Finite Group Databasebeta

← #28499180

Subgroups — #28499180

Order 15104.

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

normal subgroups only

This group has 436 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
1#21423829C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic
3#21422120C212yesmaximal
4#21422096C212yesmaximal
5C59×#2611C211yesmaximal, characteristic
6C2×#14273686—21no—
7#16837459—21no—
8#16837169—21no—
9#16837169C2212yes—
10#16837128—21no—
11#16837126C2212yes—
12C2×#125×C59C411yescharacteristic, complemented
13C2×#117×C59C2211yescharacteristic
14C59×#400C411yescharacteristic, complemented
15C59×#399C2211yescharacteristic
16C59×#398C411yescharacteristic, complemented
17C59×#397C2211yescharacteristic
18C59×#389—21no—
19C59×#388—21no—
20C59×#387—21no—
21C59×#385—21no—
22C59×#376C411yescharacteristic, complemented
23#14273686—24no—
24#14273686—42no—
25C2×C4⋊Dic59—41no—
26C2×C22⋊Dic59—22no—
27C2×C4×Dic59—41no—
28C4⋊(C59⋊C8)—41no—
29C22⋊(C59⋊C8)—22no—
30#14273638—41no—
31C23×C4×C59D411yescharacteristic
32#125×C59—41no—
33#125×C59—41no—
34#117×C59—41no—
35#117×C59—41no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59D411yescharacteristic
47C2×C22⋊C4×C59C2311yescharacteristic
48C2×C42×C59—21no—
49C2×C42×C59C2×C411yescharacteristic
50C2×C42×C59D411yescharacteristic
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56#95×C59C2×C411yescharacteristic
57#95×C59C2×C411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—42no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—42no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—42no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59—21no—
92C42×C59—21no—
93C42×C59—42no—
94C22⋊Dic59—24no—
95C22⋊Dic59—42no—
96C4⋊Dic59—44no—
97C4×Dic59—44no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104C23×C59AGL(1,ℤ/8)11yescharacteristic
105C23×C59#13711yescharacteristic
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—22no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
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#9911yescharacteristic
135C2×C4×C59#9911yescharacteristic
136C2×C4×C59#10412yes—
137C2×Dic59—44no—
138C2×Dic59—82no—
139C2×Dic59—82no—
140C59⋊C8—82no—
141#11760—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—81no—
151C22×C59C2×#9911yescharacteristic
152C22×C59#41711yescharacteristic
153C22×C59C2×#10411yescharacteristic
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—22no—
157C4×C59—42no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—81no—
163C4×C59—81no—
164C4×C59—81no—
165Dic59—84no—
166#2611D5911yescharacteristic
167#1740—591no—
168#1464—591no—
169#1352—592no—
170#1344—592no—
171C2×C59—21no—
172C2×C59—21no—
173C2×C59—41no—
174C2×C59—41no—
175C2×C59#150411yescharacteristic
176C2×C59#212311yescharacteristic
177C2×C59#212611yescharacteristic
178C2×#125Dic5911yescharacteristic, complemented
179C2×#117—1181no—
180C2×#117C2×D5911yescharacteristic
181#432—1181no—
182#400Dic5911yescharacteristic, complemented
183#399C2×D5911yescharacteristic
184#398Dic5911yescharacteristic, complemented
185#397C2×D5911yescharacteristic
186#389—21no—
187#388—21no—
188#387—21no—
189#385—21no—
190#376Dic5911yescharacteristic, complemented
191#342—592no—
192#342—1181no—
193#324—1181no—
194#323—592no—
195C59#1176011yescharacteristic, complemented, Sylow
196C23×C4C22⋊D5911yescharacteristic
197#125—41no—
198#125—41no—
199#117—41no—
200#117—41no—
201#117—1184no—
202#117—2362no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—2361no—
208C2×C4⋊C4C2×Dic5911yescharacteristic
209C2×C4⋊C4C2×Dic5911yescharacteristic
210C2×C4⋊C4C2×Dic5911yescharacteristic
211C2×C4⋊C4C22⋊D5911yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—1182no—
215C2×C22⋊C4C22⋊D5911yescharacteristic
216C2×C22⋊C4C22×D5911yescharacteristic
217C2×C42—21no—
218C2×C42—2361no—
219C2×C42C2×Dic5911yescharacteristic
220C2×C42C22⋊D5911yescharacteristic
221C4⋊C8—2361no—
222C22⋊C8—1182no—
223#97—2361no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C2×Dic5911yescharacteristic
230#95C2×Dic5911yescharacteristic
231C24C2×C22⋊D5911yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—2362no—
242C22×C4C22⋊Dic5911yescharacteristic
243C22×C4C22⋊Dic5911yescharacteristic
244C22×C4C22⋊Dic5911yescharacteristic
245C22×C4C22⋊Dic5911yescharacteristic
246C22×C4C22×Dic5911yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248C2×C8—2362no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—2364no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—1184no—
265C22⋊C4—2362no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—42no—
271C42—2364no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×C22⋊Dic5911yescharacteristic
277C23#1427375211yescharacteristic
278C23#1427375711yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—4722no—
308C2×C4—4722no—
309C2×C4—2364no—
310C2×C4#1427364211yescharacteristic
311C2×C4#1427364211yescharacteristic
312C2×C4#1427365412yes—
313C8—4722no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1427364211yescharacteristic
324C22#1683740311yescharacteristic
325C22C2×#1427365411yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—4724no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#2142260011yescharacteristic
343C2#2142575011yescharacteristic
344C2#2142576111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2611C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2611D59
hypocenterC59#11760
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×#125×C59C4semidirect
14. C59×#400C4semidirect
16. C59×#398C4semidirect
22. C59×#376C4semidirect
178. C2×#125Dic59semidirect
182. #400Dic59semidirect
184. #398Dic59semidirect
190. #376Dic59semidirect
195. C59#11760semidirect

Maximal subgroups

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

subgroupindex
#214238292
#214224302
#214221202
#214220962
C59×#26112
#1176059