Finite Group Databasebeta

← #28588424

Subgroups — #28588424

Order 15104.

1510 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273683C211yesmaximal, characteristic
2#21426474C212yesmaximal
3#21424172C214yesmaximal, complemented
4#21424170C211yesmaximal, characteristic
5#21424168C212yesmaximal
6#21423827C214yesmaximal, complemented
7C59×#1819C211yesmaximal, characteristic
8C24×Dic59C2211yescharacteristic
9C2×#14273683—24no—
10C2×#14273683—24no—
11C2×#14273683C412yescomplemented
12C2×#14273683C412yescomplemented
13C2×#14273683C2214yes—
14C2×#14273683C2212yes—
15C2×#14273683C2212yes—
16C22×Dic59⋊C4C412yescomplemented
17C22×C22⋊Dic59C2211yescharacteristic
18C22×C4×Dic59C412yescomplemented
19#16837263C2214yes—
20#16837260—24no—
21#16837258C2212yes—
22C22⋊C4×Dic59—24no—
23#16837169C2214yes—
24#16837168C2218yes—
25C22×C22⋊C4×C59C2211yescharacteristic
26C59×#379C2212yes—
27C59×#342C2214yes—
28C23×Dic59—28no—
29C23×Dic59C2×C411yescharacteristic
30C23×Dic59C2×C411yescharacteristic
31C23×Dic59C2×C411yescharacteristic
32C23×Dic59C2×C411yescharacteristic
33C23×Dic59C2×C412yes—
34C23×Dic59D414yes—
35C23×Dic59D414yes—
36C23×Dic59C2311yescharacteristic
37C23×Dic59C2311yescharacteristic
38#14273683—28no—
39#14273683—24no—
40#14273683—24no—
41#14273683—28no—
42#14273683—24no—
43#14273683—24no—
44#14273683—216no—
45C2×Dic59⋊C4—24no—
46C2×Dic59⋊C4—24no—
47C2×Dic59⋊C4—24no—
48C2×Dic59⋊C4C2×C414yes—
49C2×Dic59⋊C4C2×C414yes—
50C2×C22⋊Dic59—24no—
51C2×C22⋊Dic59—24no—
52C2×C22⋊Dic59C2×C412yes—
53C2×C22⋊Dic59C2314yes—
54C2×C22⋊Dic59C2312yes—
55C2×C4×Dic59—22no—
56C2×C4×Dic59—22no—
57C2×C4×Dic59—22no—
58C2×C4×Dic59—22no—
59C2×C4×Dic59—24no—
60C2×C4×Dic59C2×C414yes—
61C2×C4×Dic59C2×C414yes—
62C59⋊#95—28no—
63C25×C59C2311yescharacteristic
64C23×C4×C59C2×C412yes—
65C2×C22⋊C4×C59—24no—
66C2×C22⋊C4×C59—24no—
67C2×C22⋊C4×C59C2×C412yes—
68C2×C22⋊C4×C59C2314yes—
69C2×C22⋊C4×C59C2312yes—
70C24×C59—24no—
71C24×C59—44no—
72C24×C59C22×C411yescharacteristic
73C24×C59C22×C411yescharacteristic
74C24×C59C2×D414yes—
75C24×C59C2411yescharacteristic
76C22×C4×C59—22no—
77C22×C4×C59—22no—
78C22×C4×C59—24no—
79C22×C4×C59—44no—
80C22×C4×C59C22×C414yes—
81C22×Dic59—24no—
82C22×Dic59—24no—
83C22×Dic59—24no—
84C22×Dic59—24no—
85C22×Dic59—24no—
86C22×Dic59—216no—
87C22×Dic59—216no—
88C22×Dic59—22no—
89C22×Dic59—22no—
90C22×Dic59—24no—
91C22×Dic59—44no—
92C22×Dic59C22⋊C412yes—
93C22×Dic59C22⋊C412yes—
94C22×Dic59C22⋊C414yes—
95C22×Dic59C22⋊C414yes—
96C22×Dic59C22⋊C412yes—
97C22×Dic59C22⋊C412yes—
98C22×Dic59C22×C411yescharacteristic
99C22×Dic59C22×C411yescharacteristic
100C22×Dic59C22×C414yes—
101C22×Dic59C22×C411yescharacteristic
102C22×Dic59C22×C411yescharacteristic
103C22×Dic59C2×D414yes—
104C22×Dic59C2×D412yes—
105C22×Dic59C2×D412yes—
106C22×Dic59#4118yes—
107C22⋊C4×C59—24no—
108C22⋊C4×C59—24no—
109C22⋊C4×C59—28no—
110C22⋊C4×C59—48no—
111C22⋊Dic59—24no—
112C22⋊Dic59—24no—
