Finite Group Databasebeta

← #28486154

Subgroups — #28486154

Order 15104.

720 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 103 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#21429027C211yesmaximal, characteristic
2#21422611C212yesmaximal, complemented
3#21422423C211yesmaximal, characteristic, complemented
4#21421982C212yesmaximal, complemented
5C59×#2312C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273701C2211yescharacteristic
8C2×#14273696—21no—
9#16837366C2212yes—
10#16837338C412yescomplemented
11#16837311C412yescomplemented
12#16837302—22no—
13C4×C22⋊Dic59C2211yescharacteristic
14#16837135—24no—
15#16837126—22no—
16C22×C22⋊C4×C59C2211yescharacteristic
17C59×#388—24no—
18C59×#380—21no—
19C59×#379—21no—
20C4×C22⋊C4×C59C2212yes—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—21no—
23C22×C22⋊D59—22no—
24C2×#11745664—22no—
25#14273710—44no—
26#14273701—22no—
27#14273701—22no—
28#14273696—44no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—22no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—22no—
33C2×(C2×D59)⋊C4—22no—
34C2×(C2×D59)⋊C4—21no—
35C2×C4×Dic59—21no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—22no—
38C59⋊#95C2×C412yes—
39C25×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C23×C4×C59C2311yescharacteristic
42C2×C4⋊C4×C59—22no—
43C2×C22⋊C4×C59—22no—
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59—22no—
48C2×C22⋊C4×C59C2×C412yes—
49C2×C22⋊C4×C59D412yes—
50C2×C42×C59—22no—
51#95×C59—22no—
52#95×C59D414yescomplemented
53C24×C59—21no—
54C24×C59—21no—
55C24×C59—42no—
56C24×C59—42no—
57C24×C59—81no—
58C24×C59C22⋊C412yes—
59C24×C59C2×D411yescharacteristic
60C23×D59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—24no—
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—
71C2×C22⋊D59—44no—
72C2×C22⋊D59—42no—
73C2×C22⋊D59—42no—
74C2×C22⋊D59—44no—
75C2×C22⋊D59—41no—
76C2×C22⋊D59—41no—
77C2×C22⋊D59—82no—
78C22×Dic59—22no—
79C22×Dic59—21no—
80C2×D236—22no—
81C2×D236—81no—
82C4⋊C4×C59—82no—
83C22⋊C4×C59—24no—
84C22⋊C4×C59—24no—
85C22⋊C4×C59—44no—
86C22⋊C4×C59—82no—
87C22⋊C4×C59—82no—
88C42×C59—82no—
89C22⋊Dic59—24no—
90C22⋊Dic59—44no—
91(C2×D59)⋊C4—44no—
92(C2×D59)⋊C4—42no—
93C4⋊Dic59—22no—
94Dic59⋊C4—44no—
95C4×Dic59—81no—
96#11745664—82no—
97C2×C59⋊C8—42no—
98C23×C59—22no—
99C23×C59—22no—
100C23×C59—22no—
101C23×C59—22no—
102C23×C59—22no—
103C23×C59—41no—
104C23×C59—42no—
105C23×C59—42no—
106C23×C59—42no—
107C23×C59—42no—
108C23×C59—42no—
109C23×C59—42no—
110C23×C59—81no—
111C23×C59—81no—
112C23×C59—88no—
113C23×C59C2×C22⋊C411yescharacteristic
114C23×C59#12011yescharacteristic
115C23×C59#12311yescharacteristic
116C22×D59—41no—
117C22×D59—42no—
118C22×D59—41no—
119C22×D59—41no—
120C22×D59—42no—
121C2×C4×C59—22no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—44no—
129C2×C4×C59—44no—
130C2×C4×C59—44no—
131C2×C4×C59—81no—
132C2×C4×C59C4×D412yes—
133C2×C4×C59#12112yes—
134C22⋊D59—88no—
135C22⋊D59—82no—
136C22⋊D59—82no—
137C2×Dic59—24no—
138C2×Dic59—42no—
139C2×Dic59—42no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143D236—82no—
144C59⋊C8—82no—
145#9413—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—44no—
154C22×C59—44no—
155C22×C59—42no—
156C22×C59—42no—
157C22×C59—42no—
158C22×C59—82no—
159C22×C59—84no—
160C22×C59—84no—
161C22×C59#38611yescharacteristic
162C22×C59#42112yes—
163C22×C59#45312yes—
164C2×D59—82no—
165C2×D59—82no—
166C2×D59—81no—
167C2×D59—81no—
168C2×D59—81no—
169C4×C59—22no—
170C4×C59—22no—
171C4×C59—42no—
172C4×C59—84no—
173Dic59—44no—
174Dic59—81no—
175#2539—591no—
176#2312D5911yescharacteristic, complemented
177#1507—592no—
