Finite Group Databasebeta

← #28499778

Subgroups — #28499778

Order 15104.

416 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425707C211yesmaximal, characteristic, complemented
2#21424125C211yesmaximal, characteristic, complemented
3#21422108C212yesmaximal, complemented
4#21422105C212yesmaximal, complemented
5C59×#3050C211yesmaximal, characteristic
6C2×#14273667—21no—
7C2×#14273661—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—
25#14273667—42no—
26C2×C59⋊C16—22no—
27#14273661—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—21no—
104C23×C59—21no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59#10211yescharacteristic
109C23×C59#10211yescharacteristic
110C23×C59C2×D811yescharacteristic
111Q8×C59—41no—
112Q8×C59—41no—
113D4×C59—41no—
114D4×C59—41no—
115D4×C59—82no—
116D4×C59—82no—
117D4×C59—81no—
118D4×C59—81no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—82no—
132C2×C4×C59#10211yescharacteristic
133C2×C4×C59#10211yescharacteristic
134C2×C4×C59C2×C22⋊C411yescharacteristic
135C2×C4×C59C2×SD1611yescharacteristic
136C2×Dic59—82no—
137C2×Dic59—82no—
138C2×Dic59—82no—
139C8×C59—21no—
140C8×C59—21no—
141C8×C59—21no—
142C8×C59#9912yes—
143#11890—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—21no—
152C22×C59—42no—
153C22×C59—42no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59—81no—
157C22×C59C2×#10211yescharacteristic
158C22×C59#50911yescharacteristic
159C22×C59#51011yescharacteristic
160C4×C59—21no—
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—81no—
164C4×C59—81no—
165C4×C59—81no—
166C4×C59C2×#9911yescharacteristic
167C4×C59#42111yescharacteristic
168Dic59—162no—
169#3050D5911yescharacteristic
170#2112—591no—
171#1808—591no—
172#1348—592no—
173#1347—592no—
174C2×C59—21no—
175C2×C59—21no—
176C2×C59—41no—
177C2×C59—41no—
178C2×C59—161no—
179C2×C59#151911yescharacteristic
180C2×C59#213911yescharacteristic
181C2×C59#214511yescharacteristic
182C2×#135Dic5911yescharacteristic, complemented
183C23×C8C2×D5911yescharacteristic
184#525—21no—
185C2×M32—1181no—
186#467Dic5911yescharacteristic, complemented
187#467C2×D5911yescharacteristic
188#466Dic5911yescharacteristic, complemented
189#466C2×D5911yescharacteristic
190#465—21no—
191#465—21no—
192#427Dic5911yescharacteristic, complemented
193C2×#107—1181no—
194#416—21no—
195#380—1181no—
196C22⋊C16—592no—
197C22⋊C16—1181no—
198#340—592no—
199C59#1189011yescharacteristic, complemented, Sylow
200C2×#41—21no—
201C23×C4C22⋊D5911yescharacteristic
202#135—42no—
203C2×Q16C2×Dic5911yescharacteristic
204C2×SD16—21no—
205C2×D8C2×Dic5911yescharacteristic
206C22×C8—42no—
207C22×C8C2×Dic5911yescharacteristic
208C22×C8C2×Dic5911yescharacteristic
209C22×C8C22⋊D5911yescharacteristic
210C22×C8C22⋊D5911yescharacteristic
211C22×C8C22⋊D5911yescharacteristic
212C22×C8C22×D5911yescharacteristic
213#123—41no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216#121—21no—
217#121—21no—
218#120—41no—
219C4×D4—41no—
220C4×D4—41no—
221#117—21no—
222C2×C4⋊C4—1182no—
223C2×C22⋊C4—2361no—
224M32—2362no—
225C2×C16—1182no—
226#107—2362no—
227#107C2×Dic5911yescharacteristic
228#107C2×Dic5911yescharacteristic
229#106—21no—
230#103—21no—
231#103—21no—
232#102—21no—
233#102—21no—
234#95—2361no—
235C24#1174566911yescharacteristic
236#41—42no—
237#41—42no—
238C2×Q8—21no—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—2362no—
247C22×C4#1174566911yescharacteristic
248C22×C4#1174567011yescharacteristic
249C22×C4#1174567011yescharacteristic
250C22×C4C22⋊Dic5911yescharacteristic
251C22×C4C22⋊Dic5911yescharacteristic
252C22×C4C2×C22⋊D5911yescharacteristic
253Q16—41no—
254SD16—42no—
255D8—41no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8C22⋊Dic5911yescharacteristic
263C2×C8C22⋊Dic5911yescharacteristic
264C2×C8C22×Dic5911yescharacteristic
265C2×C8C2×C22⋊D5911yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—2362no—
271C4⋊C4—2362no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—4722no—
278C42—41no—
279C16—2362no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#1427364811yescharacteristic
286C23#1427364811yescharacteristic
287C23C2×#1174566911yescharacteristic
288Q8—41no—
289Q8—41no—
290D4—41no—
291D4—41no—
292D4—82no—
293D4—82no—
294D4—81no—
295D4—81no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—82no—
309C2×C4—4722no—
310C2×C4—4722no—
311C2×C4—4722no—
312C2×C4#1427364811yescharacteristic
313C2×C4#1427364811yescharacteristic
314C2×C4C2×C22⋊Dic5911yescharacteristic
315C2×C4C2×#1174567011yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8#1427364212yes—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22C2×#1427364811yescharacteristic
334C22#1683783711yescharacteristic
335C22#1683784211yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—9442no—
343C4C2×#1427364211yescharacteristic
344C4#1683741911yescharacteristic
345C2—21no—
346C2—21no—
347C2—41no—
348C2—41no—
349C2—161no—
350C2#2142266411yescharacteristic
351C2#2142580811yescharacteristic
352C2#2142583211yescharacteristic
subgroupG/H
centerC22C2×#14273648
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
Fitting, prime radicalC59×#3050C2
socleC22×C59C2×#102
cosocleC22×C8×C59C23
hypercenter#3050D59
hypocenterC59#11890
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. #21425707C2semidirect
2. #21424125C2semidirect
3. #21422108C2semidirect
4. #21422105C2semidirect
12. C2×#135×C59C4semidirect
15. C59×#467C4semidirect
17. C59×#466C4semidirect
21. C59×#427C4semidirect
182. C2×#135Dic59semidirect
186. #467Dic59semidirect
188. #466Dic59semidirect
192. #427Dic59semidirect
199. C59#11890semidirect

Maximal subgroups

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

subgroupindex
#214257072
#214241252
#214221082
#214221052
C59×#30502
#1189059