Finite Group Databasebeta

← #28486479

Subgroups — #28486479

Order 15104.

556 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 123 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#21433125C211yesmaximal, characteristic
2#21424186C212yesmaximal, complemented
3#21424084C211yesmaximal, characteristic
4#21421963C212yesmaximal, complemented
5C59×#2487C211yesmaximal, characteristic
6#16837474—24no—
7#16837459—21no—
8C2×#14273642—21no—
9#16837366—21no—
10C2×C22⋊(C59⋊C8)C2212yes—
11#16837262C2212yes—
12#16837169—21no—
13#16837164—24no—
14C2×C4×D4×C59C2211yescharacteristic
15C22×C4⋊C4×C59C2211yescharacteristic
16C59×#400C412yescomplemented
17C59×#399C412yescomplemented
18C59×#389—21no—
19C59×#386—21no—
20C59×#385—21no—
21C59×#380—21no—
22C59×#380C2211yescharacteristic
23C2×#11745664—22no—
24C22×C59⋊C8D414yes—
25C2×C4⋊Dic59—22no—
26C2×C22⋊Dic59—22no—
27C4⋊(C59⋊C8)—22no—
28#14273642—42no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32#14273638—22no—
33C59⋊#95—22no—
34C22×D4×C59D411yescharacteristic
35C23×C4×C59—21no—
36C23×C4×C59D411yescharacteristic
37C23×C4×C59C2311yescharacteristic
38C4×D4×C59—22no—
39C4×D4×C59—22no—
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59—22no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—22no—
45C2×C4⋊C4×C59C2×C412yes—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—22no—
49C2×C22⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52#95×C59—22no—
53#95×C59—22no—
54#95×C59C2×C412yes—
55C24×C59C22⋊C411yescharacteristic
56C24×C59C2×D411yescharacteristic
57C2×D4×C59—22no—
58C2×D4×C59—21no—
59C2×D4×C59—21no—
60C2×D4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—42no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59—82no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D412yes—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59#4112yes—
76C22×Dic59—42no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—24no—
79C4⋊C4×C59—42no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—82no—
83C22⋊C4×C59—22no—
84C22⋊C4×C59—22no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—82no—
88C42×C59—21no—
89C42×C59—21no—
90C22⋊Dic59—42no—
91C22⋊Dic59—42no—
92C4⋊Dic59—44no—
93#11745664—82no—
94C2×C59⋊C8—24no—
95C2×C59⋊C8—44no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104C23×C59#12112yes—
105C23×C59AGL(1,ℤ/8)11yescharacteristic
106C23×C59#13711yescharacteristic
107C23×C59#13712yes—
108D4×C59—42no—
109D4×C59—42no—
110C2×C4×C59—24no—
111C2×C4×C59—22no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—41no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—81no—
129C2×C4×C59—84no—
130C2×C4×C59C4×D412yes—
131C2×C4×C59#12011yescharacteristic
132C2×C4×C59#12311yescharacteristic
133C2×Dic59—82no—
134C2×Dic59—82no—
135C2×Dic59—82no—
136C59⋊C8—44no—
137#9473—591nomaximal, Sylow
138C22×C59—22no—
139C22×C59—21no—
140C22×C59—22no—
141C22×C59—21no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59—42no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—81no—
150C22×C59#38611yescharacteristic
151C22×C59#41711yescharacteristic
152C22×C59#41911yescharacteristic
153C22×C59#46812yes—
154C22×C59#47012yes—
155C4×C59—21no—
156C4×C59—22no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59—41no—
160C4×C59—82no—
161C4×C59—82no—
162C4×C59#45812yes—
163Dic59—162no—
164#2973—591no—
165#2487D5911yescharacteristic
166#1824—592no—
167#1799—591no—
168#1289—592no—
169C2×C59—22no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59—22no—
173C2×C59—81no—
174C2×C59#189011yescharacteristic
175C2×C59#189411yescharacteristic
176C2×C59#191011yescharacteristic
177C2×C4×D4C2×D5911yescharacteristic
178C22×C4⋊C4C2×D5911yescharacteristic
179#435—1184no—
180#432—1181no—
181C2×#99—1181no—
182#407—1181no—
183C2×C22⋊C8—592no—
184#400Dic5912yescomplemented
185#399Dic5912yescomplemented
186#389—21no—
187#386—21no—
188#385—21no—
189#380—21no—
190#380—592no—
191#380C2×D5911yescharacteristic
192#342—1181no—
193#340—1184no—
194C59#947311yescharacteristic, complemented, Sylow
195C22×D4C22⋊D5911yescharacteristic
196C23×C4—21no—
197C23×C4C22⋊D5911yescharacteristic
198C23×C4C22×D5911yescharacteristic
199C2×M16—1182no—
200C22×C8—594no—
201C4×D4—22no—
202C4×D4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—1182no—
209C2×C4⋊C4C2×Dic5912yes—
210C2×C4⋊C4C2×Dic5911yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1182no—
214C2×C22⋊C4C2×Dic5911yescharacteristic
215C2×C22⋊C4C22⋊D5911yescharacteristic
216C2×C42C22⋊D5911yescharacteristic
217C4⋊C8—1182no—
218#99—2362no—
219C22⋊C8—1182no—
220C22⋊C8—1182no—
221C22⋊C8—1182no—
222#97—1182no—
223#95—22no—
224#95—22no—
225#95—1182no—
226#95C2×Dic5912yes—
227C24C22⋊Dic5911yescharacteristic
228C24C2×C22⋊D5911yescharacteristic
229C2×D4—22no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—82no—
241C22×C4—2362no—
242C22×C4C22⋊Dic5911yescharacteristic
243C22×C4C22⋊Dic5911yescharacteristic
244C22×C4C22⋊Dic5911yescharacteristic
245C22×C4D4×D5912yes—
246C22×C4#1174568712yes—
247C22×C4C22×Dic5911yescharacteristic
248C22×C4C2×C22⋊D5911yescharacteristic
249M16—4722no—
250C2×C8—1184no—
251C2×C8—2364no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—82no—
258C4⋊C4—2364no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—82no—
264C22⋊C4—2362no—
265C22⋊C4—2362no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23C2×C22⋊Dic5911yescharacteristic
276C23#1427369912yes—
277C23#1427375211yescharacteristic
278C23#1427375711yescharacteristic
279C23#1427375812yes—
280D4—42no—
281D4—42no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—84no—
302C2×C4—4722no—
303C2×C4—4722no—
304C2×C4—4722no—
305C2×C4D4×Dic5912yes—
306C2×C4#1427369511yescharacteristic
307C2×C4#1427370811yescharacteristic
308C8—2364no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1683728211yescharacteristic
322C22#1683740311yescharacteristic
323C22#1683741111yescharacteristic
324C22#1683765012yes—
325C22#1683765812yes—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4—9442no—
334C4#1683759112yes—
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#2142451011yescharacteristic
341C2#2142453011yescharacteristic
342C2#2142462411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2487C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC23×C4×C59C23
hypercenter#2487D59
hypocenterC59#9473
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #21424186C2semidirect
4. #21421963C2semidirect
16. C59×#400C4semidirect
17. C59×#399C4semidirect
184. #400Dic59semidirect
185. #399Dic59semidirect
194. C59#9473semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214241862
#214240842
#214219632
C59×#24872
#947359