Finite Group Databasebeta

← #28498917

Subgroups — #28498917

Order 15104.

466 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
2#21423813C211yesmaximal, characteristic, complemented
3#21422108C212yesmaximal, complemented
4#21422105C212yesmaximal, complemented
5C59×#3050C211yesmaximal, characteristic
6C2×#14273686—21no—
7C22×C59⋊C16—21no—
8C22⋊(C59⋊C16)—21no—
9C22⋊(C59⋊C16)C2212yes—
10#16837165—21no—
11#16837165C2212yes—
12C2×#135×C59C411yescharacteristic, complemented
13C23×C8×C59C2211yescharacteristic
14C59×#525—21no—
15C59×#467C411yescharacteristic, complemented
16C59×#467C2211yescharacteristic
17C59×#466C411yescharacteristic, complemented
18C59×#466C2211yescharacteristic
19C59×#465—21no—
20C59×#465—21no—
21C59×#427C411yescharacteristic, complemented
22C59×#416—21no—
23#14273686—24no—
24#14273686—42no—
25C2×C4⋊Dic59—22no—
26C2×C22⋊Dic59—41no—
27C2×C4×Dic59—41no—
28C2×C59⋊C16—22no—
29C2×C59⋊C16—42no—
30C2×#41×C59—21no—
31C23×C4×C59D411yescharacteristic
32#135×C59—42no—
33C2×Q16×C59C2×C411yescharacteristic
34C2×SD16×C59—21no—
35C2×D8×C59C2×C411yescharacteristic
36C22×C8×C59—42no—
37C22×C8×C59C2×C411yescharacteristic
38C22×C8×C59C2×C411yescharacteristic
39C22×C8×C59D411yescharacteristic
40C22×C8×C59D411yescharacteristic
41C22×C8×C59D411yescharacteristic
42C22×C8×C59C2311yescharacteristic
43#123×C59—41no—
44C22⋊Q8×C59—21no—
45C22⋊Q8×C59—21no—
46#121×C59—21no—
47#121×C59—21no—
48#120×C59—41no—
49C4×D4×C59—41no—
50C4×D4×C59—41no—
51#117×C59—21no—
52#107×C59C2×C411yescharacteristic
53#107×C59C2×C411yescharacteristic
54#106×C59—21no—
55#103×C59—21no—
56#103×C59—21no—
57#102×C59—21no—
58#102×C59—21no—
59C24×C59D811yescharacteristic
60#41×C59—42no—
61#41×C59—42no—
62C2×Q8×C59—21no—
63C2×D4×C59—21no—
64C2×D4×C59—41no—
65C2×D4×C59—41no—
66C2×D4×C59—41no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—41no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59D811yescharacteristic
73C22×C4×C59SD1611yescharacteristic
74C22×C4×C59SD1611yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×Dic59—42no—
77Q16×C59—41no—
78SD16×C59—42no—
79D8×C59—41no—
80C2×C8×C59—22no—
81C2×C8×C59—22no—
82C2×C8×C59—41no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59—41no—
86C2×C8×C59C22⋊C411yescharacteristic
87C2×C8×C59C22⋊C411yescharacteristic
88C2×C8×C59C22×C411yescharacteristic
89C2×C8×C59C2×D411yescharacteristic
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C42×C59—41no—
100C22⋊Dic59—44no—
101C4⋊Dic59—24no—
102C4⋊Dic59—42no—
103C4×Dic59—44no—
104C59⋊C16—42no—
105C23×C59—22no—
106C23×C59—22no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59#10211yescharacteristic
110C23×C59#10211yescharacteristic
111C23×C59C2×D811yescharacteristic
112Q8×C59—41no—
113Q8×C59—41no—
114D4×C59—41no—
115D4×C59—41no—
116D4×C59—82no—
117D4×C59—82no—
118D4×C59—82no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—82no—
130C2×C4×C59#9912yes—
131C2×C4×C59#10211yescharacteristic
132C2×C4×C59#10211yescharacteristic
133C2×C4×C59#10412yes—
134C2×C4×C59C2×C22⋊C411yescharacteristic
135C2×C4×C59C2×SD1611yescharacteristic
136C2×Dic59—44no—
137C2×Dic59—82no—
138C2×Dic59—82no—
139C8×C59—21no—
140C8×C59—21no—
141C8×C59—21no—
142C8×C59—21no—
143#11703—591nomaximal, Sylow
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—44no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59#32712yes—
156C22×C59C2×#10211yescharacteristic
157C22×C59#50812yes—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—81no—
161C4×C59—81no—
162C4×C59—81no—
163C4×C59#32712yes—
164C4×C59C2×#9911yescharacteristic
165C4×C59C2×#10411yescharacteristic
166Dic59—84no—
167#3050D5911yescharacteristic
168C2×C22⋊C16—591no—
169#1735—591no—
170#1348—592no—
171#1347—592no—
172C2×C59—22no—
173C2×C59—22no—
174C2×C59—41no—
175C2×C59—161no—
176C2×C59C2×#32711yescharacteristic
177C2×C59#213811yescharacteristic
