Finite Group Databasebeta

← #28499757

Subgroups — #28499757

Order 15104.

422 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
2#21424121C211yesmaximal, characteristic, complemented
3#21422108C212yesmaximal, complemented
4#21422105C212yesmaximal, complemented
5C59×#3050C211yesmaximal, characteristic
6C22×C59⋊C16—21no—
7C2×#14273659—21no—
8#16837262—21no—
9C22⋊(C59⋊C16)—21no—
10C22⋊(C59⋊C16)C2212yes—
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—
23C2×C4⋊Dic59—22no—
24C2×C22⋊Dic59—41no—
25C2×C59⋊C16—22no—
26C2×C59⋊C16—42no—
27#14273659—42no—
28C59⋊#95—41no—
29C2×#41×C59—21no—
30C23×C4×C59D411yescharacteristic
31#135×C59—42no—
32C2×Q16×C59C2×C411yescharacteristic
33C2×SD16×C59—21no—
34C2×D8×C59C2×C411yescharacteristic
35C22×C8×C59—42no—
36C22×C8×C59C2×C411yescharacteristic
37C22×C8×C59C2×C411yescharacteristic
38C22×C8×C59D411yescharacteristic
39C22×C8×C59D411yescharacteristic
40C22×C8×C59D411yescharacteristic
41C22×C8×C59C2311yescharacteristic
42#123×C59—41no—
43C22⋊Q8×C59—21no—
44C22⋊Q8×C59—21no—
45#121×C59—21no—
46#121×C59—21no—
47#120×C59—41no—
48C4×D4×C59—41no—
49C4×D4×C59—41no—
50#117×C59—21no—
51#107×C59C2×C411yescharacteristic
52#107×C59C2×C411yescharacteristic
53#106×C59—21no—
54#103×C59—21no—
55#103×C59—21no—
56#102×C59—21no—
57#102×C59—21no—
58C24×C59D811yescharacteristic
59#41×C59—42no—
60#41×C59—42no—
61C2×Q8×C59—21no—
62C2×D4×C59—21no—
63C2×D4×C59—41no—
64C2×D4×C59—41no—
65C2×D4×C59—41no—
66C22×C4×C59—41no—
67C22×C4×C59—41no—
68C22×C4×C59—41no—
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59D811yescharacteristic
72C22×C4×C59SD1611yescharacteristic
73C22×C4×C59SD1611yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—42no—
76Q16×C59—41no—
77SD16×C59—42no—
78D8×C59—41no—
79C2×C8×C59—22no—
80C2×C8×C59—22no—
81C2×C8×C59—22no—
82C2×C8×C59—41no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C2×C8×C59C22⋊C411yescharacteristic
86C2×C8×C59C22⋊C411yescharacteristic
87C2×C8×C59C22×C411yescharacteristic
88C2×C8×C59C2×D411yescharacteristic
89C4⋊C4×C59—21no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C42×C59—41no—
99C22⋊Dic59—82no—
100C4⋊Dic59—42no—
101C4⋊Dic59—42no—
102C59⋊C16—42no—
103C23×C59—22no—
104C23×C59—22no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59#10211yescharacteristic
108C23×C59#10211yescharacteristic
109C23×C59C2×D811yescharacteristic
110Q8×C59—41no—
111Q8×C59—41no—
112D4×C59—41no—
113D4×C59—41no—
114D4×C59—82no—
115D4×C59—82no—
116D4×C59—82no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—82no—
130C2×C4×C59#10211yescharacteristic
131C2×C4×C59#10211yescharacteristic
132C2×C4×C59C2×C22⋊C411yescharacteristic
133C2×C4×C59C2×SD1611yescharacteristic
134C2×Dic59—82no—
135C2×Dic59—82no—
136C2×Dic59—82no—
137C8×C59—21no—
138C8×C59—21no—
139C8×C59—21no—
140C8×C59#9912yes—
141#11885—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—21no—
146C22×C59—21no—
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—44no—
150C22×C59—81no—
151C22×C59—81no—
152C22×C59—81no—
153C22×C59#36012yes—
154C22×C59C2×#10211yescharacteristic
155C22×C59#50812yes—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—21no—
159C4×C59—81no—
160C4×C59—81no—
161C4×C59—81no—
162C4×C59C2×#9911yescharacteristic
163C4×C59#42111yescharacteristic
164Dic59—162no—
165#3050D5911yescharacteristic
166C2×C22⋊C16—591no—
167#1807—591no—
168#1348—592no—
169#1347—592no—
170C2×C59—22no—
171C2×C59—22no—
172C2×C59—41no—
173C2×C59—161no—
174C2×C59#151911yescharacteristic
175C2×C59#213811yescharacteristic
176C2×C59C2×#36011yescharacteristic
177C2×#135Dic5911yescharacteristic, complemented
178C23×C8C2×D5911yescharacteristic
179#525—21no—
180C22×C16—1181no—