113C22⋊Dic59—28no—
114C22⋊Dic59—48no—
115Dic59⋊C4—28no—
116Dic59⋊C4—28no—
117Dic59⋊C4—48no—
118C4×Dic59—24no—
119C4×Dic59—24no—
120C4×Dic59—24no—
121C4×Dic59—24no—
122C4×Dic59—48no—
123C23×C59—22no—
124C23×C59—24no—
125C23×C59—22no—
126C23×C59—28no—
127C23×C59—28no—
128C23×C59—42no—
129C23×C59—44no—
130C23×C59—42no—
131C23×C59—416no—
132C23×C59C2×C22⋊C412yes—
133C23×C59C2×C22⋊C412yes—
134C23×C59C23×C411yescharacteristic
135C23×C59C22×D412yes—
136C23×C59C2×#4114yes—
137C2×C4×C59—24no—
138C2×C4×C59—44no—
139C2×C4×C59—44no—
140C2×C4×C59—44no—
141C2×C4×C59—48no—
142C2×C4×C59—44no—
143C2×Dic59—22no—
144C2×Dic59—22no—
145C2×Dic59—28no—
146C2×Dic59—24no—
147C2×Dic59—24no—
148C2×Dic59—28no—
149C2×Dic59—28no—
150C2×Dic59—24no—
151C2×Dic59—24no—
152C2×Dic59—24no—
153C2×Dic59—44no—
154C2×Dic59—44no—
155C2×Dic59—44no—
156C2×Dic59—48no—
157C2×Dic59—44no—
158C2×Dic59C2×C22⋊C412yes—
159C2×Dic59C2×C22⋊C412yes—
160C2×Dic59C2×C22⋊C411yescharacteristic
161C2×Dic59C2×C22⋊C411yescharacteristic
162C2×Dic59C2×C22⋊C412yes—
163C2×Dic59#11714yes—
164C2×Dic59#11714yes—
165C2×Dic59#12318yes—
166C2×Dic59#12318yes—
167#20688—591nomaximal, Sylow
168C22×C59—28no—
169C22×C59—22no—
170C22×C59—22no—
171C22×C59—24no—
172C22×C59—22no—
173C22×C59—24no—
174C22×C59—22no—
175C22×C59—44no—
176C22×C59—416no—
177C22×C59—44no—
178C22×C59#41014yes—
179C22×C59C22×C22⋊C411yescharacteristic
180C22×C59C2×#11712yes—
181C22×C59C2×#12314yes—
182C4×C59—44no—
183C4×C59—84no—
184Dic59—22no—
185Dic59—22no—
186Dic59—44no—
187Dic59—84no—
188Dic59#38112yes—
189Dic59#38112yes—
190Dic59#38114yes—
191C22×#117—591no—
192#2280—592no—
193#1820—594no—
194#1819—592no—
195#1819—591no—
196#1819D5911yescharacteristic
197#1740—594no—
198C2×C59—24no—
199C2×C59—22no—
200C2×C59—24no—
201C2×C59—42no—
202C2×C59C2×#38111yescharacteristic
203C2×C59C2×#41012yes—
204C24×C4—591no—
205C2×#117—592no—
206C2×#117—592no—
207C2×#117—592no—
208C2×#117—592no—
209C2×#117—594no—
210C2×#117—1184no—
211C2×#117—1184no—
212C22×C4⋊C4—592no—
213C22×C22⋊C4—591no—
214C22×C22⋊C4C2×D5911yescharacteristic
215C22×C42—592no—
216#381—594no—
217#380—1184no—
218#379—592no—
219#379C2×D5912yes—
220C4×C22⋊C4—1184no—
221#342—598no—
222#342—594no—
223#342C2×D5914yes—
224C59#2068811yescharacteristic, complemented, Sylow
225C25C22×D5911yescharacteristic
226C23×C4—591no—
227C23×C4—591no—
228C23×C4—591no—
229C23×C4—591no—
230C23×C4—591no—
231C23×C4—594no—
232C23×C4—594no—
233C23×C4—591no—
234C23×C4—592no—
235C23×C4—1188no—
236C23×C4C4×D5912yes—
237#117—1184no—
238#117—1184no—
239#117—1188no—
240#117—1184no—
241#117—1184no—
242#117—1188no—
243#117—11816no—
244C2×C4⋊C4—594no—
245C2×C4⋊C4—594no—
246C2×C4⋊C4—1184no—
247C2×C4⋊C4—1184no—
248C2×C4⋊C4—1184no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—592no—
252C2×C22⋊C4—592no—
253C2×C22⋊C4—594no—
254C2×C22⋊C4—1184no—
255C2×C22⋊C4—1184no—
256C2×C22⋊C4C4×D5912yes—
257C2×C22⋊C4C22×D5914yes—
258C2×C22⋊C4C22×D5912yes—
