Finite Group Databasebeta

← #28598022

Subgroups — #28598022

Order 15104.

1374 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837952C211yesmaximal, characteristic
2#21426494C211yesmaximal, characteristic, complemented
3#21425022C212yesmaximal, complemented
4#21425018C212yesmaximal, complemented
5#21424907C214yesmaximal, complemented
6#21424037C214yesmaximal, complemented
7C2×C59×#431C211yesmaximal, characteristic, complemented
8C22×#11745685C2212yes—
9#16837952—24no—
10#16837952C418yescomplemented
11C2×#14273727C2211yescharacteristic
12C2×#14273724C2211yescharacteristic
13C2×#14273715C2212yes—
14C2×#14273701—21no—
15C2×#14273701—21no—
16C2×C4×D236—21no—
17C2×C4×D236—21no—
18C2×#14273652C2218yes—
19C2×#14273649C2218yes—
20#16837270—21no—
21#16837268C2212yes—
22#16837265—21no—
23#16837262C2212yes—
24C4×C4⋊Dic59C2212yes—
25C4×C22⋊Dic59—21no—
26#16837248—21no—
27C22×C42×C59C2211yescharacteristic
28C59×#431—24no—
29C2×C4⋊C8×C59C2212yes—
30C2×C22⋊C8×C59C2212yes—
31C2×C4×C8×C59C2212yes—
32C2×#11745685—24no—
33C2×#11745685—48no—
34C2×#11745685C2×C418yes—
35C22×Dic118C2312yes—
36C22×D236—21no—
37C22×D236—21no—
38C22×D236C2312yes—
39C22×C22⋊D59—22no—
40C22×C4×D59—22no—
41#14273727—22no—
42#14273727C2×C414yes—
43#14273724—22no—
44#14273724C2×C414yes—
45#14273715—24no—
46#14273715C2×C418yes—
47#14273701—28no—
48#14273701—28no—
49C4×D236—28no—
50C4×D236—28no—
51C2×C4⋊Dic59—21no—
52C2×C4⋊Dic59—21no—
53C2×C4⋊Dic59C2314yes—
54C2×C22⋊Dic59—22no—
55C2×(C2×D59)⋊C4—22no—
56C2×(C2×D59)⋊C4—22no—
57C2×C4×Dic59—22no—
58#14273652—48no—
59#14273649—48no—
60C59⋊#95—22no—
61C59⋊#95—22no—
62C59⋊#95—22no—
63C59⋊#95—22no—
64C23×C4×C59D411yescharacteristic
65C23×C4×C59D411yescharacteristic
66C23×C4×C59C2311yescharacteristic
67C22×C8×C59C2314yes—
68C2×C42×C59C2×C414yes—
69C2×C42×C59D411yescharacteristic
70C2×C42×C59D411yescharacteristic
71C2×C42×C59D414yes—
72C2×C42×C59C2311yescharacteristic
73C2×C42×C59C2311yescharacteristic
74C4⋊C8×C59—24no—
75C22⋊C8×C59—24no—
76C4×C8×C59—24no—
77C24×C59C2×D411yescharacteristic
78C23×D59—22no—
79C22×C4×C59—24no—
80C22×C4×C59—24no—
81C22×C4×C59—42no—
82C22×C4×C59C22⋊C414yes—
83C22×C4×C59C22⋊C414yes—
84C22×C4×C59C22×C412yes—
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D412yes—
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2411yescharacteristic
92C2×C22⋊D59—28no—
93C2×C22⋊D59—48no—
94C22×Dic59—22no—
95C22×Dic59—24no—
96#11745685—416no—
97#11745685—416no—
98C2×D236—28no—
99C2×D236—24no—
100C2×D236—24no—
101C2×D236—22no—
102C2×D236—44no—
103C2×D236—44no—
104C2×D236C22×C414yes—
105C2×C4×D59—28no—
106C2×C4×D59—48no—
107C2×Dic118—28no—
108C2×Dic118—22no—
109C2×Dic118C22×C414yes—
110C2×C8×C59—24no—
111C2×C8×C59—28no—
112C42×C59C22⋊C414yes—
113C42×C59C22⋊C414yes—
114C42×C59C22×C412yes—
115C42×C59C22×C412yes—
116C42×C59C2×D412yes—
117C42×C59C2×D412yes—
118C22⋊Dic59—44no—
119(C2×D59)⋊C4—28no—
120(C2×D59)⋊C4—28no—
121C4⋊Dic59—24no—
122C4⋊Dic59—24no—
123C4⋊Dic59—24no—
124C4⋊Dic59—24no—
