Finite Group Databasebeta

← #28499573

Subgroups — #28499573

Order 15104.

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

normal subgroups only

This group has 432 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#21424173C211yesmaximal, characteristic, complemented
3#21422118C212yesmaximal, complemented
4#21422094C212yesmaximal
5C59×#1458C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic, complemented
7#16837322C411yescharacteristic, complemented
8#16837319C411yescharacteristic, complemented
9#16837314C2211yescharacteristic
10#16837311C2211yescharacteristic
11#16837302—21no—
12#16837293—21no—
13#16837292—21no—
14#16837281—21no—
15#16837265—21no—
16#16837248C411yescharacteristic, complemented
17#16837169—21no—
18#16837169C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26#14273715—41no—
27#14273715—41no—
28C2×Dic59⋊C4—21no—
29C2×C22⋊Dic59—22no—
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—
37C59⋊#95—21no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—41no—
41C59⋊#95—41no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45#117×C59—41no—
46#117×C59—41no—
47C2×C4⋊C4×C59C2×C411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59C2311yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59D411yescharacteristic
53C4⋊C8×C59—41no—
54C22⋊C8×C59—22no—
55C4×C8×C59—41no—
56C24×C59C2×D411yescharacteristic
57C23×D59—21no—
58C22×C4×C59—22no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—41no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22×C411yescharacteristic
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—
74C22×Dic59—42no—
75C2×D236—41no—
76C2×D236—81no—
77C2×Dic118—41no—
78C2×Dic118—81no—
79C2×C8×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C42×C59—21no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C22⋊Dic59—81no—
91(C2×D59)⋊C4—42no—
92(C2×D59)⋊C4—42no—
93(C2×D59)⋊C4—42no—
94(C2×D59)⋊C4—41no—
95(C2×D59)⋊C4—41no—
96(C2×D59)⋊C4—81no—
97Dic59⋊C4—81no—
98C4×Dic59—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104C23×C59#13512yes—
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—21no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59#9911yescharacteristic
128C2×C4×C59#9911yescharacteristic
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—
139C2×Dic59—82no—
140C2×Dic59—82no—
141C2×Dic59—82no—
142D236—82no—
143Dic118—82no—
144C8×C59—82no—
145#11844—591nomaximal, Sylow
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—22no—
149C22×C59—42no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—81no—
153C22×C59C2×#9911yescharacteristic
154C22×C59#41611yescharacteristic
155C22×C59#42111yescharacteristic
156C2×D59—81no—
157C2×D59—81no—
158C2×D59—81no—
159C2×D59—81no—
160C2×D59—81no—
161C2×D59—81no—
162C2×D59—81no—
163C4×C59—21no—
164C4×C59—21no—
165C4×C59—41no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—41no—
170Dic59—42no—
171Dic59—81no—
172Dic59—81no—
173Dic59—162no—
174#2610—591no—
175#1820—591no—
176#1458D5911yescharacteristic, complemented
177#1351—592no—
178#1343—592no—
179C2×C59—21no—
180C2×C59—21no—
181C2×C59—42no—
182C2×C59#151611yescharacteristic
183C2×C59#212211yescharacteristic
184C2×C59#212611yescharacteristic
185D59—161no—
186C2×#124—591no—
187C2×#117C2×D5911yescharacteristic
188#431—21no—
189#395—591no—
190#394—591no—
191#393—591no—
192#392—591no—
193#390—1181no—
194#388—1181no—
195#388—1181no—
196#386—1181no—
197#381—1181no—
198#376—591no—
199#342—592no—
200#342—1181no—
201#324—21no—
202#323C2×D5912yes—
203C59#1184411yescharacteristic, complemented, Sylow
204C22×Q8—591no—
205C22×D4—591no—
206C22×D4—1181no—
207C23×C4D23611yescharacteristic
208#124—2361no—
209#124—2361no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4C4×D5911yescharacteristic
214C2×C4⋊C4C22⋊D5911yescharacteristic
215C2×C22⋊C4—1181no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4—1181no—
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1181no—
221C2×C22⋊C4—1182no—
222C2×C22⋊C4D23611yescharacteristic
223C2×C22⋊C4C22×D5911yescharacteristic
224C2×C42—1181no—
225C2×C42C4×D5911yescharacteristic
226C2×C42C22⋊D5911yescharacteristic
227C4⋊C8—41no—
228C22⋊C8—22no—
229C4×C8—41no—
230#95—591no—
231#95—591no—
232#95—1181no—
233#95—1181no—
234#95—1181no—
235#95—2361no—
236#95—2361no—
237C24—1181no—
238C24C2×D23611yescharacteristic
239C2×Q8—4721no—
240C2×Q8—2361no—
241C2×D4—4721no—
242C2×D4—4721no—
243C2×D4—2361no—
244C2×D4—2362no—
245C2×D4—2362no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—1181no—
251C22×C4—1181no—
252C22×C4—1181no—
253C22×C4—2362no—
254C22×C4(C2×D59)⋊C411yescharacteristic
255C22×C4(C2×D59)⋊C411yescharacteristic
256C22×C4(C2×D59)⋊C411yescharacteristic
257C22×C4(C2×D59)⋊C411yescharacteristic
258C22×C4C2×C4×D5911yescharacteristic
259C22×C4C2×C22⋊D5911yescharacteristic
260C2×C8—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—4721no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—4721no—
267C22⋊C4—4721no—
268C22⋊C4—2362no—
269C22⋊C4—2362no—
270C22⋊C4—2362no—
271C22⋊C4—2362no—
272C22⋊C4—2362no—
273C22⋊C4—2361no—
274C22⋊C4—2361no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—2362no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—2361no—
285C23—2361no—
286C23—2361no—
287C23—2361no—
288C23—2361no—
289C23—2361no—
290C23—2361no—
291C23C2×(C2×D59)⋊C411yescharacteristic
292C23#1427374912yes—
293Q8—4722no—
294D4—4722no—
295D4—4724no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—1182no—
312C2×C4—4722no—
313C2×C4—4722no—
314C2×C4—4722no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4—2361no—
322C2×C4—2361no—
323C2×C4#1427364111yescharacteristic
324C2×C4#1427364111yescharacteristic
325C8—82no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—4721no—
334C22—4721no—
335C22—4721no—
336C22—4721no—
337C22—4721no—
338C22—4721no—
339C22—4721no—
340C22C2×#1427364111yescharacteristic
341C22#1683739911yescharacteristic
342C22#1683741811yescharacteristic
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—9442no—
351C4—4721no—
352C4—4721no—
353C4—2362no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—9441no—
358C2#2142265311yescharacteristic
359C2#2142574711yescharacteristic
360C2#2142575911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1458C2
socleC22×C59C2×#99
prime radical#16837311C22
cosocleC2×C22⋊C4×C59C23
hypercenter#1458D59
hypocenterC59#11844
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. #21424173C2semidirect
3. #21422118C2semidirect
5. C59×#1458C2semidirect
6. C2×#14273715C4semidirect
7. #16837322C4semidirect
8. #16837319C4semidirect
16. #16837248C4semidirect
176. #1458D59semidirect
203. C59#11844semidirect

Maximal subgroups

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

subgroupindex
#214297832
#214241732
#214221182
#214220942
C59×#14582
#1184459