Finite Group Databasebeta

← #28481199

Subgroups — #28481199

Order 15104.

976 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16837973C211yesmaximal, characteristic
2#21424024C211yesmaximal, characteristic
3#21422628C214yesmaximal, complemented
4C59×#1782C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic, complemented
6#16837973—28no—
7C2×#14273725—21no—
8C2×#14273714—21no—
9C2×#14273708—21no—
10C2×#14273708—21no—
11C2×#14273706C412yescomplemented
12C2×#14273706C2212yes—
13C2×#14273686C411yescharacteristic, complemented
14C2×#14273646—21no—
15C2×#14273644—21no—
16#16837366C2212yes—
17#16837138—28no—
18C22×#41×C59C2211yescharacteristic
19C2×#99×C59—21no—
20C2×#99×C59—21no—
21C59×#381C2212yes—
22C22×Dic118—21no—
23C22×Dic118—24no—
24C22×Dic118—42no—
25C22×Dic118—42no—
26C22×Dic118C2×C412yes—
27C23×Dic59—21no—
28C2×#11745664—22no—
29#14273725—28no—
30#14273714—28no—
31#14273708—28no—
32#14273708—28no—
33#14273706—24no—
34#14273706—24no—
35#14273706—24no—
36#14273706D418yes—
37#14273686—24no—
38C2×C4⋊Dic59C2×C412yes—
39C2×Dic59⋊C4—22no—
40C2×Dic59⋊C4—22no—
41C2×C22⋊Dic59—22no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—22no—
44C2×C4×Dic59—21no—
45#14273646—42no—
46#14273644—42no—
47C22⋊(C59⋊C8)—24no—
48C2×#41×C59—24no—
49C2×#41×C59—42no—
50C2×#41×C59—42no—
51C22×Q8×C59D411yescharacteristic
52C22×D4×C59D411yescharacteristic
53C22×D4×C59D411yescharacteristic
54C22×D4×C59D411yescharacteristic
55C23×C4×C59C2×C411yescharacteristic
56C23×C4×C59C2×C411yescharacteristic
57C23×C4×C59C2311yescharacteristic
58C2×C22⋊C4×C59—22no—
59#99×C59—42no—
60#99×C59—42no—
61#95×C59—24no—
62C24×C59C22⋊C411yescharacteristic
63C24×C59C22⋊C411yescharacteristic
64C24×C59C2×D411yescharacteristic
65#41×C59—416no—
66C2×Q8×C59—22no—
67C2×Q8×C59—22no—
68C2×Q8×C59—41no—
69C2×D4×C59—22no—
70C2×D4×C59—22no—
71C2×D4×C59—22no—
72C2×D4×C59—22no—
73C2×D4×C59—22no—
74C2×D4×C59—22no—
75C2×D4×C59—41no—
76C2×D4×C59—41no—
77C2×D4×C59—41no—
78C22×C4×C59—22no—
79C22×C4×C59—22no—
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59—42no—
87C22×C4×C59C22⋊C411yescharacteristic
88C22×C4×C59C22⋊C411yescharacteristic
89C22×C4×C59C22×C411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D414yes—
92C22×Dic59—24no—
93C22×Dic59—22no—
94C22×Dic59—22no—
95C22×Dic59—42no—
96C22×Dic59—42no—
97C2×Dic118—24no—
98C2×Dic118—22no—
99C2×Dic118—42no—
100C2×Dic118—416no—
101C2×Dic118—42no—
102C2×Dic118—42no—
103C2×Dic118—42no—
104C2×Dic118C22⋊C414yes—
105C22⋊C4×C59—42no—
106C22⋊C4×C59—42no—
107C22⋊Dic59—24no—
108C22⋊Dic59—28no—
109C22⋊Dic59—28no—
110C4⋊Dic59—22no—
111C4⋊Dic59C22⋊C414yes—
112Dic59⋊C4—28no—
113Dic59⋊C4—28no—
114C4×Dic59—24no—
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—
132D4×C59—44no—
133D4×C59—44no—
134D4×C59—44no—
135C2×C4×C59—28no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
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—44no—
146C2×C4×C59—44no—
147C2×C4×C59—82no—
148C2×C4×C59—84no—
149C2×C4×C59#9912yes—
150C2×C4×C59C2×C22⋊C412yes—
151C2×C4×C59#12012yes—
152C2×Dic59—24no—
153C2×Dic59—24no—
154C2×Dic59—48no—
155C2×Dic59—44no—
156C2×Dic59—42no—
157C2×Dic59—42no—
158Dic118—44no—
159Dic118—48no—
160Dic118—44no—
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—41no—
183C4×C59—162no—
184Dic59—44no—
185Dic59—44no—
186C2×#536—591no—
187#1785—591no—
188#1782D5911yescharacteristic
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×Q8—591no—