178C2×C59#214511yescharacteristic
179C2×#135Dic5911yescharacteristic, complemented
180C23×C8C2×D5911yescharacteristic
181#525—21no—
182C2×#117—1181no—
183C22×C16—1181no—
184#467Dic5911yescharacteristic, complemented
185#467C2×D5911yescharacteristic
186#466Dic5911yescharacteristic, complemented
187#466C2×D5911yescharacteristic
188#465—21no—
189#465—21no—
190#427Dic5911yescharacteristic, complemented
191#416—21no—
192C22⋊C16—592no—
193C22⋊C16—1181no—
194#340—592no—
195#340—1181no—
196C59#1170311yescharacteristic, complemented, Sylow
197C2×#41—21no—
198C23×C4C22⋊D5911yescharacteristic
199#135—42no—
200C2×Q16C2×Dic5911yescharacteristic
201C2×SD16—21no—
202C2×D8C2×Dic5911yescharacteristic
203C22×C8—42no—
204C22×C8C2×Dic5911yescharacteristic
205C22×C8C2×Dic5911yescharacteristic
206C22×C8C22⋊D5911yescharacteristic
207C22×C8C22⋊D5911yescharacteristic
208C22×C8C22⋊D5911yescharacteristic
209C22×C8C22×D5911yescharacteristic
210#123—41no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213#121—21no—
214#121—21no—
215#120—41no—
216C4×D4—41no—
217C4×D4—41no—
218#117—21no—
219#117—1184no—
220#117—2362no—
221C2×C4⋊C4—1182no—
222C2×C22⋊C4—2361no—
223C2×C42—2361no—
224C2×C16—1182no—
225C2×C16—2362no—
226#107C2×Dic5911yescharacteristic
227#107C2×Dic5911yescharacteristic
228#106—21no—
229#103—21no—
230#103—21no—
231#102—21no—
232#102—21no—
233C24#1174566911yescharacteristic
234#41—42no—
235#41—42no—
236C2×Q8—21no—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—2362no—
245C22×C4#1174566911yescharacteristic
246C22×C4#1174567011yescharacteristic
247C22×C4#1174567011yescharacteristic
248C22×C4C22⋊Dic5911yescharacteristic
249C22×C4C22⋊Dic5911yescharacteristic
250C22×C4C2×C22⋊D5911yescharacteristic
251Q16—41no—
252SD16—42no—
253D8—41no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22⋊Dic5911yescharacteristic
261C2×C8C22⋊Dic5911yescharacteristic
262C2×C8C22×Dic5911yescharacteristic
263C2×C8C2×C22⋊D5911yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—1184no—
269C4⋊C4—2362no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—2364no—
276C42—41no—
277C42—2364no—
278C16—2362no—
279C23—22no—
280C23—22no—
281C23—41no—
282C23—41no—
283C23#1427364811yescharacteristic
284C23#1427364811yescharacteristic
285C23C2×#1174566911yescharacteristic
286Q8—41no—
287Q8—41no—
288D4—41no—
289D4—41no—
290D4—82no—
291D4—82no—
292D4—82no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—82no—
304C2×C4—4722no—
305C2×C4—4722no—
306C2×C4—2364no—
307C2×C4#1427364212yes—
308C2×C4#1427364811yescharacteristic
309C2×C4#1427364811yescharacteristic
310C2×C4#1427365412yes—
311C2×C4C2×C22⋊Dic5911yescharacteristic
312C2×C4C2×#1174567011yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—44no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#1683713612yes—
329C22C2×#1427364811yescharacteristic
330C22#1683783212yes—
331C4—21no—
332C4—21no—
333C4—81no—
334C4—81no—
335C4—81no—
336C4—4724no—
337C4#1683713612yes—
338C4C2×#1427364211yescharacteristic
339C4C2×#1427365411yescharacteristic
340C2—22no—
341C2—22no—
342C2—41no—
343C2—161no—
344C2C2×#1683713611yescharacteristic
345C2#2142580311yescharacteristic
346C2#2142583211yescharacteristic
subgroupG/H
centerC2×C4#14273648
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#3050C2
socleC22×C59C2×#102
cosocleC22×C8×C59C23
hypercenter#3050D59
hypocenterC59#11703
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(C59⋊C16)C2semidirect
2. #21423813C2semidirect
3. #21422108C2semidirect
4. #21422105C2semidirect
12. C2×#135×C59C4semidirect
15. C59×#467C4semidirect
17. C59×#466C4semidirect
21. C59×#427C4semidirect
179. C2×#135Dic59semidirect
184. #467Dic59semidirect
186. #466Dic59semidirect
190. #427Dic59semidirect
196. C59#11703semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C59⋊C16)2
#214238132
#214221082
#214221052
C59×#30502
#1170359