Finite Group Databasebeta

← #28486499

Subgroups — #28486499

Order 15104.

734 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 115 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#21426449C211yesmaximal, characteristic
2#21424195C212yesmaximal, complemented
3#21422423C211yesmaximal, characteristic
4#21421948C212yesmaximal, complemented
5C59×#2482C211yesmaximal, characteristic
6C22×C22⋊Dic59—21no—
7#16837366C2212yes—
8C22⋊C4×Dic59—24no—
9C4×C22⋊Dic59—21no—
10C4×C22⋊Dic59C2212yes—
11#16837161—24no—
12#16837126—22no—
13C2×#117×C59C2211yescharacteristic
14C22×C22⋊C4×C59C2211yescharacteristic
15C59×#399C412yescomplemented
16C59×#397C412yescomplemented
17C59×#388—22no—
18C59×#385—21no—
19C59×#381C2211yescharacteristic
20C59×#379—21no—
21C23×Dic59—22no—
22C2×#11745664—22no—
23C2×C22⋊Dic59—22no—
24C2×C22⋊Dic59—28no—
25C2×C22⋊Dic59—22no—
26C2×C4×Dic59—22no—
27C2×C4×Dic59D414yes—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)—22no—
30C59⋊#95—22no—
31C59⋊#95—22no—
32C25×C59D411yescharacteristic
33C23×C4×C59D411yescharacteristic
34C23×C4×C59C2311yescharacteristic
35#117×C59—22no—
36#117×C59—22no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59C2×C411yescharacteristic, complemented
39C2×C4⋊C4×C59C2×C411yescharacteristic, complemented
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59C2×C412yes—
47C2×C22⋊C4×C59D412yes—
48C2×C42×C59—21no—
49#95×C59—22no—
50#95×C59—21no—
51#95×C59C2×C412yescomplemented
52C24×C59—21no—
53C24×C59—24no—
54C24×C59—21no—
55C24×C59—42no—
56C24×C59—81no—
57C24×C59C22⋊C412yes—
58C24×C59C2×D411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—42no—
65C22×C4×C59—81no—
66C22×C4×C59C22⋊C412yes—
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D412yes—
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59#4112yes—
71C22×Dic59—24no—
72C22×Dic59—28no—
73C22×Dic59—82no—
74C22×Dic59—82no—
75C4⋊C4×C59—22no—
76C4⋊C4×C59—22no—
77C4⋊C4×C59—82no—
78C22⋊C4×C59—24no—
79C22⋊C4×C59—24no—
80C22⋊C4×C59—42no—
81C22⋊C4×C59—44no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—82no—
85C42×C59—42no—
86C22⋊Dic59—28no—
87C22⋊Dic59—28no—
88C4×Dic59—44no—
89C4×Dic59—82no—
90#11745664—44no—
91C2×C59⋊C8—42no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—24no—
95C23×C59—22no—
96C23×C59—22no—
97C23×C59—22no—
98C23×C59—22no—
99C23×C59—41no—
100C23×C59—42no—
101C23×C59—42no—
102C23×C59—42no—
103C23×C59—42no—
104C23×C59—81no—
105C23×C59—81no—
106C23×C59—88no—
107C23×C59#10414yes—
108C23×C59C2×C22⋊C411yescharacteristic
109C23×C59#12011yescharacteristic
110C23×C59#12311yescharacteristic
111C2×C4×C59—24no—
112C2×C4×C59—22no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—81no—
126C2×C4×C59C4×D412yes—
127C2×C4×C59#12112yes—
128C2×Dic59—28no—
129C2×Dic59—44no—
130C2×Dic59—44no—
131C2×Dic59—88no—
132C59⋊C8—82no—
133#9477—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—24no—
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—44no—
143C22×C59—44no—
144C22×C59—44no—
145C22×C59—44no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—82no—
149C22×C59—84no—
150C22×C59#38611yescharacteristic
151C22×C59C2×#10412yes—
152C22×C59#45812yes—
153C4×C59—22no—
154C4×C59—22no—
155C4×C59—42no—
156C4×C59—82no—
157C4×C59—81no—
158C4×C59—81no—
159Dic59—48no—
160#2482D5911yescharacteristic
161#2273—591no—
