Finite Group Databasebeta

← #28498970

Subgroups — #28498970

Order 15104.

468 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 107 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#21427364C211yesmaximal, characteristic
2#21423829C211yesmaximal, characteristic
3#21422121C212yesmaximal
4#21422118C212yesmaximal
5C59×#1458C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic, complemented
7C2×#14273686—21no—
8#16837341C411yescharacteristic, complemented
9#16837338C2211yescharacteristic
10#16837335—21no—
11#16837325—21no—
12#16837322C411yescharacteristic, complemented
13#16837314C2211yescharacteristic
14C4×Dic59⋊C4—21no—
15C4×C4⋊Dic59C411yescharacteristic, complemented
16C4×C22⋊Dic59—21no—
17#16837169—21no—
18#16837169C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C4×Dic118—41no—
25C4×Dic118—41no—
26#14273686—24no—
27#14273686—42no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59—41no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×C22⋊Dic59—22no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C4×Dic59—21no—
38C2×C4×Dic59—21no—
39C2×C4×Dic59—41no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47#117×C59—41no—
48#117×C59—41no—
49C2×C4⋊C4×C59C2×C411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59C2311yescharacteristic
53C2×C42×C59C2×C411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C4⋊C8×C59—41no—
56C22⋊C8×C59—22no—
57C4×C8×C59—41no—
58C24×C59C2×D411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—22no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59SD1612yes—
67C22×C4×C59Q1612yes—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—42no—
75C2×Dic118—41no—
76C2×Dic118—42no—
77C2×C8×C59—42no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C42×C59—21no—
83C42×C59—21no—
84C42×C59—21no—
85C42×C59—21no—
86C22⋊Dic59—24no—
87C22⋊Dic59—42no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C4⋊Dic59—42no—
91C4⋊Dic59—44no—
92C4⋊Dic59—41no—
93Dic59⋊C4—42no—
94Dic59⋊C4—42no—
95Dic59⋊C4—42no—
96C4×Dic59—44no—
97C4×Dic59—41no—
98C4×Dic59—41no—
99C4×Dic59—42no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59C2×C22⋊C411yescharacteristic
105C23×C59C2×SD1611yescharacteristic
106C23×C59C2×Q1611yescharacteristic
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59#9911yescharacteristic
122C2×C4×C59#9911yescharacteristic
123C2×C4×C59#10312yes—
124C2×C4×C59#10312yes—
125C2×C4×C59#10412yes—
126C2×Dic59—22no—
127C2×Dic59—22no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—44no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—82no—
140C2×Dic59—82no—
141Dic118—42no—
142Dic118—42no—
143C8×C59—82no—
144#11715—591nomaximal, Sylow
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—41no—
153C22×C59—81no—
154C22×C59C2×#9911yescharacteristic
155C22×C59C2×#10311yescharacteristic
156C22×C59C2×#10411yescharacteristic
157C4×C59—22no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59#32812yes—
163C4×C59#32812yes—
164Dic59—42no—
165Dic59—42no—
166Dic59—81no—
167Dic59—84no—
168Dic59—81no—
169#2396—591no—
170#1740—591no—
171#1458D5911yescharacteristic
172#1352—592no—
173#1351—592no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—41no—
177C2×C59—41no—
178C2×C59C2×#32811yescharacteristic
179C2×C59#212011yescharacteristic
180C2×C59#212611yescharacteristic
181C2×C4×Q8—591no—
182C2×#117—1181no—
183C2×#117C2×D5911yescharacteristic
184#431—21no—
185#400—591no—
186#399—591no—
187#398—1181no—
188#396—1181no—
189#395—591no—
190#393—591no—
191C4×C4⋊C4—591no—
192C4×C4⋊C4—1181no—
193C4×C22⋊C4—1181no—
194#342—592no—
195#342—1181no—
196#324—21no—
197#323C2×D5912yes—
198C59#1171511yescharacteristic, complemented, Sylow
199C22×Q8—591no—
200C23×C4D23611yescharacteristic
201C4×Q8—2361no—
202C4×Q8—2361no—
203#117—41no—
204#117—41no—
205#117—1184no—
206#117—2362no—
207C2×C4⋊C4—591no—
208C2×C4⋊C4—1181no—
209C2×C4⋊C4—1181no—
210C2×C4⋊C4—1181no—
211C2×C4⋊C4—1181no—
212C2×C4⋊C4—2361no—
213C2×C4⋊C4C4×D5911yescharacteristic
214C2×C4⋊C4C22⋊D5911yescharacteristic
215C2×C22⋊C4—1182no—
216C2×C22⋊C4—1181no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4D23611yescharacteristic
219C2×C22⋊C4C22×D5911yescharacteristic
220C2×C42—1181no—
221C2×C42—1181no—
222C2×C42—2361no—
223C2×C42C4×D5911yescharacteristic
224C2×C42C22⋊D5911yescharacteristic
225C4⋊C8—41no—
226C22⋊C8—22no—
227C4×C8—41no—
228#95—591no—
229#95—591no—
230#95—1181no—
231#95—1181no—
232#95—1181no—
233#95—1181no—
234C24C2×D23611yescharacteristic
235C2×Q8—2361no—
236C2×Q8—2362no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—1181no—
241C22×C4—1181no—
242C22×C4—1181no—
243C22×C4—1181no—
244C22×C4—2362no—
245C22×C4#1174566012yes—
246C22×C4Dic23612yes—
247C22×C4(C2×D59)⋊C411yescharacteristic
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4C2×C4×D5911yescharacteristic
252C22×C4C2×C22⋊D5911yescharacteristic
253C2×C8—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2361no—
257C4⋊C4—2362no—
258C4⋊C4—2364no—
259C4⋊C4—2362no—
260C4⋊C4—2362no—
261C4⋊C4—2362no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—1184no—
265C22⋊C4—2362no—
266C22⋊C4—2362no—
267C22⋊C4—2362no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—2364no—
273C42—2361no—
274C42—2361no—
275C42—2362no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D59)⋊C411yescharacteristic
281C23C2×#1174566011yescharacteristic
282C23C2×Dic23611yescharacteristic
283Q8—2362no—
284Q8—2362no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—1182no—
300C2×C4—1182no—
301C2×C4—4722no—
302C2×C4—4722no—
303C2×C4—2364no—
304C2×C4—2361no—
305C2×C4—2361no—
306C2×C4—2361no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4#1427364111yescharacteristic
315C2×C4#1427364111yescharacteristic
316C2×C4#1427365212yes—
317C2×C4#1427365212yes—
318C2×C4#1427365312yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#1427364111yescharacteristic
330C22C2×#1427365211yescharacteristic
331C22C2×#1427365311yescharacteristic
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—4721no—
338C4—4724no—
339C4—4721no—
340C4—2362no—
341C4—2362no—
342C4#1683714012yes—
343C4#1683714012yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1683714011yescharacteristic
349C2#2142573811yescharacteristic
350C2#2142575911yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1458C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1458D59
hypocenterC59#11715
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic118C4semidirect
8. #16837341C4semidirect
12. #16837322C4semidirect
15. C4×C4⋊Dic59C4semidirect
198. C59#11715semidirect

Maximal subgroups

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

subgroupindex
#214273642
#214238292
#214221212
#214221182
C59×#14582
#1171559