1174568911yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—591no—
227C22×C4—591no—
228C22×C4—592no—
229C22×C4—1182no—
230C22×C4—1182no—
231C22×C4—1181no—
232C22×C4—1181no—
233C22×C4—4722no—
234C22×C4—2362no—
235C22×C4C2×C4×D5911yescharacteristic
236C4⋊C4—41no—
237C4⋊C4—4722no—
238C4⋊C4—2361no—
239C4⋊C4—2362no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—1184no—
244C22⋊C4—1184no—
245C22⋊C4—1184no—
246C22⋊C4—2362no—
247C22⋊C4—2362no—
248C42—41no—
249C42—592no—
250C42—1182no—
251C42—1181no—
252C42—4724no—
253C42—2362no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23C22⋊C4×D5911yescharacteristic
259C23(C22⋊D59)⋊C412yes—
260C23#1427368912yes—
261Q8—2362no—
262Q8—2362no—
263D4—42no—
264D4—41no—
265D4—41no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—592no—
277C2×C4—1182no—
278C2×C4—1182no—
279C2×C4—1181no—
280C2×C4—1182no—
281C2×C4—4722no—
282C2×C4—4722no—
283C2×C4—4722no—
284C2×C4—2361no—
285C2×C4—2364no—
286C2×C4—2364no—
287C2×C4—2361no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4—2361no—
291C2×C4—2362no—
292C2×C4—2361no—
293C2×C4C42×D5911yescharacteristic
294C2×C4#1427368511yescharacteristic
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22#1683724911yescharacteristic
305C22#99×D5911yescharacteristic
306C22#1683737411yescharacteristic
307C4—21no—
308C4—21no—
309C4—81no—
310C4—84no—
311C4—81no—
312C4—1182no—
313C4—4724no—
314C4—2362no—
315C4—2362no—
316C4—2362no—
317C4—2361no—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—42no—
322C2#2142388311yescharacteristic
323C2#2142576211yescharacteristic
324C2#2142576911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C2×C42
FrattiniC22×D4C22×D59
FittingC59×#1393C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenter#1393D59
hypocenterC59#12627
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
7. C2×#14273692C4semidirect
10. #16837308C4semidirect
13. #16837271C4semidirect
15. #16837248C4semidirect
16. #16837196C4semidirect
72. C2×Dic118C42semidirect
90. C4×Dic59C42semidirect
182. C59#12627semidirect

Maximal subgroups

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

subgroupindex
#214268062
C2×#168371962
#99×Dic592
#214222212
C59×#13932
#1262759