181#467Dic5911yescharacteristic, complemented
182#467C2×D5911yescharacteristic
183#466Dic5911yescharacteristic, complemented
184#466C2×D5911yescharacteristic
185#465—21no—
186#465—21no—
187#427Dic5911yescharacteristic, complemented
188C2×#106—1181no—
189#416—21no—
190#380—1181no—
191C22⋊C16—592no—
192C22⋊C16—1181no—
193#340—592no—
194C59#1188511yescharacteristic, complemented, Sylow
195C2×#41—21no—
196C23×C4C22⋊D5911yescharacteristic
197#135—42no—
198C2×Q16C2×Dic5911yescharacteristic
199C2×SD16—21no—
200C2×D8C2×Dic5911yescharacteristic
201C22×C8—42no—
202C22×C8C2×Dic5911yescharacteristic
203C22×C8C2×Dic5911yescharacteristic
204C22×C8C22⋊D5911yescharacteristic
205C22×C8C22⋊D5911yescharacteristic
206C22×C8C22⋊D5911yescharacteristic
207C22×C8C22×D5911yescharacteristic
208#123—41no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211#121—21no—
212#121—21no—
213#120—41no—
214C4×D4—41no—
215C4×D4—41no—
216#117—21no—
217C2×C4⋊C4—1182no—
218C2×C22⋊C4—2361no—
219C2×C16—1182no—
220C2×C16—2362no—
221#107C2×Dic5911yescharacteristic
222#107C2×Dic5911yescharacteristic
223#106—21no—
224#106—2362no—
225#103—21no—
226#103—21no—
227#102—21no—
228#102—21no—
229#95—2361no—
230C24#1174566911yescharacteristic
231#41—42no—
232#41—42no—
233C2×Q8—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—2362no—
242C22×C4#1174566911yescharacteristic
243C22×C4#1174567011yescharacteristic
244C22×C4#1174567011yescharacteristic
245C22×C4C22⋊Dic5911yescharacteristic
246C22×C4C22⋊Dic5911yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248Q16—41no—
249SD16—42no—
250D8—41no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8C22⋊Dic5911yescharacteristic
258C2×C8C22⋊Dic5911yescharacteristic
259C2×C8C22×Dic5911yescharacteristic
260C2×C8C2×C22⋊D5911yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—2362no—
266C4⋊C4—2362no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—4722no—
273C42—41no—
274C16—2362no—
275C23—22no—
276C23—22no—
277C23—41no—
278C23—41no—
279C23#1427364811yescharacteristic
280C23#1427364811yescharacteristic
281C23C2×#1174566911yescharacteristic
282Q8—41no—
283Q8—41no—
284D4—41no—
285D4—41no—
286D4—82no—
287D4—82no—
288D4—82no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—4722no—
303C2×C4—4722no—
304C2×C4—4722no—
305C2×C4#1427364811yescharacteristic
306C2×C4#1427364811yescharacteristic
307C2×C4C2×C22⋊Dic5911yescharacteristic
308C2×C4C2×#1174567011yescharacteristic
309C8—21no—
310C8—21no—
311C8—21no—
312C8#1427364212yes—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—44no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#1683721212yes—
325C22C2×#1427364811yescharacteristic
326C22#1683783212yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—9442no—
334C4C2×#1427364211yescharacteristic
335C4#1683741911yescharacteristic
336C2—22no—
337C2—22no—
338C2—41no—
339C2—161no—
340C2#2142266411yescharacteristic
341C2#2142580311yescharacteristic
342C2C2×#1683721211yescharacteristic
subgroupG/H
centerC22C2×#14273648
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#3050C2
socleC22×C59C2×#102
cosocleC22×C8×C59C23
hypercenter#3050D59
hypocenterC59#11885
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ 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. #21424121C2semidirect
3. #21422108C2semidirect
4. #21422105C2semidirect
12. C2×#135×C59C4semidirect
15. C59×#467C4semidirect
17. C59×#466C4semidirect
21. C59×#427C4semidirect
177. C2×#135Dic59semidirect
181. #467Dic59semidirect
183. #466Dic59semidirect
187. #427Dic59semidirect
194. C59#11885semidirect

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
#214241212
#214221082
#214221052
C59×#30502
#1188559