162#1826—592no—
163#1462—591no—
164#1284—592no—
165C2×C59—22no—
166C2×C59—22no—
167C2×C59—41no—
168C2×C59—44no—
169C2×C59#189011yescharacteristic
170C2×C59#189611yescharacteristic
171C2×C59#191111yescharacteristic
172C2×#117C2×D5911yescharacteristic
173C22×C22⋊C4—1181no—
174C22×C22⋊C4C2×D5911yescharacteristic
175#407—592no—
176#399Dic5912yescomplemented
177#397Dic5912yescomplemented
178#388—22no—
179#385—21no—
180#381C2×D5911yescharacteristic
181#379—21no—
182C4×C22⋊C4—592no—
183C4×C22⋊C4—1184no—
184C4×C22⋊C4—1181no—
185#339—1184no—
186#323—1182no—
187C59#947711yescharacteristic, complemented, Sylow
188C25C22⋊D5911yescharacteristic
189C23×C4—1182no—
190C23×C4C22⋊D5911yescharacteristic
191C23×C4C22×D5911yescharacteristic
192C2×M16—1182no—
193#117—22no—
194#117—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C2×Dic5911yescharacteristic, complemented
197C2×C4⋊C4C2×Dic5911yescharacteristic, complemented
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1182no—
205C2×C22⋊C4—1188no—
206C2×C22⋊C4—1182no—
207C2×C22⋊C4C2×Dic5912yes—
208C2×C22⋊C4C22⋊D5912yes—
209C2×C42—21no—
210C2×C42—594no—
211C2×C42—1182no—
212C22⋊C8—1182no—
213C22⋊C8—1182no—
214#95—22no—
215#95—21no—
216#95—1182no—
217#95—1182no—
218#95C2×Dic5912yescomplemented
219C24—21no—
220C24—24no—
221C24—21no—
222C24—42no—
223C24—81no—
224C24C22⋊Dic5912yes—
225C24C2×C22⋊D5911yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4—1188no—
234C22×C4—1184no—
235C22×C4—4722no—
236C22×C4—4722no—
237C22×C4C22⋊Dic5912yes—
238C22×C4D4×D5912yes—
239C22×C4#1174568712yes—
240C22×C4C22×Dic5911yescharacteristic
241C22×C4C2×C22⋊D5911yescharacteristic
242M16—2364no—
243C2×C8—2362no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—82no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—42no—
250C22⋊C4—44no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—82no—
254C22⋊C4—1188no—
255C22⋊C4—1188no—
256C42—42no—
257C42—4722no—
258C42—2364no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—88no—
274C23#1427365414yes—
275C23C2×C22⋊Dic5911yescharacteristic
276C23#1427369511yescharacteristic
277C23#1427370811yescharacteristic
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—1188no—
294C2×C4—4728no—
295C2×C4—2364no—
296C2×C4—2364no—
297C2×C4D4×Dic5912yes—
298C2×C4#1427369912yes—
299C8—4722no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22#1683728211yescharacteristic
317C22C2×#1427365412yes—
318C22#1683759112yes—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4—2368no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#2142451011yescharacteristic
331C2#2142454011yescharacteristic
332C2#2142462911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2482C2
socleC22×C59#386
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenter#2482D59
hypocenterC59#9477
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
2. #21424195C2semidirect
4. #21421948C2semidirect
15. C59×#399C4semidirect
16. C59×#397C4semidirect
38. C2×C4⋊C4×C59C2×C4semidirect
39. C2×C4⋊C4×C59C2×C4semidirect
51. #95×C59C2×C4semidirect
176. #399Dic59semidirect
177. #397Dic59semidirect
187. C59#9477semidirect
196. C2×C4⋊C4C2×Dic59semidirect
197. C2×C4⋊C4C2×Dic59semidirect
218. #95C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214264492
#214241952
#214224232
#214219482
C59×#24822
#947759