Finite Group Databasebeta

← #28481586

Subgroups — #28481586

Order 15104.

550 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 121 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#21424123C212yesmaximal, complemented
3#21424081C212yesmaximal
4#21421964C211yesmaximal, characteristic, complemented
5C59×#1289C211yesmaximal, characteristic, complemented
6C23×C59⋊C8C2211yescharacteristic
7C22×C8×D59C411yescharacteristic, complemented
8C2×C4×C22⋊D59C2211yescharacteristic
9(C22⋊D59)⋊C8—22no—
10(C22⋊D59)⋊C8—22no—
11C2×#14273660—22no—
12C2×Dic59⋊C8C411yescharacteristic, complemented
13C2×#14273641—22no—
14C2×(C2×D59)⋊C8C411yescharacteristic, complemented
15C2×C8×Dic59C411yescharacteristic, complemented
16#16837262C2212yes—
17#16837168—22no—
18#16837164—22no—
19C2×C22⋊C8×C59C2211yescharacteristic
20C59×#380C2212yes—
21C22×C22⋊D59D411yescharacteristic
22C22×C4×D59C2×C411yescharacteristic
23C22×C59⋊C8—42no—
24C22×C59⋊C8C2×C411yescharacteristic
25C22×C59⋊C8C2×C411yescharacteristic
26C22×C59⋊C8D412yes—
27C22×C59⋊C8D412yes—
28C2×C8×D59—21no—
29C2×C8×D59—21no—
30C2×C8×D59—42no—
31C4×C22⋊D59—22no—
32C4×C22⋊D59—22no—
33C2×C4⋊Dic59—22no—
34C2×Dic59⋊C4Q811yescharacteristic
35C2×C22⋊Dic59—22no—
36C2×C22⋊Dic59D411yescharacteristic
37C2×(C2×D59)⋊C4Q811yescharacteristic
38C2×C4×Dic59C2×C411yescharacteristic
39#14273660—44no—
40Dic59⋊C8—22no—
41#14273641—44no—
42(C2×D59)⋊C8—22no—
43C8×Dic59—21no—
44C8×Dic59—21no—
45C59⋊#95—22no—
46C23×C4×C59C2311yescharacteristic
47C22×C8×C59C2×C411yescharacteristic
48C22×C8×C59C2×C411yescharacteristic
49C2×C4⋊C4×C59—22no—
50C2×C22⋊C4×C59—22no—
51C22⋊C8×C59—22no—
52#95×C59—22no—
53C24×C59C2×D411yescharacteristic
54C23×D59C4⋊C411yescharacteristic
55C22×C4×C59—41no—
56C22×C4×C59—41no—
57C22×C4×C59—42no—
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D412yes—
60C22×C4×C59C2×Q811yescharacteristic
61C22×C4×C59#4112yes—
62C2×C22⋊D59—22no—
63C2×C22⋊D59—22no—
64C2×C22⋊D59D812yes—
65C2×C22⋊D59Q1612yes—
66C22×Dic59—42no—
67C22×Dic59C4⋊C411yescharacteristic
68C22×Dic59C4⋊C411yescharacteristic
69C22×Dic59C4⋊C411yescharacteristic
70C2×C4×D59—21no—
71C2×C4×D59—21no—
72C2×C4×D59—41no—
73C2×C4×D59—41no—
74C2×C8×C59—21no—
75C2×C8×C59—21no—
76C2×C8×C59—41no—
77C2×C8×C59—41no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59—42no—
80C22⋊C4×C59—42no—
81C22⋊C4×C59—42no—
82C22⋊Dic59—22no—
83C22⋊Dic59—42no—
84C22⋊Dic59—42no—
85(C2×D59)⋊C4—22no—
86C4⋊Dic59—42no—
87C4⋊Dic59—42no—
88Dic59⋊C4—22no—
89C4×Dic59—21no—
90C4×Dic59—21no—
91C2×C59⋊C8—24no—
92C2×C59⋊C8—22no—
93C2×C59⋊C8—41no—
94C2×C59⋊C8—42no—
95C2×C59⋊C8C22⋊C411yescharacteristic
96C2×C59⋊C8C22⋊C412yes—
97C2×C59⋊C8C22⋊C411yescharacteristic
98C8×D59—42no—
99C8×D59—42no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59C2×C4⋊C411yescharacteristic