125C4×Dic59—44no—
126C23×C59—22no—
127C23×C59—22no—
128C23×C59—22no—
129C23×C59C2×C22⋊C412yes—
130C23×C59C22×D411yescharacteristic
131C22×D59—24no—
132C22×D59—44no—
133C22×D59—44no—
134C22×D59—44no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—24no—
140C2×C4×C59—24no—
141C2×C4×C59—28no—
142C2×C4×C59—44no—
143C2×C4×C59C2×C22⋊C412yes—
144C2×C4×C59C2×C22⋊C412yes—
145C2×C4×C59C2×C22⋊C412yes—
146C2×C4×C59C2×C22⋊C412yes—
147C2×C4×C59C2×C22⋊C412yes—
148C2×C4×C59#13514yes—
149C2×C4×C59C23×C411yescharacteristic
150C2×C4×C59C22×D411yescharacteristic
151C2×C4×C59H(5,𝔽2)14yes—
152C22⋊D59—416no—
153C2×Dic59—24no—
154C2×Dic59—44no—
155C2×Dic59—44no—
156C2×Dic59—44no—
157C2×Dic59—44no—
158D236—28no—
159D236—48no—
160D236—44no—
161D236—48no—
162C4×D59—416no—
163Dic118—28no—
164Dic118—44no—
165C8×C59—44no—
166#21807—591nomaximal, Sylow
167C22×C59—22no—
168C22×C59—22no—
169C22×C59—22no—
170C22×C59—44no—
171C22×C59C22×C22⋊C411yescharacteristic
172C22×C59#53512yes—
173C22×C59C2×#13512yes—
174C22×C59#57512yes—
175C2×D59—44no—
176C2×D59—44no—
177C2×D59—44no—
178C2×D59—88no—
179C4×C59—22no—
180C4×C59—22no—
181C4×C59—22no—
182C4×C59—22no—
183C4×C59#41614yes—
184C4×C59#51812yes—
185C4×C59#53512yes—
186Dic59—44no—
187Dic59—84no—
188C2×#532—591no—
189C2×#431D5911yescharacteristic, complemented
190#2283—591no—
191#1982—592no—
192#1981—592no—
193#1958—594no—
194#1788—594no—
195C2×C59—42no—
196C2×C59#230911yescharacteristic
197C2×C59C2×#41611yescharacteristic
198C2×C59#289611yescharacteristic
199C2×C59#321214yes—
200D59—84no—
201C22×#41—592no—
202#532—598no—
203#532—1184no—
204C2×C4⋊Q8—591no—
205C2×#127—591no—
206C2×#124—592no—
207C2×#121—1181no—
208C2×#121—1181no—
209C2×C4×D4—1181no—
210C2×C4×D4—1181no—
211C22×C42C2×D5911yescharacteristic
212#431—24no—
213C2×C4⋊C8C2×D5912yes—
214C2×#103—598no—
215C2×#102—598no—
216C2×C22⋊C8C2×D5912yes—
217C2×C4×C8C2×D5912yes—
218#383—1181no—
219#382—592no—
220#381—1181no—
221#380—592no—
222C4×C4⋊C4—592no—
223C4×C22⋊C4—1181no—
224#376—1181no—
225C59#2180711yescharacteristic, complemented, Sylow
226C2×#41—598no—
227C2×#41—1184no—
228C2×#41—2368no—
229C22×Q8—592no—
230C22×D4—592no—
231C22×D4—1182no—
232C22×D4—1181no—
233C22×D4—1181no—
234C23×C4—1182no—
235C23×C4D23611yescharacteristic
236C23×C4D23611yescharacteristic
237C23×C4C22×D5911yescharacteristic
238C22×C8C22×D5914yes—
239C4⋊Q8—594no—
240C4⋊Q8—1182no—
241#127—594no—
242#127—1182no—
243#124—598no—
244#124—1184no—
245#121—1188no—
246#121—1188no—
247C4×D4—1188no—
248C4×D4—1188no—
249C2×C4⋊C4—594no—
250C2×C4⋊C4—1181no—
251C2×C4⋊C4—1181no—
252C2×C22⋊C4—1182no—
253C2×C22⋊C4—1182no—
254C2×C22⋊C4—1182no—
255C2×C42—1182no—
256C2×C42C4×D5914yes—
257C2×C42D23611yescharacteristic
258C2×C42D23611yescharacteristic
259C2×C42C22⋊D5914yes—
260C2×C42C22×D5911yescharacteristic
261C2×C42C22×D5911yescharacteristic
262C4⋊C8—24no—
263#103—2368no—
264#102—2368no—
265C22⋊C8—24no—
266C4×C8—24no—
267#95—1182no—
268#95—1182no—
269#95—1182no—
270#95—1182no—
