Finite Group Databasebeta

← #28481200

Subgroups — #28481200

Order 15104.

976 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 139 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#21424024C211yesmaximal, characteristic
2#21424014C211yesmaximal, characteristic
3#21422629C214yesmaximal, complemented
4C2×C59×#536C211yesmaximal, characteristic
5C2×#14273646—21no—
6C2×#14273644—21no—
7C2×#14273642—21no—
8C2×#14273642—21no—
9#16837366C2212yes—
10#16837265C2212yes—
11#16837139—28no—
12C22×#41×C59C2211yescharacteristic
13C23×Q8×C59C411yescharacteristic, complemented
14C59×#536—28no—
15C2×C4⋊Q8×C59—21no—
16C2×#124×C59—21no—
17C2×#123×C59—21no—
18C2×#123×C59—21no—
19C2×C22⋊Q8×C59C412yescomplemented
20C2×C22⋊Q8×C59C2212yes—
21C2×#117×C59C411yescharacteristic, complemented
22C2×#11745664—22no—
23C2×C22⋊Dic59—22no—
24#14273646—42no—
25#14273644—42no—
26#14273642—42no—
27#14273642—42no—
28C22⋊(C59⋊C8)—24no—
29C59⋊#95—24no—
30C2×#41×C59—24no—
31C2×#41×C59—42no—
32C2×#41×C59—42no—
33C22×Q8×C59—24no—
34C22×Q8×C59—21no—
35C22×Q8×C59—42no—
36C22×Q8×C59—42no—
37C22×Q8×C59C2×C412yes—
38C22×Q8×C59D411yescharacteristic
39C22×D4×C59D411yescharacteristic
40C22×D4×C59D411yescharacteristic
41C22×D4×C59D411yescharacteristic
42C23×C4×C59—21no—
43C23×C4×C59C2×C411yescharacteristic
44C23×C4×C59C2×C411yescharacteristic
45C23×C4×C59C2311yescharacteristic
46C4⋊Q8×C59—28no—
47#124×C59—28no—
48#123×C59—28no—
49#123×C59—28no—
50C22⋊Q8×C59—24no—
51C22⋊Q8×C59—24no—
52C22⋊Q8×C59—24no—
53C22⋊Q8×C59D418yes—
54#117×C59—24no—
55C2×C4⋊C4×C59—22no—
56C2×C4⋊C4×C59—22no—
57C2×C4⋊C4×C59C2×C412yes—
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—22no—
60C2×C22⋊C4×C59—22no—
61C2×C42×C59—21no—
62C24×C59C22⋊C411yescharacteristic
63C24×C59C22⋊C411yescharacteristic
64C24×C59C2×D411yescharacteristic
65#41×C59—416no—
66C2×Q8×C59—24no—
67C2×Q8×C59—22no—
68C2×Q8×C59—22no—
69C2×Q8×C59—22no—
70C2×Q8×C59—41no—
71C2×Q8×C59—416no—
72C2×Q8×C59—42no—
73C2×Q8×C59—42no—
74C2×Q8×C59—42no—
75C2×Q8×C59—42no—
76C2×Q8×C59C22⋊C414yes—
77C2×D4×C59—22no—
78C2×D4×C59—22no—
79C2×D4×C59—22no—
80C2×D4×C59—22no—
81C2×D4×C59—22no—
82C2×D4×C59—22no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C22×C4×C59—22no—
87C22×C4×C59—22no—
88C22×C4×C59—22no—
89C22×C4×C59—24no—
90C22×C4×C59—22no—
91C22×C4×C59—22no—
92C22×C4×C59—22no—
93C22×C4×C59—41no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59—42no—
98C22×C4×C59—42no—
99C22×C4×C59C22⋊C411yescharacteristic
100C22×C4×C59C22⋊C411yescharacteristic
101C22×C4×C59C22×C411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59C2×D414yes—
104C22×Dic59—42no—
105C4⋊C4×C59—28no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—28no—
108C4⋊C4×C59C22⋊C414yes—
109C22⋊C4×C59—24no—
110C22⋊C4×C59—28no—
111C22⋊C4×C59—28no—
112C42×C59—24no—
113C22⋊Dic59—42no—
114C22⋊Dic59—42no—
115#11745664—42no—
116#11745664—42no—
117C2×C59⋊C8—42no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—42no—
126C23×C59—42no—
127C23×C59—42no—
128C23×C59#9912yes—
129C23×C59C2×C22⋊C411yescharacteristic
130C23×C59#12012yes—
131Q8×C59—44no—
132Q8×C59—44no—
133Q8×C59—44no—
134Q8×C59—48no—
135D4×C59—44no—
136D4×C59—44no—
137D4×C59—44no—
138C2×C4×C59—28no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—24no—
146C2×C4×C59—24no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—44no—
151C2×C4×C59—44no—
152C2×C4×C59—44no—
153C2×C4×C59—48no—
154C2×C4×C59—42no—
155C2×C4×C59—42no—
156C2×C4×C59#9912yes—
157C2×C4×C59C2×C22⋊C412yes—
158C2×C4×C59#12012yes—
159C2×Dic59—82no—
160C2×Dic59—84no—
161C59⋊C8—82no—
162#8520—591nomaximal, Sylow
163C22×C59—22no—
164C22×C59—24no—
165C22×C59—22no—
166C22×C59—22no—
167C22×C59—42no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—41no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—81no—