103C23×C59#12112yes—
104C23×C59C2×D811yescharacteristic
105C23×C59C2×Q1611yescharacteristic
106C23×C59#13512yes—
107C22×D59—21no—
108C22×D59—21no—
109C22×D59—41no—
110C22×D59—41no—
111C22×D59#10712yescomplemented
112C2×C4×C59—22no—
113C2×C4×C59—21no—
114C2×C4×C59—22no—
115C2×C4×C59—21no—
116C2×C4×C59—41no—
117C2×C4×C59—42no—
118C2×C4×C59—82no—
119C2×C4×C59—82no—
120C2×C4×C59—82no—
121C2×C4×C59C2×C22⋊C411yescharacteristic
122C2×C4×C59#11711yescharacteristic
123C2×C4×C59C22⋊Q812yes—
124C22⋊D59—24no—
125C22⋊D59—24no—
126C2×Dic59—21no—
127C2×Dic59—21no—
128C2×Dic59—21no—
129C2×Dic59—21no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—82no—
133C2×Dic59—82no—
134C2×Dic59—82no—
135C2×Dic59#10712yescomplemented
136C4×D59—42no—
137C4×D59—41no—
138C4×D59—41no—
139C8×C59—41no—
140C8×C59—41no—
141C59⋊C8—21no—
142C59⋊C8—22no—
143C59⋊C8#9912yescomplemented
144#8609—591nomaximal, Sylow
145C22×C59—21no—
146C22×C59—22no—
147C22×C59—21no—
148C22×C59—22no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—42no—
152C22×C59#38011yescharacteristic
153C22×C59C2×#10711yescharacteristic
154C22×C59#42711yescharacteristic
155C22×C59#46612yes—
156C22×C59#46712yes—
157C2×D59—22no—
158C2×D59—42no—
159C2×D59—41no—
160C2×D59—41no—
161C2×D59—42no—
162C4×C59—21no—
163C4×C59—22no—
164C4×C59—162no—
165C4×C59C2×#9911yescharacteristic
166C4×C59#41011yescharacteristic
167Dic59—22no—
168Dic59—41no—
169Dic59—41no—
170Dic59—162no—
171C2×C8×D4—591no—
172#1808—592no—
173#1799—592no—
174#1289—591no—
175#1289D5911yescharacteristic, complemented
176C2×C59—22no—
177C2×C59—21no—
178C2×C59—21no—
179C2×C59—22no—
180C2×C59#179911yescharacteristic
181C2×C59#180811yescharacteristic
182C2×C59#181211yescharacteristic
183D59—42no—
184C23×C8—591no—
185C23×C8—591no—
186C2×C4×D4—591no—
187C8×D4—1182no—
188C8×D4—1182no—
189C2×#107—1182no—
190C2×C4⋊C8—591no—
191C2×#99—1182no—
192C2×C22⋊C8—591no—
193C2×C22⋊C8C2×D5911yescharacteristic
194C2×C4×C8—591no—
195#380—592no—
196#380C2×D5912yes—
197#342—1182no—
198#340—1182no—
199C59#860911yescharacteristic, complemented, Sylow
200C22×D4—591no—
201C23×C4—591no—
202C23×C4C22×D5911yescharacteristic
203C22×C8—591no—
204C22×C8—592no—
205C22×C8—591no—
206C22×C8—592no—
207C22×C8—1181no—
208C22×C8—1181no—
209C22×C8—2362no—
210C22×C8—2362no—
211C22×C8C4×D5911yescharacteristic
212C22×C8C4×D5911yescharacteristic
213C4×D4—1182no—
214C4×D4—1182no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—591no—
217C2×C4⋊C4—1182no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—591no—
220C2×C22⋊C4—591no—
221C2×C22⋊C4—1182no—
222C2×C42—591no—
223#107—2364no—
224C4⋊C8—1182no—
225#99—2364no—
226C22⋊C8—22no—
227C22⋊C8—1182no—
228C4×C8—1181no—
229C4×C8—1181no—
230#95—22no—
231#95—1182no—
232C24—591no—
233C24D4×D5911yescharacteristic
234C2×D4—592no—
