Finite Group Databasebeta

← #28599278

Subgroups — #28599278

Order 15104.

762 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426719C211yesmaximal, characteristic
2#21426705C211yesmaximal, characteristic
3#21424866C212yesmaximal, complemented
4#21424860C212yesmaximal, complemented
5#21424297C212yesmaximal
6#21424291C212yesmaximal
7#21424264C211yesmaximal, characteristic, complemented
8#21424253C211yesmaximal, characteristic, complemented
9#21424230C211yesmaximal, characteristic, complemented
10#21424227C211yesmaximal, characteristic, complemented
11C2×C59×#433C211yesmaximal, characteristic
12C2×#14273686C2211yescharacteristic
13C22×C4⋊Dic59C2211yescharacteristic
14C2×#14273661C2212yes—
15C2×#14273659C2212yes—
16C2×#14273659C2212yes—
17C2×#14273659C2212yes—
18#16837301—22no—
19#16837295—22no—
20#16837288—22no—
21#16837277—22no—
22#16837272C2211yescharacteristic
23#16837272C2211yescharacteristic
24#16837272C2211yescharacteristic
25#16837268C2211yescharacteristic
26#16837268C2211yescharacteristic
27#16837268C2211yescharacteristic
28#16837262C2212yes—
29C4×C4⋊Dic59C2212yes—
30#16837165C2212yes—
31#16837165C2212yes—
32#16837165C2212yes—
33#16837165C2212yes—
34C2×#117×C59C2211yescharacteristic
35C59×#433C418yescomplemented
36C2×C4⋊C8×C59C2212yes—
37C2×C22⋊C8×C59C2212yes—
38C2×#97×C59C2211yescharacteristic
39C2×C4×C8×C59C2211yescharacteristic
40C23×Dic59—21no—
41#14273686—42no—
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59—22no—
44C2×C4⋊Dic59—22no—
45C2×C4⋊Dic59—22no—
46C2×C4⋊Dic59—42no—
47C2×C4⋊Dic59C2312yes—
48C2×C4⋊Dic59C2311yescharacteristic
49C2×C4⋊Dic59C2311yescharacteristic
50C2×C4⋊Dic59C2311yescharacteristic
51C2×C4⋊Dic59C2311yescharacteristic
52C2×C4⋊Dic59C2312yes—
53C2×C22⋊Dic59—21no—
54C2×C22⋊Dic59—22no—
55C2×C4×Dic59—21no—
56C2×C4×Dic59—22no—
57#14273661—42no—
58#14273659—42no—
59#14273659—42no—
60#14273659—42no—
61C59⋊#95—22no—
62C59⋊#95—22no—
63C59⋊#95—22no—
64C59⋊#95—22no—
65C59⋊#95—22no—
66C23×C4×C59C2311yescharacteristic
67C22×C8×C59C2312yes—
68C22×C8×C59C2312yes—
69#117×C59C2×C414yes—
70#117×C59Q814yes—
71C2×C4⋊C4×C59D412yes—
72C2×C22⋊C4×C59D412yes—
73C2×C42×C59C2311yescharacteristic
74C2×C42×C59C2311yescharacteristic
75C4⋊C8×C59C2×C418yescomplemented
76C22⋊C8×C59C2×C418yes—
77#97×C59C2×C414yescomplemented
78C4×C8×C59C2×C414yescomplemented
79C24×C59C2×D411yescharacteristic
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59C22×C412yes—
83C22×C4×C59C2×D412yes—
84C22×C4×C59C2×D412yes—
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×Q812yes—
87C22×C4×C59C2411yescharacteristic
88C22×Dic59—21no—
89C22×Dic59—22no—
90C22×Dic59—22no—
91C22×Dic59—21no—
92C22×Dic59—21no—
93C22×Dic59—21no—
94C22×Dic59—42no—
95C22×Dic59—82no—
96C2×C8×C59—22no—
97C2×C8×C59—22no—
98C2×C8×C59—22no—
99C2×C8×C59—22no—
100C2×C8×C59C22×C414yes—
101C2×C8×C59C22×C414yes—
102C4⋊C4×C59C4⋊C418yes—
103C22⋊C4×C59C4⋊C418yes—
104C42×C59C22×C412yes—
105C42×C59C22×C412yes—
106C42×C59C2×Q812yes—
107C42×C59C2×Q812yes—
108C22⋊Dic59—42no—
