Finite Group Databasebeta

← #28589261

Subgroups — #28589261

Order 15104.

810 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426705C212yesmaximal
2#21424260C211yesmaximal, characteristic, complemented
3#21424256C211yesmaximal, characteristic, complemented
4#21424230C212yesmaximal, complemented
5#21424129C212yesmaximal
6#21424125C212yesmaximal
7#21423993C212yesmaximal, complemented
8C4×#14273661C212yesmaximal, complemented
9C2×C59×#405C211yesmaximal, characteristic
10C2×#14273661C2212yes—
11C2×#14273661C2212yes—
12C2×#14273661C2212yes—
13C2×#14273659C2212yes—
14#16837295—24no—
15#16837288—24no—
16#16837272C2212yes—
17#16837268C2212yes—
18#16837262C2214yes—
19C4×C4⋊Dic59C2214yes—
20#16837165C2214yes—
21#16837165C2214yes—
22C22×M16×C59C2211yescharacteristic
23C23×C8×C59C2211yescharacteristic
24C2×#117×C59C2211yescharacteristic
25C59×#405C418yescomplemented
26C2×#97×C59C2211yescharacteristic
27C2×#97×C59C2211yescharacteristic
28C2×C4×C8×C59C2211yescharacteristic
29C2×C4×C8×C59C2211yescharacteristic
30C2×C4⋊Dic59—24no—
31C2×C4⋊Dic59C2314yes—
32C2×C4⋊Dic59C2314yes—
33C2×C22⋊Dic59—24no—
34C2×C4×Dic59—24no—
35#14273661—24no—
36#14273661—24no—
37#14273661—24no—
38#14273659—24no—
39C59⋊#95—24no—
40C59⋊#95—24no—
41C59⋊#95—24no—
42C23×C4×C59C2311yescharacteristic
43C2×M16×C59—22no—
44C2×M16×C59—22no—
45C2×M16×C59C2×C414yes—
46C22×C8×C59—22no—
47C22×C8×C59—22no—
48C22×C8×C59C2×C414yes—
49C22×C8×C59C2311yescharacteristic
50C22×C8×C59C2311yescharacteristic
51C22×C8×C59C2311yescharacteristic
52C22×C8×C59C2311yescharacteristic
53#117×C59C2×C414yes—
54#117×C59Q814yes—
55C2×C4⋊C4×C59D412yes—
56C2×C22⋊C4×C59D412yes—
57C2×C42×C59C2311yescharacteristic
58C2×C42×C59C2311yescharacteristic
59#97×C59C2×C414yes—
60#97×C59C2×C414yes—
61C4×C8×C59C2×C414yes—
62C4×C8×C59C2×C414yes—
63C24×C59C2×D411yescharacteristic
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59C22×C412yes—
69C22×C4×C59C2×D412yes—
70C22×C4×C59C2×D412yes—
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×Q812yes—
73C22×C4×C59C2411yescharacteristic
74C22×Dic59—24no—
75C22×Dic59—24no—
76M16×C59—28no—
77C2×C8×C59—21no—
78C2×C8×C59—21no—
79C2×C8×C59—21no—
80C2×C8×C59—21no—
81C2×C8×C59—28no—
82C2×C8×C59C22×C412yes—
83C2×C8×C59C22×C412yes—
84C2×C8×C59C22×C412yes—
85C2×C8×C59C22×C412yes—
86C2×C8×C59#4112yes—
87C2×C8×C59#4112yes—
88C2×C8×C59#4112yes—
89C2×C8×C59#4112yes—
90C4⋊C4×C59C4⋊C418yes—
91C22⋊C4×C59C4⋊C418yes—
92C42×C59C22×C412yes—
93C42×C59C22×C412yes—
94C42×C59C2×Q812yes—
95C42×C59C2×Q812yes—
96C22⋊Dic59—84no—
97C4⋊Dic59—24no—
98C4⋊Dic59—24no—
99C4⋊Dic59—24no—
100C4⋊Dic59—24no—
101C4⋊Dic59—84no—
102C4×Dic59—84no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—22no—
106C23×C59—22no—
107C23×C59C2×C4⋊C412yes—
108C23×C59C22×D411yescharacteristic
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59C2×C4⋊C414yes—
120C2×C4×C59C2×C4⋊C414yes—
121C2×C4×C59C2×C4⋊C412yes—
122C2×C4×C59C23×C411yescharacteristic
123C2×C4×C59C22×Q811yescharacteristic
124C2×C4×C59C2×#4111yescharacteristic
125C2×C4×C59C2×#4111yescharacteristic
126C2×C4×C59C2×#4111yescharacteristic
127C2×C4×C59C2×#4111yescharacteristic
128C2×Dic59—44no—
129C2×Dic59—44no—
130C2×Dic59—44no—
131C2×Dic59—44no—
132C2×Dic59—44no—
133C2×Dic59—44no—
134C8×C59—22no—
135C8×C59—22no—
136C8×C59#11714yes—
137C8×C59#11714yes—
138#20821—591nomaximal, Sylow
139C22×C59—24no—
140C22×C59—22no—
141C22×C59—42no—
142C22×C59—41no—
143C22×C59—41no—
144C22×C59C22×C4⋊C411yescharacteristic
145C22×C59#53211yescharacteristic
146C22×C59#53211yescharacteristic
147C22×C59#57612yes—
148C22×C59#57812yes—
149C4×C59—24no—
150C4×C59—24no—
151C4×C59—22no—
152C4×C59C2×#11711yescharacteristic
153C4×C59C2×#11711yescharacteristic
154C4×C59#53311yescharacteristic
155C4×C59#53311yescharacteristic
156Dic59—84no—
157Dic59—84no—
158C2×#405D5911yescharacteristic
159#2314—592no—
160#1846—591no—
161#1845—591no—
162#1836—592no—
163#1809—592no—
164#1808—592no—
165#1776—592no—
166C4×#107—592no—
167C2×C59—42no—
168C2×C59#228511yescharacteristic