235C2×D4—592no—
236C2×D4—1182no—
237C2×D4—1182no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—591no—
242C22×C4—591no—
243C22×C4—591no—
244C22×C4—1181no—
245C22×C4—1181no—
246C22×C4—2361no—
247C22×C4—2361no—
248C22×C4—2362no—
249C22×C4C2×C4×D5911yescharacteristic
250C22×C4#1174568512yes—
251C22×C4D4×D5912yes—
252C22×C4Q8×D5911yescharacteristic
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—591no—
258C2×C8—591no—
259C2×C8—592no—
260C2×C8—1184no—
261C2×C8—1182no—
262C2×C8—2362no—
263C2×C8—2362no—
264C2×C8—2361no—
265C2×C8—2362no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—1182no—
269C4⋊C4—2362no—
270C4⋊C4—2362no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—1182no—
274C22⋊C4—1182no—
275C22⋊C4—2362no—
276C22⋊C4—2362no—
277C42—1181no—
278C42—1181no—
279C23—41no—
280C23—41no—
281C23—592no—
282C23—1181no—
283C23—1181no—
284C23—2361no—
285C23—2361no—
286C23C4⋊C4×D5911yescharacteristic
287C23#1427370012yes—
288C23D8×D5911yescharacteristic
289C23Q16×D5911yescharacteristic
290C23#1427374812yes—
291D4—1184no—
292D4—1184no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—592no—
303C2×C4—1181no—
304C2×C4—1181no—
305C2×C4—1181no—
306C2×C4—1181no—
307C2×C4—4722no—
308C2×C4—4722no—
309C2×C4—4722no—
310C2×C4—2362no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4C22⋊C4×D5911yescharacteristic
316C2×C4#1427368411yescharacteristic
317C2×C4#1427370512yes—
318C8—41no—
319C8—41no—
320C8—592no—
321C8—1182no—
322C8—1181no—
323C22—22no—
324C22—21no—
325C22—22no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—1182no—
331C22—2362no—
332C22—2362no—
333C22—2361no—
334C22—2361no—
335C22#1683726111yescharacteristic
336C22#107×D5911yescharacteristic
337C22#1683743711yescharacteristic
338C22#1683763712yes—
339C22#1683764212yes—
340C4—21no—
341C4—22no—
342C4—162no—
343C4—1182no—
344C4—9442no—
345C4—2361no—
346C4—2361no—
347C4#99×D5911yescharacteristic
348C4#1683737411yescharacteristic
349C2—22no—
350C2—21no—
351C2—21no—
352C2—22no—
353C2—2362no—
354C2#2142408311yescharacteristic
355C2#2142412411yescharacteristic
356C2#2142414111yescharacteristic
subgroupG/H
centerC22#16837261
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1289C2
socleC22×C59#380
prime radicalC22×C22⋊D59D4
cosocleC23×C4×C59C23
hypercenter#1289D59
hypocenterC59#8609
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #21424123C2semidirect
4. #21421964C2semidirect
5. C59×#1289C2semidirect
7. C22×C8×D59C4semidirect
12. C2×Dic59⋊C8C4semidirect
14. C2×(C2×D59)⋊C8C4semidirect
15. C2×C8×Dic59C4semidirect
111. C22×D59#107semidirect
135. C2×Dic59#107semidirect
143. C59⋊C8#99semidirect
175. #1289D59semidirect
199. C59#8609semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D59)⋊C82
#214241232
#214240812
#214219642
C59×#12892
#860959