178#1462—591no—
179#1297—592no—
180C2×C59—22no—
181C2×C59—22no—
182C2×C59—41no—
183C2×C59—82no—
184C2×C59#188711yescharacteristic
185C2×C59#189711yescharacteristic
186C2×C59#191311yescharacteristic
187D59—161no—
188C2×#123—1181no—
189C2×#121—591no—
190C2×#120—1181no—
191C22×C22⋊C4C2×D5911yescharacteristic
192#407—592no—
193#399—592no—
194#392—592no—
195#390—1182no—
196#388—24no—
197#380—21no—
198#379—21no—
199C4×C22⋊C4—591no—
200C4×C22⋊C4C2×D5912yes—
201#327—1184no—
202#323—1182no—
203C59#941311yescharacteristic, complemented, Sylow
204C25C22⋊D5911yescharacteristic
205C22×D4—591no—
206C22×D4—1182no—
207C22×D4—1181no—
208C23×C4C22⋊D5911yescharacteristic
209C23×C4C22×D5911yescharacteristic
210C2×M16—1182no—
211#123—2364no—
212#121—1182no—
213#121—1182no—
214#120—2364no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—591no—
217C2×C4⋊C4—1182no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—1181no—
224C2×C22⋊C4—1182no—
225C2×C22⋊C4—1182no—
226C2×C22⋊C4—1181no—
227C2×C22⋊C4C4×D5912yes—
228C2×C22⋊C4D23612yes—
229C2×C42—22no—
230C2×C42—1181no—
231C22⋊C8—1182no—
232C22⋊C8—1182no—
233#95—22no—
234#95—592no—
235#95C22⋊D5914yescomplemented
236C24—21no—
237C24—21no—
238C24—42no—
239C24—42no—
240C24—81no—
241C24—1181no—
242C24(C2×D59)⋊C412yes—
243C24C2×C22⋊D5911yescharacteristic
244C2×D4—1182no—
245C2×D4—4721no—
246C2×D4—4722no—
247C2×D4—2364no—
248C2×D4—2362no—
249C2×D4—2362no—
250C2×D4—2364no—
251C2×D4—2361no—
252C2×D4—2361no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4—1182no—
259C22×C4—1181no—
260C22×C4(C2×D59)⋊C412yes—
261C22×C4C2×C4×D5911yescharacteristic
262C22×C4C2×D23611yescharacteristic
263C22×C4#1174568512yes—
264C22×C4C2×C22⋊D5912yes—
265M16—4722no—
266C2×C8—2362no—
267C4⋊C4—82no—
268C4⋊C4—1182no—
269C4⋊C4—2364no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—44no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C22⋊C4—1184no—
276C22⋊C4—2364no—
277C22⋊C4—2364no—
278C22⋊C4—2362no—
279C42—82no—
280C42—4721no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—41no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—81no—
294C23—81no—
295C23—88no—
296C23—2361no—
297C23—2362no—
298C23—2361no—
299C23—2362no—
300C23—2361no—
301C23C2×(C2×D59)⋊C411yescharacteristic
302C23#1427369611yescharacteristic
303C23#1427371011yescharacteristic
304D4—4722no—
305D4—4722no—
306D4—4722no—
307D4—4728no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—81no—
319C2×C4—1184no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4—2362no—
324C2×C4—2362no—
325C2×C4C4×C22⋊D5912yes—
326C2×C4#1427370112yes—
327C8—4722no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—82no—
341C22—84no—
342C22—84no—
343C22—4722no—
344C22—4722no—
345C22—4721no—
346C22—4721no—
347C22—4721no—
348C22#1683728111yescharacteristic
349C22#1683742012yes—
350C22#1683757012yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—84no—
355C4—4721no—
356C4—2364no—
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—9441no—
362C2#2142449411yescharacteristic
363C2#2142454411yescharacteristic
364C2#2142463811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2312C2
socleC22×C59#386
prime radical#21429027C2
cosocleC23×C4×C59C23
hypercenter#2312D59
hypocenterC59#9413
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #21422611C2semidirect
3. #21422423C2semidirect
4. #21421982C2semidirect
5. C59×#2312C2semidirect
10. #16837338C4semidirect
11. #16837311C4semidirect
52. #95×C59D4semidirect
176. #2312D59semidirect
203. C59#9413semidirect
235. #95C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214290272
#214226112
#214224232
#214219822
C59×#23122
#941359