271C24—1182no—
272C24C2×D23611yescharacteristic
273#41—23616no—
274#41—23616no—
275C2×Q8—594no—
276C2×Q8—1182no—
277C2×Q8—1188no—
278C2×D4—594no—
279C2×D4—1188no—
280C2×D4—1184no—
281C2×D4—1184no—
282C2×D4—1182no—
283C2×D4—1188no—
284C2×D4—2364no—
285C2×D4—2368no—
286C2×D4—2364no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—42no—
290C22×C4—1182no—
291C22×C4—1188no—
292C22×C4—1184no—
293C22×C4—2368no—
294C22×C4(C2×D59)⋊C414yes—
295C22×C4(C2×D59)⋊C414yes—
296C22×C4C2×C4×D5912yes—
297C22×C4C2×D23611yescharacteristic
298C22×C4C2×D23611yescharacteristic
299C22×C4C2×D23611yescharacteristic
300C22×C4C2×D23611yescharacteristic
301C22×C4C2×D23611yescharacteristic
302C22×C4C2×C22⋊D5912yes—
303C22×C4C23×D5911yescharacteristic
304C2×C8—24no—
305C2×C8—28no—
306C4⋊C4—1184no—
307C4⋊C4—1184no—
308C4⋊C4—1184no—
309C4⋊C4—1184no—
310C22⋊C4—1188no—
311C22⋊C4—1188no—
312C22⋊C4—2364no—
313C42—2364no—
314C42(C2×D59)⋊C414yes—
315C42(C2×D59)⋊C414yes—
316C42C2×C4×D5912yes—
317C42C2×C4×D5912yes—
318C42C2×C22⋊D5912yes—
319C42C2×C22⋊D5912yes—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—1184no—
324C23—2364no—
325C23—2364no—
326C23—2364no—
327C23C2×(C2×D59)⋊C412yes—
328C23C22×D23611yescharacteristic
329Q8—1188no—
330Q8—2364no—
331D4—1188no—
332D4—2368no—
333D4—23616no—
334D4—2368no—
335D4—2364no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—28no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—44no—
344C2×C4—1184no—
345C2×C4—23616no—
346C2×C4—2364no—
347C2×C4—2364no—
348C2×C4—2364no—
349C2×C4—2364no—
350C2×C4C2×(C2×D59)⋊C412yes—
351C2×C4C2×(C2×D59)⋊C412yes—
352C2×C4C2×(C2×D59)⋊C412yes—
353C2×C4C2×(C2×D59)⋊C412yes—
354C2×C4C2×(C2×D59)⋊C412yes—
355C2×C4#1427374914yes—
356C2×C4C22×C4×D5911yescharacteristic
357C2×C4C22×C22⋊D5911yescharacteristic
358C2×C4#1427377314yes—
359C8—44no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—4728no—
365C22—2364no—
366C22—2364no—
367C22—2364no—
368C22C22×(C2×D59)⋊C411yescharacteristic
369C22#1683797112yes—
370C22C2×#1427374912yes—
371C22#1683820212yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—4724no—
377C4—2364no—
378C4#1683739914yes—
379C4#1683787612yes—
380C4#1683796612yes—
381C2—42no—
382C2—4724no—
383C2#2142667611yescharacteristic
384C2C2×#1683739911yescharacteristic
385C2#2143261611yescharacteristic
386C2#2143507414yes—
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#431C2
socleC23×C59C22×D4
prime radicalC2×#16837952C2
cosocleC22×C4×C59C24
hypercenterC2×#431D59
hypocenterC59#21807
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
2. #21426494C2semidirect
3. #21425022C2semidirect
4. #21425018C2semidirect
5. #21424907C2semidirect
6. #21424037C2semidirect
7. C2×C59×#431C2semidirect
10. #16837952C4semidirect
189. C2×#431D59semidirect
225. C59#21807semidirect

Maximal subgroups

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

subgroupindex
C2×#168379522
#214264942
#214250222
#214250182
#214249072
#214240372
C2×C59×#4312
#2180759