199#536—1188no—
200C2×C4⋊Q8—1181no—
201C2×#124—1181no—
202C2×#123—1181no—
203C2×#123—1181no—
204C2×C22⋊Q8—592no—
205C2×C22⋊Q8—592no—
206C2×#117—591no—
207C2×#101—1181no—
208C2×#100—1181no—
209C2×#99—21no—
210C2×#99—21no—
211#407—592no—
212#381C2×D5912yes—
213#328—1188no—
214C59#852011yescharacteristic, complemented, Sylow
215C2×#41—24no—
216C2×#41—42no—
217C2×#41—42no—
218C22×Q8—592no—
219C22×Q8—1184no—
220C22×Q8—1181no—
221C22×Q8—2362no—
222C22×Q8—2362no—
223C22×Q8D23611yescharacteristic
224C22×D4D23611yescharacteristic
225C22×D4C22⋊D5911yescharacteristic
226C22×D4C22⋊D5911yescharacteristic
227C23×C4—1181no—
228C23×C4C4×D5911yescharacteristic
229C23×C4C4×D5911yescharacteristic
230C23×C4C22×D5911yescharacteristic
231C2×M16—1182no—
232C4⋊Q8—1188no—
233#124—1188no—
234#123—1188no—
235#123—1188no—
236C22⋊Q8—598no—
237C22⋊Q8—1184no—
238C22⋊Q8—1184no—
239C22⋊Q8—1184no—
240#117—1184no—
241C2×C4⋊C4—592no—
242C2×C4⋊C4—1182no—
243C2×C4⋊C4—1182no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—1181no—
246C2×C22⋊C4—1182no—
247C2×C22⋊C4—1182no—
248C2×C42—1181no—
249#101—2362no—
250#100—2362no—
251#99—42no—
252#99—42no—
253C22⋊C8—1184no—
254#95—24no—
255C24(C2×D59)⋊C411yescharacteristic
256C24(C2×D59)⋊C411yescharacteristic
257C24C2×C22⋊D5911yescharacteristic
258#41—416no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—594no—
263C2×Q8—1184no—
264C2×Q8—1182no—
265C2×Q8—23616no—
266C2×Q8—2362no—
267C2×Q8—2362no—
268C2×Q8—2362no—
269C2×Q8—2362no—
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—22no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—1184no—
289C22×C4—1182no—
290C22×C4—1182no—
291C22×C4—2362no—
292C22×C4—2362no—
293C22×C4(C2×D59)⋊C411yescharacteristic
294C22×C4(C2×D59)⋊C411yescharacteristic
295C22×C4C2×C4×D5911yescharacteristic
296C22×C4C2×D23611yescharacteristic
297C22×C4D4×D5914yes—
298M16—2362no—
299M16—2362no—
300C2×C8—2362no—
301C4⋊C4—594no—
302C4⋊C4—1188no—
303C4⋊C4—1182no—
304C4⋊C4—1188no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—1184no—
308C22⋊C4—1188no—
309C22⋊C4—1188no—
310C42—1184no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23#1427364112yes—
322C23C2×(C2×D59)⋊C411yescharacteristic
323C23#1427369512yes—
324Q8—44no—
325Q8—2364no—
326Q8—2364no—
327Q8—2368no—
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—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—82no—
344C2×C4—84no—
345C2×C4—1184no—
346C2×C4—1184no—
347C2×C4—2364no—
348C2×C4—2368no—
349C2×C4—2362no—
350C2×C4—2362no—
351C2×C4#1427364112yes—
352C2×C4C22⋊C4×D5912yes—
353C2×C4#1427369712yes—
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#1683725711yescharacteristic
369C22C2×#1427364111yescharacteristic
370C22C2×#1427364111yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—41no—
375C4—162no—
376C4—2364no—
377C4—2364no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2#2142401011yescharacteristic
384C2#2142406912yes—
subgroupG/H
centerC22#16837257
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1782C2
socleC22×C59#379
prime radicalC22×#41×C59C22
cosocleC23×C4×C59C23
hypercenter#1782D59
hypocenterC59#8520
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. #21422628C2semidirect
5. C23×Dic118C4semidirect
11. C2×#14273706C4semidirect
13. C2×#14273686C4semidirect
214. C59#8520semidirect

Maximal subgroups

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

subgroupindex
C2×#168379732
#214240242
#214226282
C59×#17822
#852059