169C2×C59#291312yes—
170C2×C59#317512yes—
171C2×C59#317611yescharacteristic
172C2×C59#317711yescharacteristic
173C22×M16C2×D5911yescharacteristic
174C23×C8C2×D5911yescharacteristic
175C2×#117C2×D5911yescharacteristic
176C2×#107—592no—
177C2×#107—592no—
178C2×#107—592no—
179C2×#106—592no—
180#405Dic5918yescomplemented
181C2×#97C2×D5911yescharacteristic
182C2×#97C2×D5911yescharacteristic
183C2×C4×C8C2×D5911yescharacteristic
184C2×C4×C8C2×D5911yescharacteristic
185#388—1184no—
186#387—1184no—
187#384—592no—
188#382—592no—
189#380—594no—
190C4×C4⋊C4—594no—
191#340—594no—
192#340—594no—
193C59#2082111yescharacteristic, complemented, Sylow
194C23×C4C22×D5911yescharacteristic
195C2×M16—22no—
196C2×M16—22no—
197C2×M16C2×Dic5914yes—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C2×Dic5914yes—
201C22×C8C22×D5911yescharacteristic
202C22×C8C22×D5911yescharacteristic
203C22×C8C22×D5911yescharacteristic
204C22×C8C22×D5911yescharacteristic
205#117Dic11814yes—
206#117C2×Dic5914yes—
207C2×C4⋊C4—594no—
208C2×C4⋊C4—594no—
209C2×C4⋊C4—1184no—
210C2×C4⋊C4D23612yes—
211C2×C22⋊C4—1184no—
212C2×C22⋊C4D23612yes—
213C2×C42—1184no—
214C2×C42C22×D5911yescharacteristic
215C2×C42C22×D5911yescharacteristic
216#107—1184no—
217#107—1184no—
218#107—1184no—
219#106—1184no—
220#97C2×Dic5914yes—
221#97C2×Dic5914yes—
222C4×C8C2×Dic5914yes—
223C4×C8C2×Dic5914yes—
224#95—1184no—
225#95—1184no—
226#95—1184no—
227C24C2×D23611yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—1184no—
233C22×C4—1184no—
234C22×C4C2×Dic11812yes—
235C22×C4C2×D23611yescharacteristic
236C22×C4C2×D23612yes—
237C22×C4C2×D23612yes—
238C22×C4C22×Dic5912yes—
239C22×C4C23×D5911yescharacteristic
240M16—28no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—28no—
246C2×C8#1174568512yes—
247C2×C8#1174568512yes—
248C2×C8#1174568512yes—
249C2×C8#1174568512yes—
250C2×C8C22×Dic5912yes—
251C2×C8C22×Dic5912yes—
252C2×C8C22×Dic5912yes—
253C2×C8C22×Dic5912yes—
254C4⋊C4—1184no—
255C4⋊C4—1184no—
256C4⋊C4—1184no—
257C4⋊C4—1184no—
258C4⋊C4—4724no—
259C4⋊C4C4⋊Dic5918yes—
260C22⋊C4—4724no—
261C22⋊C4C4⋊Dic5918yes—
262C42—4724no—
263C42C2×Dic11812yes—
264C42C2×Dic11812yes—
265C42C22×Dic5912yes—
266C42C22×Dic5912yes—
267C23—22no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23C2×C4⋊Dic5912yes—
272C23C22×D23611yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—2364no—
284C2×C4—2364no—
285C2×C4—2364no—
286C2×C4—2364no—
287C2×C4—2364no—
288C2×C4—2364no—
289C2×C4C2×C4⋊Dic5914yes—
290C2×C4C2×C4⋊Dic5914yes—
291C2×C4C2×C4⋊Dic5912yes—
292C2×C4C23×Dic5911yescharacteristic
293C2×C4C22×Dic11811yescharacteristic
294C2×C4C2×#1174568511yescharacteristic
295C2×C4C2×#1174568511yescharacteristic
296C2×C4C2×#1174568511yescharacteristic
297C2×C4C2×#1174568511yescharacteristic
298C8—22no—
299C8—22no—
300C8#1427368614yes—
301C8#1427368614yes—
302C22—22no—
303C22—24no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22C22×C4⋊Dic5911yescharacteristic
308C22#1683795211yescharacteristic
309C22#1683795211yescharacteristic
310C22#1683820712yes—
311C22#1683821912yes—
312C4—22no—
313C4—24no—
314C4—24no—
315C4—4724no—
316C4—4724no—
317C4C2×#1427368611yescharacteristic
318C4C2×#1427368611yescharacteristic
319C4#1683795911yescharacteristic
320C4#1683795911yescharacteristic
321C2—42no—
322C2#2142650311yescharacteristic
323C2#2143269612yes—
324C2#2143473512yes—
325C2#2143474611yescharacteristic
326C2#2143475511yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#405C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC2×#405D59
hypocenterC59#20821
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. #21424260C2semidirect
3. #21424256C2semidirect
4. #21424230C2semidirect
7. #21423993C2semidirect
8. C4×#14273661C2semidirect
25. C59×#405C4semidirect
180. #405Dic59semidirect
193. C59#20821semidirect

Maximal subgroups

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

subgroupindex
#214267052
#214242602
#214242562
#214242302
#214241292
#214241252
#214239932
C4×#142736612
C2×C59×#4052
#2082159