Finite Group Databasebeta

← #28481595

Subgroups — #28481595

Order 15104.

486 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×(C22⋊D59)⋊C8C211yesmaximal, characteristic
2#21424132C212yesmaximal, complemented
3#21422432C212yesmaximal
4#21421973C211yesmaximal, characteristic, complemented
5C59×#1293C211yesmaximal, characteristic, complemented
6C23×C59⋊C8C2211yescharacteristic
7C22×C8×D59C411yescharacteristic
8C2×C4×C22⋊D59C2211yescharacteristic
9(C22⋊D59)⋊C8—22no—
10(C22⋊D59)⋊C8—22no—
11C2×#14273662—22no—
12C2×Dic59⋊C8C411yescharacteristic
13#16837366C2212yes—
14C2×(C2×D59)⋊C8C411yescharacteristic
15C2×C8×Dic59C411yescharacteristic
16#16837166—22no—
17#16837127—22no—
18#16837125—22no—
19C59×#407C2212yes—
20C2×C22⋊C8×C59C2211yescharacteristic
21C22×C22⋊D59Q811yescharacteristic
22C22×C4×D59C2×C411yescharacteristic
23C2×#11745664—22no—
24C22×C59⋊C8—42no—
25C22×C59⋊C8C2×C411yescharacteristic
26C22×C59⋊C8C2×C411yescharacteristic
27C22×C59⋊C8D412yes—
28C22×C59⋊C8D412yes—
29C2×C8×D59—21no—
30C2×C8×D59—21no—
31C2×C8×D59—42no—
32C4×C22⋊D59—22no—
33C4×C22⋊D59—22no—
34C2×Dic59⋊C4D411yescharacteristic
35C2×C22⋊Dic59Q811yescharacteristic
36C2×(C2×D59)⋊C4D411yescharacteristic
37C2×C4×Dic59C2×C411yescharacteristic
38#14273662—44no—
39Dic59⋊C8—22no—
40C22⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)—22no—
42(C2×D59)⋊C8—22no—
43C8×Dic59—21no—
44C8×Dic59—21no—
45C23×C4×C59C2311yescharacteristic
46C2×M16×C59—22no—
47C22×C8×C59C2×C411yescharacteristic
48C22×C8×C59C2×C411yescharacteristic
49C22⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51C22⋊C8×C59—22no—
52C24×C59C2×Q811yescharacteristic
53C23×D59C4⋊C411yescharacteristic
54C22×C4×C59—41no—
55C22×C4×C59—41no—
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D412yes—
58C22×C4×C59C2×D411yescharacteristic
59C22×C4×C59#4112yes—
60C2×C22⋊D59—22no—
61C2×C22⋊D59—22no—
62C2×C22⋊D59—22no—
63C22×Dic59C4⋊C411yescharacteristic
64C22×Dic59C4⋊C411yescharacteristic
65C22×Dic59C4⋊C411yescharacteristic
66C2×C4×D59—21no—
67C2×C4×D59—21no—
68C2×C4×D59—41no—
69C2×C4×D59—41no—
70M16×C59—42no—
71M16×C59—42no—
72C2×C8×C59—21no—
73C2×C8×C59—21no—
74C2×C8×C59—41no—
75C2×C8×C59—41no—
76C2×C8×C59—42no—
77C22⋊Dic59—22no—
78(C2×D59)⋊C4—22no—
79Dic59⋊C4—22no—
80C4×Dic59—21no—
81C4×Dic59—21no—
82#11745664—42no—
83#11745664—42no—
84C2×C59⋊C8—24no—
85C2×C59⋊C8—22no—
86C2×C59⋊C8—41no—
87C2×C59⋊C8—42no—
88C2×C59⋊C8—42no—
89C2×C59⋊C8C22⋊C411yescharacteristic
90C2×C59⋊C8C22⋊C412yes—
91C2×C59⋊C8C22⋊C411yescharacteristic
92C8×D59—42no—
93C8×D59—42no—
94C23×C59—21no—
95C23×C59—21no—
96C23×C59—41no—
97C23×C59—41no—
98C23×C59C2×C4⋊C411yescharacteristic
99C23×C59C22⋊Q812yes—
100C22×D59—21no—
101C22×D59—21no—
102C22×D59—41no—
103C22×D59—41no—
104C22×D59#10812yescomplemented
105C2×C4×C59—22no—
106C2×C4×C59—21no—
107C2×C4×C59—22no—
108C2×C4×C59—21no—
109C2×C4×C59—41no—
110C2×C4×C59—42no—
111C2×C4×C59C2×C22⋊C411yescharacteristic
112C2×C4×C59#11711yescharacteristic
113C2×C4×C59#12112yes—
114C22⋊D59—28no—
115C2×Dic59—21no—
116C2×Dic59—21no—
117C2×Dic59—21no—
118C2×Dic59—21no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121C2×Dic59#10812yescomplemented
122C4×D59—42no—
123C4×D59—41no—
124C4×D59—41no—
125C8×C59—41no—
126C8×C59—41no—
127C8×C59—82no—
128C59⋊C8—21no—
129C59⋊C8—22no—
130C59⋊C8—82no—
131C59⋊C8#9912yes—
132#8611—591nomaximal, Sylow
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—22no—
136C22×C59—22no—
137C22×C59—22no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59#38011yescharacteristic
141C22×C59C2×#10811yescharacteristic
142C22×C59C2×#10811yescharacteristic
143C2×D59—22no—
144C2×D59—42no—
145C2×D59—41no—
146C2×D59—41no—
147C2×D59—42no—
148C4×C59—22no—
149C4×C59—21no—
