Finite Group Databasebeta

← #28499608

Subgroups — #28499608

Order 15104.

418 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429783C211yesmaximal, characteristic
2#21424179C211yesmaximal, characteristic, complemented
3#21422127C212yesmaximal, complemented
4#21422100C212yesmaximal, complemented
5C59×#1458C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7#16837366—21no—
8#16837322C2211yescharacteristic
9#16837319C2211yescharacteristic
10#16837314C411yescharacteristic
11#16837311C411yescharacteristic
12#16837302—21no—
13#16837293—21no—
14#16837292—21no—
15#16837281—21no—
16#16837248C411yescharacteristic
17#16837171—21no—
18#16837171C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324C2212yes—
22C59×#323—21no—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C2×#11745664—22no—
27#14273715—41no—
28#14273715—41no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×C4×Dic59—21no—
37C22⋊(C59⋊C8)—41no—
38C22⋊(C59⋊C8)—41no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45#117×C59—41no—
46#117×C59—41no—
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59C2311yescharacteristic
49C2×C22⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C4⋊C8×C59—41no—
54C22⋊C8×C59—22no—
55C4×C8×C59—41no—
56C24×C59C22⋊C411yescharacteristic
57C23×D59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59—41no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22×C411yescharacteristic
66C22×C4×C59C2×D411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C2×C22⋊D59—42no—
69C2×C22⋊D59—42no—
70C2×C22⋊D59—81no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C2×D236—41no—
75C2×D236—81no—
76C2×Dic118—41no—
77C2×Dic118—81no—
78C2×C8×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C22⋊Dic59—81no—
88(C2×D59)⋊C4—42no—
89(C2×D59)⋊C4—42no—
90(C2×D59)⋊C4—41no—
91(C2×D59)⋊C4—41no—
92(C2×D59)⋊C4—42no—
93(C2×D59)⋊C4—81no—
94Dic59⋊C4—81no—
95C4×Dic59—42no—
96#11745664—42no—
97#11745664—42no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59C2×C22⋊C411yescharacteristic
105C22×D59—41no—
106C22×D59—41no—
107C22×D59—41no—
108C22×D59—41no—
109C22×D59—41no—
110C22×D59—41no—
111C22×D59—41no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59#9911yescharacteristic
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59#13512yes—
129C22⋊D59—84no—
130C2×Dic59—22no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139D236—82no—
140Dic118—82no—
141C8×C59—82no—
142C59⋊C8—82no—
143#11851—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—42no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—81no—
151C22×C59C2×#9911yescharacteristic
152C22×C59#41611yescharacteristic
153C22×C59#42111yescharacteristic
154C2×D59—81no—
155C2×D59—81no—
156C2×D59—81no—
157C2×D59—81no—
158C2×D59—81no—
159C2×D59—81no—
160C2×D59—81no—
161C4×C59—22no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—41no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168Dic59—42no—
169Dic59—81no—
170Dic59—81no—
171#2610—591no—
172#1822—591no—
173#1458D5911yescharacteristic, complemented
174#1354—592no—
175#1345—592no—
176C2×C59—21no—
177C2×C59—21no—
178C2×C59—81no—
179C2×C59#151611yescharacteristic
180C2×C59#212211yescharacteristic
181C2×C59#213211yescharacteristic
182D59—161no—
183C2×#124—591no—
184C2×#117C2×D5911yescharacteristic
185#431—21no—
186#407—1181no—
187#395—591no—
188#394—591no—
189#393—591no—
190#392—591no—
191#390—1181no—
192#388—1181no—
193#388—1181no—
194#386—1181no—
195#376—591no—
196#343—592no—
197#343—1181no—
198#324C2×D5912yes—
199#323—21no—
200C59#1185111yescharacteristic, complemented, Sylow
201C22×Q8—591no—
202C22×D4—591no—
203C22×D4—1181no—
204C23×C4D23611yescharacteristic
205C2×M16—1182no—
206#124—2361no—
207#124—2361no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4D23611yescharacteristic
212C2×C4⋊C4C22×D5911yescharacteristic
213C2×C22⋊C4—1181no—
214C2×C22⋊C4—1181no—
215C2×C22⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4—1181no—
219C2×C22⋊C4C4×D5911yescharacteristic
220C2×C22⋊C4C22⋊D5911yescharacteristic
221C2×C42—1181no—
222C2×C42C4×D5911yescharacteristic
223C2×C42C22⋊D5911yescharacteristic
224C4⋊C8—41no—
225C22⋊C8—22no—
226C22⋊C8—2361no—
227C22⋊C8—2361no—
228C4×C8—41no—
229#95—591no—
230#95—591no—
231#95—1181no—
232#95—1181no—
233#95—1181no—
234C24—1181no—
235C24(C2×D59)⋊C411yescharacteristic
236C2×Q8—4721no—
237C2×Q8—2361no—
238C2×D4—4721no—
239C2×D4—4721no—
240C2×D4—2361no—
241C2×D4—2362no—
242C2×D4—2362no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4—1181no—
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4C2×C4×D5911yescharacteristic
254C22×C4C2×D23611yescharacteristic
255C22×C4C2×C22⋊D5911yescharacteristic
256M16—2362no—
257M16—2362no—
258C2×C8—42no—
259C2×C8—2362no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—4721no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—4721no—
266C22⋊C4—4721no—
267C22⋊C4—2362no—
268C22⋊C4—2362no—
269C22⋊C4—2361no—
270C22⋊C4—2361no—
271C22⋊C4—2362no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—2362no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—2361no—
283C23—2361no—
284C23—2361no—
285C23—2361no—
286C23—2361no—
287C23—2361no—
288C23—2361no—
289C23C2×(C2×D59)⋊C411yescharacteristic
290Q8—4722no—
291D4—4722no—
292D4—4724no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—1182no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4#1427364111yescharacteristic
317C2×C4#1427364111yescharacteristic
318C2×C4#1427374912yes—
319C8—82no—
320C8—4722no—
321C22—21no—
322C22—21no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—4721no—
329C22—4721no—
330C22—4721no—
331C22—4721no—
332C22—4721no—
333C22—4721no—
334C22—4721no—
335C22C2×#1427364111yescharacteristic
336C22#1683739911yescharacteristic
337C22#1683741811yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—4721no—
346C4—4721no—
347C4—2362no—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—9441no—
352C2#2142265311yescharacteristic
353C2#2142574511yescharacteristic
354C2#2142577911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1458C2
socleC22×C59C2×#99
prime radical#16837311C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1458D59
hypocenterC59#11851
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #21424179C2semidirect
3. #21422127C2semidirect
4. #21422100C2semidirect
5. C59×#1458C2semidirect
173. #1458D59semidirect
200. C59#11851semidirect

Maximal subgroups

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

subgroupindex
#214297832
#214241792
#214221272
#214221002
C59×#14582
#1185159