176C22×C59#37911yescharacteristic
177C22×C59C2×#9911yescharacteristic
178C22×C59C2×#9911yescharacteristic
179C4×C59—22no—
180C4×C59—22no—
181C4×C59—22no—
182C4×C59—44no—
183C4×C59—41no—
184C4×C59—44no—
185Dic59—162no—
186C2×#536D5911yescharacteristic
187#1785—591no—
188#1782—591no—
189#1511—594no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—42no—
193C2×C59—42no—
194C2×C59—81no—
195C2×C59#178111yescharacteristic
196C2×C59#179612yes—
197C22×#41C2×D5911yescharacteristic
198C23×Q8Dic5911yescharacteristic, complemented
199#536—28no—
200C2×C4⋊Q8—21no—
201C2×#124—21no—
202C2×#123—21no—
203C2×#123—21no—
204C2×C22⋊Q8Dic5912yescomplemented
205C2×C22⋊Q8C2×D5912yes—
206C2×#117Dic5911yescharacteristic, complemented
207C2×#101—1181no—
208C2×#100—1181no—
209C2×#99—1181no—
210C2×#99—1181no—
211#407—592no—
212#381—592no—
213#328—1188no—
214C59#852011yescharacteristic, complemented, Sylow
215C2×#41—24no—
216C2×#41—42no—
217C2×#41—42no—
218C22×Q8—24no—
219C22×Q8—21no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C2×Dic5912yes—
223C22×Q8C22⋊D5911yescharacteristic
224C22×D4C22⋊D5911yescharacteristic
225C22×D4C22⋊D5911yescharacteristic
226C22×D4C22⋊D5911yescharacteristic
227C23×C4—21no—
228C23×C4C2×Dic5911yescharacteristic
229C23×C4C2×Dic5911yescharacteristic
230C23×C4C22×D5911yescharacteristic
231C2×M16—1182no—
232C4⋊Q8—28no—
233#124—28no—
234#123—28no—
235#123—28no—
236C22⋊Q8—24no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8C22⋊D5918yes—
240#117—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C2×Dic5912yes—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—1182no—
248C2×C42—21no—
249#101—2362no—
250#100—2362no—
251#99—2362no—
252#99—2362no—
253C22⋊C8—1184no—
254#95—1184no—
255C24C22⋊Dic5911yescharacteristic
256C24C22⋊Dic5911yescharacteristic
257C24C2×C22⋊D5911yescharacteristic
258#41—416no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8C22⋊Dic5914yes—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—2362no—
293C22×C4C22⋊Dic5911yescharacteristic
294C22×C4C22⋊Dic5911yescharacteristic
295C22×C4C22×Dic5911yescharacteristic
296C22×C4C2×C22⋊D5911yescharacteristic
297C22×C4C2×C22⋊D5914yes—
298M16—2362no—
299M16—2362no—
300C2×C8—2362no—
301C4⋊C4—28no—
302C4⋊C4—22no—
303C4⋊C4—28no—
304C4⋊C4C22⋊Dic5914yes—
305C22⋊C4—24no—
306C22⋊C4—28no—
307C22⋊C4—28no—
308C22⋊C4—2362no—
309C22⋊C4—2362no—
310C42—24no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23#1427364212yes—
322C23C2×C22⋊Dic5911yescharacteristic
323C23#1427369612yes—
324Q8—44no—
325Q8—44no—
326Q8—44no—
327Q8—48no—
328D4—44no—
329D4—44no—
330D4—44no—
331C2×C4—28no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—4722no—
350C2×C4—4724no—
351C2×C4#1427364212yes—
352C2×C4C2×C22⋊Dic5912yes—
353C2×C4#1427369612yes—
354C8—4722no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#1683725811yescharacteristic
369C22C2×#1427364211yescharacteristic
370C22C2×#1427364211yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—41no—
375C4—44no—
376C4—44no—
377C4—9442no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2#2142401111yescharacteristic
384C2#2142407112yes—
subgroupG/H
centerC22#16837258
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#536C2
socleC22×C59#379
prime radicalC22×#41×C59C22
cosocleC23×C4×C59C23
hypercenterC2×#536D59
hypocenterC59#8520
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
3. #21422629C2semidirect
13. C23×Q8×C59C4semidirect
19. C2×C22⋊Q8×C59C4semidirect
21. C2×#117×C59C4semidirect
198. C23×Q8Dic59semidirect
204. C2×C22⋊Q8Dic59semidirect
206. C2×#117Dic59semidirect
214. C59#8520semidirect

Maximal subgroups

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

subgroupindex
#214240242
#214240142
#214226292
C2×C59×#5362
#852059