Finite Group Databasebeta

← #28499404

Subgroups — #28499404

Order 15104.

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

normal subgroups only

This group has 402 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#21427364C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic
3#21422119C212yesmaximal
4#21422116C212yesmaximal
5C59×#1820C211yesmaximal, characteristic
6C2×C4×Dic118C411yescharacteristic, complemented
7#16837459—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#16837128—21no—
18#16837126C2212yes—
19C2×#117×C59C2211yescharacteristic
20C59×#381—21no—
21C59×#342—21no—
22C59×#342C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C4×Dic118—41no—
25C4×Dic118—41no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C4×Dic59—21no—
34C2×C4×Dic59—21no—
35C4⋊(C59⋊C8)—41no—
36C22⋊(C59⋊C8)—22no—
37#14273638—41no—
38C59⋊#95—21no—
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×C59C2×C411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59—22no—
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C22⋊C4×C59C2311yescharacteristic
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59D411yescharacteristic
54#95×C59—41no—
55#95×C59—41no—
56C24×C59C2×D411yescharacteristic
57C22×C4×C59—22no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—41no—
61C22×C4×C59—42no—
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22×C411yescharacteristic
67C22×C4×C59C2×D411yescharacteristic
68C22×Dic59—21no—
69C22×Dic59—21no—
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×Dic118—41no—
73C2×Dic118—42no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—42no—
79C22⋊C4×C59—42no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C22⋊Dic59—42no—
85C22⋊Dic59—42no—
86C4⋊Dic59—42no—
87C4⋊Dic59—41no—
88Dic59⋊C4—42no—
89Dic59⋊C4—42no—
90Dic59⋊C4—42no—
91C4×Dic59—42no—
92C4×Dic59—41no—
93C4×Dic59—41no—
94C2×C59⋊C8—42no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59C2×C22⋊C411yescharacteristic
100C23×C59#13712yes—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—41no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—41no—
116C2×C4×C59—82no—
117C2×C4×C59—82no—
118C2×C4×C59—82no—
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#9911yescharacteristic
121C2×Dic59—22no—
122C2×Dic59—22no—
123C2×Dic59—41no—
124C2×Dic59—41no—
125C2×Dic59—41no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133Dic118—42no—
134Dic118—42no—
135C59⋊C8—82no—
136#11810—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—22no—
140C22×C59—42no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59—81no—
144C22×C59C2×#9911yescharacteristic
145C22×C59#41911yescharacteristic
146C22×C59#42111yescharacteristic
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—41no—
154C4×C59—162no—
155Dic59—42no—
156Dic59—42no—
157Dic59—81no—
158Dic59—81no—
159#2396—591no—
160#1820D5911yescharacteristic
161#1464—591no—
162#1352—592no—
163#1351—592no—
164C2×C59—21no—
165C2×C59—21no—
166C2×C59—42no—
167C2×C59#151111yescharacteristic
168C2×C59#212011yescharacteristic
169C2×C59#212611yescharacteristic
170C2×C4×Q8—591no—
171C2×#117C2×D5911yescharacteristic
172#432—1181no—
173#400—591no—
174#399—591no—
175#398—1181no—
176#396—1181no—
177#395—591no—
178#393—591no—
179#381—21no—
180C4×C4⋊C4—591no—
181C4×C4⋊C4—1181no—
182C4×C22⋊C4—1181no—
183#342—21no—
184#342C2×D5912yes—
185#324—1181no—
186#323—592no—
187C59#1181011yescharacteristic, complemented, Sylow
188C22×Q8—591no—
189C23×C4C22⋊D5911yescharacteristic
190C4×Q8—2361no—
191C4×Q8—2361no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—591no—
195C2×C4⋊C4—1181no—
196C2×C4⋊C4—1181no—
197C2×C4⋊C4—1181no—
198C2×C4⋊C4—1181no—
199C2×C4⋊C4C4×D5911yescharacteristic
200C2×C4⋊C4D23611yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—1181no—
203C2×C22⋊C4—1181no—
204C2×C22⋊C4C22⋊D5911yescharacteristic
205C2×C22⋊C4C22×D5911yescharacteristic
206C2×C42—1181no—
207C2×C42—1181no—
208C2×C42C4×D5911yescharacteristic
209C2×C42D23611yescharacteristic
210C4⋊C8—2361no—
211C22⋊C8—1182no—
212#97—2361no—
213#95—41no—
214#95—41no—
215#95—591no—
216#95—591no—
217#95—1181no—
218#95—1181no—
219#95—1181no—
220#95—1181no—
221C24C2×C22⋊D5911yescharacteristic
222C2×Q8—2361no—
223C2×Q8—2362no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—1181no—
230C22×C4—1181no—
231C22×C4—1181no—
232C22×C4—1181no—
233C22×C4(C2×D59)⋊C411yescharacteristic
234C22×C4(C2×D59)⋊C411yescharacteristic
235C22×C4(C2×D59)⋊C411yescharacteristic
236C22×C4(C2×D59)⋊C411yescharacteristic
237C22×C4C2×C4×D5911yescharacteristic
238C22×C4C2×D23611yescharacteristic
239C2×C8—2362no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—2361no—
243C4⋊C4—2362no—
244C4⋊C4—2362no—
245C4⋊C4—2362no—
246C4⋊C4—2362no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—2362no—
252C22⋊C4—2362no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—2362no—
258C42—2361no—
259C42—2361no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D59)⋊C411yescharacteristic
265C23#1427375712yes—
266Q8—2362no—
267Q8—2362no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—1182no—
287C2×C4—1182no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4—2361no—
293C2×C4—2361no—
294C2×C4—2361no—
295C2×C4—2361no—
296C2×C4—2361no—
297C2×C4—2361no—
298C2×C4#1427364111yescharacteristic
299C2×C4#1427364111yescharacteristic
300C8—4722no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#1427364111yescharacteristic
309C22#1683741011yescharacteristic
310C22#1683742011yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—162no—
319C4—4721no—
320C4—4721no—
321C4—2362no—
322C4—2362no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2142262811yescharacteristic
327C2#2142573511yescharacteristic
328C2#2142575911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1820C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1820D59
hypocenterC59#11810
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1820

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
187. C59#11810semidirect

Maximal subgroups

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

subgroupindex
#214273642
#214224302
#214221192
#214221162
C59×#18202
#1181059