150C4×C59C2×#9911yescharacteristic
151C4×C59#41011yescharacteristic
152Dic59—22no—
153Dic59—41no—
154Dic59—41no—
155C2×C8×D4—591no—
156#1810—592no—
157#1465—592no—
158#1293—591no—
159#1293D5911yescharacteristic, complemented
160C2×C59—21no—
161C2×C59—21no—
162C2×C59—42no—
163C2×C59#179911yescharacteristic
164C2×C59#181011yescharacteristic
165C2×C59#181311yescharacteristic
166D59—42no—
167C23×C8—591no—
168C23×C8—591no—
169C2×C4×D4—591no—
170C8×D4—1182no—
171C8×D4—1182no—
172C2×#108—1182no—
173C2×C4⋊C8—591no—
174#407—592no—
175#407C2×D5912yes—
176C2×C22⋊C8—591no—
177C2×C22⋊C8C2×D5911yescharacteristic
178C2×C4×C8—591no—
179#341—1182no—
180#324—1182no—
181#323—1182no—
182C59#861111yescharacteristic, complemented, Sylow
183C22×D4—591no—
184C23×C4—591no—
185C23×C4C22×D5911yescharacteristic
186C2×M16—22no—
187C2×M16—1182no—
188C22×C8—591no—
189C22×C8—592no—
190C22×C8—591no—
191C22×C8—592no—
192C22×C8—1181no—
193C22×C8—1181no—
194C22×C8—2362no—
195C22×C8—2362no—
196C22×C8C4×D5911yescharacteristic
197C22×C8C4×D5911yescharacteristic
198C4×D4—1182no—
199C4×D4—1182no—
200C2×C4⋊C4—591no—
201C2×C22⋊C4—591no—
202C2×C22⋊C4—591no—
203C2×C42—591no—
204#108—2364no—
205C4⋊C8—1182no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—1182no—
210C22⋊C8—1182no—
211C22⋊C8—1182no—
212C4×C8—1181no—
213C4×C8—1181no—
214C24—591no—
215C24Q8×D5911yescharacteristic
216C2×D4—1182no—
217C2×D4—1182no—
218C2×D4—1182no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—591no—
222C22×C4—591no—
223C22×C4—591no—
224C22×C4—1181no—
225C22×C4—1181no—
226C22×C4—2361no—
227C22×C4—2361no—
228C22×C4C2×C4×D5911yescharacteristic
229C22×C4#1174568512yes—
230C22×C4D4×D5912yes—
231C22×C4D4×D5911yescharacteristic
232M16—42no—
233M16—42no—
234M16—2362no—
235M16—2362no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—591no—
242C2×C8—591no—
243C2×C8—592no—
244C2×C8—1184no—
245C2×C8—1182no—
246C2×C8—2362no—
247C2×C8—2362no—
248C2×C8—2361no—
249C2×C8—2362no—
250C2×C8—2362no—
251C4⋊C4—1182no—
252C22⋊C4—1182no—
253C22⋊C4—1182no—
254C42—1181no—
255C42—1181no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—592no—
261C23—1181no—
262C23—1181no—
263C23—2361no—
264C23—2361no—
265C23C4⋊C4×D5911yescharacteristic
266C23#1427370512yes—
267D4—1188no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—592no—
275C2×C4—1181no—
276C2×C4—1181no—
277C2×C4—1181no—
278C2×C4—1181no—
279C2×C4—2362no—
280C2×C4—2361no—
281C2×C4—2361no—
282C2×C4—2361no—
283C2×C4—2361no—
284C2×C4C22⋊C4×D5911yescharacteristic
285C2×C4#1427368411yescharacteristic
286C2×C4#1427370012yes—
287C8—41no—
288C8—41no—
289C8—82no—
290C8—592no—
291C8—1182no—
292C8—1181no—
293C8—4722no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—1182no—
302C22—2362no—
303C22—2362no—
304C22—2361no—
305C22—2361no—
306C22#1683726111yescharacteristic
307C22#108×D5911yescharacteristic
308C22#108×D5911yescharacteristic
309C4—22no—
310C4—21no—
311C4—1182no—
312C4—2361no—
313C4—2361no—
314C4#99×D5911yescharacteristic
315C4#1683737411yescharacteristic
316C2—21no—
317C2—21no—
318C2—42no—
319C2—2362no—
320C2#2142408311yescharacteristic
321C2#2142413311yescharacteristic
322C2#2142414511yescharacteristic
subgroupG/H
centerC22#16837261
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1293C2
socleC22×C59#380
prime radicalC22×C22⋊D59Q8
cosocleC23×C4×C59C23
hypercenter#1293D59
hypocenterC59#8611
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #21424132C2semidirect
4. #21421973C2semidirect
5. C59×#1293C2semidirect
104. C22×D59#108semidirect
121. C2×Dic59#108semidirect
159. #1293D59semidirect
182. C59#8611semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D59)⋊C82
#214241322
#214224322
#214219732
C59×#12932
#861159