109C22⋊Dic59—82no—
110C4⋊Dic59—22no—
111C4⋊Dic59—22no—
112C4⋊Dic59—22no—
113C4⋊Dic59—22no—
114C4⋊Dic59—22no—
115C4⋊Dic59—22no—
116C4⋊Dic59—24no—
117C4⋊Dic59—42no—
118C4⋊Dic59—82no—
119C4⋊Dic59—82no—
120C4⋊Dic59—82no—
121C4×Dic59—42no—
122C4×Dic59—82no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—22no—
126C23×C59—22no—
127C23×C59C2×C4⋊C412yes—
128C23×C59C22×D411yescharacteristic
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—24no—
136C2×C4×C59—24no—
137C2×C4×C59C2×C4⋊C414yes—
138C2×C4×C59C2×C4⋊C414yes—
139C2×C4×C59C2×C4⋊C412yes—
140C2×C4×C59C23×C411yescharacteristic
141C2×C4×C59C22×Q811yescharacteristic
142C2×C4×C59H(5,𝔽2)12yes—
143C2×C4×C592−1+412yes—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—42no—
149C2×Dic59—42no—
150C2×Dic59—42no—
151C2×Dic59—42no—
152C2×Dic59—42no—
153C2×Dic59—42no—
154C2×Dic59—42no—
155C2×Dic59—42no—
156C2×Dic59—42no—
157C2×Dic59—42no—
158C2×Dic59—84no—
159C8×C59—24no—
160C8×C59—24no—
161#21990—591nomaximal, Sylow
162C22×C59—22no—
163C22×C59—24no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—42no—
167C22×C59C22×C4⋊C411yescharacteristic
168C22×C59#53511yescharacteristic
169C22×C59#53611yescharacteristic
170C22×C59#57712yes—
171C22×C59#57712yes—
172C4×C59—22no—
173C4×C59—24no—
174C4×C59—24no—
175C4×C59#52012yes—
176C4×C59#54412yes—
177Dic59—81no—
178Dic59—82no—
179Dic59—81no—
180Dic59—81no—
181Dic59—81no—
182Dic59—82no—
183C2×#433D5911yescharacteristic
184#2317—591no—
185#2314—591no—
186#1949—592no—
187#1948—592no—
188#1854—592no—
189#1853—592no—
190#1847—591no—
191#1844—591no—
192#1836—591no—
193#1835—591no—
194C2×C59—42no—
195C2×C59#231711yescharacteristic
196C2×C59#291212yes—
197C2×C59#322111yescharacteristic
198C2×C59#322211yescharacteristic
199C2×C59#324412yes—
200C2×#117—591no—
201C2×#117C2×D5911yescharacteristic
202C22×C4⋊C4—591no—
203#433Dic5918yescomplemented
204C2×#107—592no—
205C2×#106—592no—
206C2×#106—592no—
207C2×#106—592no—
208C2×C4⋊C8C2×D5912yes—
209C2×C22⋊C8C2×D5912yes—
210C2×#97C2×D5911yescharacteristic
211C2×C4×C8C2×D5911yescharacteristic
212#389—1182no—
213#388—1182no—
214#387—1182no—
215#385—1182no—
216#384—591no—
217#384—591no—
218#384—591no—
219#382—591no—
220#382—591no—
221#382—591no—
222#380—592no—
223C4×C4⋊C4—592no—
224#340—592no—
225#340—592no—
226#340—592no—
227#340—592no—
228C59#2199011yescharacteristic, complemented, Sylow
229C23×C4—1181no—
230C23×C4C22×D5911yescharacteristic
231C22×C8C22×D5912yes—
232C22×C8C22×D5912yes—
233#117—2362no—
234#117Dic11814yes—
235#117C2×Dic5914yes—
236C2×C4⋊C4—591no—
237C2×C4⋊C4—591no—
238C2×C4⋊C4—591no—
239C2×C4⋊C4—591no—
240C2×C4⋊C4—592no—
241C2×C4⋊C4—592no—
242C2×C4⋊C4—1182no—
243C2×C4⋊C4—1182no—
244C2×C4⋊C4—1182no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4—2362no—
247C2×C4⋊C4D23612yes—
248C2×C22⋊C4—1182no—
249C2×C22⋊C4—1181no—
250C2×C22⋊C4D23612yes—
251C2×C42—1182no—
252C2×C42—1181no—
253C2×C42C22×D5911yescharacteristic
254C2×C42C22×D5911yescharacteristic
255#107—2362no—
256#106—2362no—
257#106—2362no—