259C2×C42—594no—
260C2×C42—594no—
261C2×C42—1182no—
262C2×C42—1182no—
263C2×C42—1182no—
264C2×C42—1182no—
265C2×C42—1184no—
266#95—1188no—
267C24—24no—
268C24—44no—
269C24C2×C4×D5911yescharacteristic
270C24C2×C4×D5911yescharacteristic
271C24D4×D5914yes—
272C24C23×D5911yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—44no—
277C22×C4—591no—
278C22×C4—591no—
279C22×C4—592no—
280C22×C4—592no—
281C22×C4—598no—
282C22×C4—592no—
283C22×C4—592no—
284C22×C4—594no—
285C22×C4—592no—
286C22×C4—592no—
287C22×C4—594no—
288C22×C4—591no—
289C22×C4—591no—
290C22×C4—594no—
291C22×C4—594no—
292C22×C4—11816no—
293C22×C4—11816no—
294C22×C4—1184no—
295C22×C4—1184no—
296C22×C4—1184no—
297C22×C4—1184no—
298C22×C4—1184no—
299C22×C4—1182no—
300C22×C4—1182no—
301C22×C4—1184no—
302C22×C4—2364no—
303C22×C4C2×C4×D5914yes—
304C4⋊C4—1188no—
305C4⋊C4—1188no—
306C4⋊C4—2368no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—28no—
310C22⋊C4—48no—
311C22⋊C4—1184no—
312C22⋊C4—1184no—
313C22⋊C4—1188no—
314C22⋊C4—2368no—
315C42—1184no—
316C42—1184no—
317C42—1184no—
318C42—1184no—
319C42—2368no—
320C23—22no—
321C23—24no—
322C23—28no—
323C23—22no—
324C23—28no—
325C23—42no—
326C23—44no—
327C23—42no—
328C23—416no—
329C23C22⋊C4×D5912yes—
330C23C22⋊C4×D5912yes—
331C23C22×C4×D5911yescharacteristic
332C23C2×D4×D5912yes—
333C23#41×D5914yes—
334C2×C4—24no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—48no—
339C2×C4—44no—
340C2×C4—598no—
341C2×C4—598no—
342C2×C4—591no—
343C2×C4—591no—
344C2×C4—592no—
345C2×C4—594no—
346C2×C4—594no—
347C2×C4—592no—
348C2×C4—592no—
349C2×C4—1184no—
350C2×C4—1184no—
351C2×C4—1188no—
352C2×C4—1184no—
353C2×C4—1188no—
354C2×C4—1184no—
355C2×C4—1188no—
356C2×C4—1182no—
357C2×C4—1182no—
358C2×C4—1184no—
359C2×C4—2368no—
360C2×C4—2364no—
361C2×C4—2364no—
362C2×C4—2364no—
363C2×C4—2364no—
364C22—28no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—24no—
369C22—22no—
370C22—24no—
371C22—44no—
372C22—416no—
373C22—44no—
374C22#1683737414yes—
375C22C2×C22⋊C4×D5911yescharacteristic
376C22#117×D5912yes—
377C22#123×D5914yes—
378C4—44no—
379C4—84no—
380C4—592no—
381C4—594no—
382C4—592no—
383C4—1182no—
384C4—1182no—
385C4—4724no—
386C4—2364no—
387C2—24no—
388C2—22no—
389C2—24no—
390C2—42no—
391C2#381×D5911yescharacteristic
392C2C2×#1683737412yes—
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC24C23×D59
FittingC59×#1819C2
socleC22×C59C22×C22⋊C4
prime radicalC25×C59C23
cosocleC24×C59C24
hypercenter#1819D59
hypocenterC59#20688
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #21424172C2semidirect
6. #21423827C2semidirect
11. C2×#14273683C4semidirect
12. C2×#14273683C4semidirect
16. C22×Dic59⋊C4C4semidirect
18. C22×C4×Dic59C4semidirect
224. C59#20688semidirect

Maximal subgroups

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

subgroupindex
C22×#142736832
#214264742
#214241722
#214241702
#214241682
#214238272
C59×#18192
#2068859