258#106—2362no—
259C4⋊C8C2×Dic5918yescomplemented
260C22⋊C8C2×Dic5918yes—
261#97C2×Dic5914yescomplemented
262C4×C8C2×Dic5914yescomplemented
263#95—1182no—
264#95—1182no—
265#95—1182no—
266#95—1182no—
267#95—1182no—
268C24C2×D23611yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—1182no—
272C22×C4—1181no—
273C22×C4—1182no—
274C22×C4—1181no—
275C22×C4—1181no—
276C22×C4—1181no—
277C22×C4—4722no—
278C22×C4—2362no—
279C22×C4C2×Dic11812yes—
280C22×C4C2×D23612yes—
281C22×C4C2×D23611yescharacteristic
282C22×C4C2×D23612yes—
283C22×C4C22×Dic5912yes—
284C22×C4C23×D5911yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8C22×Dic5914yes—
290C2×C8C22×Dic5914yes—
291C4⋊C4—1182no—
292C4⋊C4—1182no—
293C4⋊C4—1182no—
294C4⋊C4—1182no—
295C4⋊C4—1182no—
296C4⋊C4—1182no—
297C4⋊C4—1184no—
298C4⋊C4—4722no—
299C4⋊C4—4722no—
300C4⋊C4—4722no—
301C4⋊C4—2362no—
302C4⋊C4C4⋊Dic5918yes—
303C22⋊C4—4722no—
304C22⋊C4—2362no—
305C22⋊C4C4⋊Dic5918yes—
306C42—4722no—
307C42—2362no—
308C42C2×Dic11812yes—
309C42C2×Dic11812yes—
310C42C22×Dic5912yes—
311C42C22×Dic5912yes—
312C23—22no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23C2×C4⋊Dic5912yes—
317C23C22×D23611yescharacteristic
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—4724no—
327C2×C4—2362no—
328C2×C4—2362no—
329C2×C4—2362no—
330C2×C4—2362no—
331C2×C4—2362no—
332C2×C4—2362no—
333C2×C4—2362no—
334C2×C4—2362no—
335C2×C4—2361no—
336C2×C4—2361no—
337C2×C4—2361no—
338C2×C4—2361no—
339C2×C4—2362no—
340C2×C4—2362no—
341C2×C4C2×C4⋊Dic5914yes—
342C2×C4C2×C4⋊Dic5914yes—
343C2×C4C2×C4⋊Dic5912yes—
344C2×C4C23×Dic5911yescharacteristic
345C2×C4C22×Dic11811yescharacteristic
346C2×C4#1427377312yes—
347C2×C4#1427377412yes—
348C8—24no—
349C8—24no—
350C22—22no—
351C22—24no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22C22×C4⋊Dic5911yescharacteristic
356C22#1683797111yescharacteristic
357C22#1683797811yescharacteristic
358C22#1683821412yes—
359C22#1683821412yes—
360C4—22no—
361C4—24no—
362C4—24no—
363C4—4722no—
364C4—4722no—
365C4—4721no—
366C4—4721no—
367C4—4721no—
368C4—4721no—
369C4#1683788512yes—
370C4#1683803012yes—
371C2—42no—
372C2#2142671911yescharacteristic
373C2#2143269112yes—
374C2#2143515911yescharacteristic
375C2#2143517011yescharacteristic
376C2#2143539612yes—
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#433C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC2×#433D59
hypocenterC59#21990
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #21424866C2semidirect
4. #21424860C2semidirect
7. #21424264C2semidirect
8. #21424253C2semidirect
9. #21424230C2semidirect
10. #21424227C2semidirect
35. C59×#433C4semidirect
75. C4⋊C8×C59C2×C4semidirect
77. #97×C59C2×C4semidirect
78. C4×C8×C59C2×C4semidirect
203. #433Dic59semidirect
228. C59#21990semidirect
259. C4⋊C8C2×Dic59semidirect
261. #97C2×Dic59semidirect
262. C4×C8C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214267192
#214267052
#214248662
#214248602
#214242972
#214242912
#214242642
#214242532
#214242302
#214242272
C2×C59×#4332
#2199059