Finite Group Databasebeta

← #27295181

Subgroups — #27295181

Order 13568.

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#20276326C211yesmaximal, characteristic, complemented
2#20274744C211yesmaximal, characteristic, complemented
3#20272727C212yesmaximal, complemented
4#20272724C212yesmaximal, complemented
5C53×#3050C211yesmaximal, characteristic
6C2×#13133157—21no—
7C2×#13133151—21no—
8#16810483—21no—
9C22⋊#11719908—21no—
10C22⋊#11719908C2212yes—
11#16810386C2212yes—
12C2×#135×C53C411yescharacteristic, complemented
13C23×C8×C53C2211yescharacteristic
14C53×#525—21no—
15C53×#467C411yescharacteristic, complemented
16C53×#467C2211yescharacteristic
17C53×#466C411yescharacteristic, complemented
18C53×#466C2211yescharacteristic
19C53×#465—21no—
20C53×#465—21no—
21C53×#427C411yescharacteristic, complemented
22C53×#416—21no—
23C2×C4⋊Dic53—22no—
24C2×C22⋊Dic53—41no—
25#13133157—42no—
26C2×#11719908—22no—
27#13133151—42no—
28#13133124—41no—
29C2×#41×C53—21no—
30C23×C4×C53D411yescharacteristic
31#135×C53—42no—
32C2×Q16×C53C2×C411yescharacteristic
33C2×SD16×C53—21no—
34C2×D8×C53C2×C411yescharacteristic
35C22×C8×C53—42no—
36C22×C8×C53C2×C411yescharacteristic
37C22×C8×C53C2×C411yescharacteristic
38C22×C8×C53D411yescharacteristic
39C22×C8×C53D411yescharacteristic
40C22×C8×C53D411yescharacteristic
41C22×C8×C53C2311yescharacteristic
42#123×C53—41no—
43C22⋊Q8×C53—21no—
44C22⋊Q8×C53—21no—
45#121×C53—21no—
46#121×C53—21no—
47#120×C53—41no—
48C4×D4×C53—41no—
49C4×D4×C53—41no—
50#117×C53—21no—
51#107×C53C2×C411yescharacteristic
52#107×C53C2×C411yescharacteristic
53#106×C53—21no—
54#103×C53—21no—
55#103×C53—21no—
56#102×C53—21no—
57#102×C53—21no—
58C24×C53D811yescharacteristic
59#41×C53—42no—
60#41×C53—42no—
61C2×Q8×C53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53—41no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53D811yescharacteristic
72C22×C4×C53SD1611yescharacteristic
73C22×C4×C53SD1611yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—42no—
76Q16×C53—41no—
77SD16×C53—42no—
78D8×C53—41no—
79C2×C8×C53—22no—
80C2×C8×C53—22no—
81C2×C8×C53—22no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53C22⋊C411yescharacteristic
86C2×C8×C53C22⋊C411yescharacteristic
87C2×C8×C53C22×C411yescharacteristic
88C2×C8×C53C2×D411yescharacteristic
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C42×C53—41no—
99C22⋊Dic53—82no—
100C4⋊Dic53—42no—
101C4⋊Dic53—42no—
102#11719908—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53#10211yescharacteristic
109C23×C53#10211yescharacteristic
110C23×C53C2×D811yescharacteristic
111Q8×C53—41no—
112Q8×C53—41no—
113D4×C53—41no—
114D4×C53—41no—
115D4×C53—82no—
116D4×C53—82no—
117D4×C53—81no—
118D4×C53—81no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—82no—
132C2×C4×C53#10211yescharacteristic
133C2×C4×C53#10211yescharacteristic
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136C2×Dic53—82no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139C8×C53—21no—
140C8×C53—21no—
141C8×C53—21no—
142C8×C53#9912yes—
143#11890—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53C2×#10211yescharacteristic
158C22×C53#50911yescharacteristic
159C22×C53#51011yescharacteristic
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—81no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53C2×#9911yescharacteristic
167C4×C53#42111yescharacteristic
168Dic53—162no—
169#3050D5311yescharacteristic
170#2112—531no—
171#1808—531no—
172#1348—532no—
173#1347—532no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—41no—
177C2×C53—41no—
178C2×C53—161no—
179C2×C53#151911yescharacteristic
180C2×C53#213911yescharacteristic
181C2×C53#214511yescharacteristic
182C2×#135Dic5311yescharacteristic, complemented
183C23×C8C2×D5311yescharacteristic
184#525—21no—
185C2×M32—1061no—
186#467Dic5311yescharacteristic, complemented
187#467C2×D5311yescharacteristic
188#466Dic5311yescharacteristic, complemented
189#466C2×D5311yescharacteristic
190#465—21no—
191#465—21no—
192#427Dic5311yescharacteristic, complemented
193C2×#107—1061no—
194#416—21no—
195#380—1061no—
196C22⋊C16—532no—
197C22⋊C16—1061no—
198#340—532no—
199C53#1189011yescharacteristic, complemented, Sylow
200C2×#41—21no—
201C23×C4C22⋊D5311yescharacteristic
202#135—42no—
203C2×Q16C2×Dic5311yescharacteristic
204C2×SD16—21no—
205C2×D8C2×Dic5311yescharacteristic
206C22×C8—42no—
207C22×C8C2×Dic5311yescharacteristic
208C22×C8C2×Dic5311yescharacteristic
209C22×C8C22⋊D5311yescharacteristic
210C22×C8C22⋊D5311yescharacteristic
211C22×C8C22⋊D5311yescharacteristic
212C22×C8C22×D5311yescharacteristic
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—1062no—
223C2×C22⋊C4—2121no—
224M32—2122no—
225C2×C16—1062no—
226#107—2122no—
227#107C2×Dic5311yescharacteristic
228#107C2×Dic5311yescharacteristic
229#106—21no—
230#103—21no—
231#103—21no—
232#102—21no—
233#102—21no—
234#95—2121no—
235C24#1171992211yescharacteristic
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—2122no—
247C22×C4#1171992211yescharacteristic
248C22×C4#1171992311yescharacteristic
249C22×C4#1171992311yescharacteristic
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22⋊Dic5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
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⋊Dic5311yescharacteristic
263C2×C8C22⋊Dic5311yescharacteristic
264C2×C8C22×Dic5311yescharacteristic
265C2×C8C2×C22⋊D5311yescharacteristic
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—2122no—
271C4⋊C4—2122no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—4242no—
278C42—41no—
279C16—2122no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#1313313811yescharacteristic
286C23#1313313811yescharacteristic
287C23C2×#1171992211yescharacteristic
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—4242no—
310C2×C4—4242no—
311C2×C4—4242no—
312C2×C4#1313313811yescharacteristic
313C2×C4#1313313811yescharacteristic
314C2×C4C2×C22⋊Dic5311yescharacteristic
315C2×C4C2×#1171992311yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8#1313313212yes—
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×#1313313811yescharacteristic
334C22#1681105811yescharacteristic
335C22#1681106311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—8482no—
343C4C2×#1313313211yescharacteristic
344C4#1681064011yescharacteristic
345C2—21no—
346C2—21no—
347C2—41no—
348C2—41no—
349C2—161no—
350C2#2027328311yescharacteristic
351C2#2027642711yescharacteristic
352C2#2027645111yescharacteristic
subgroupG/H
centerC22C2×#13133138
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3050C2
socleC22×C53C2×#102
cosocleC22×C8×C53C23
hypercenter#3050D53
hypocenterC53#11890
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. #20276326C2semidirect
2. #20274744C2semidirect
3. #20272727C2semidirect
4. #20272724C2semidirect
12. C2×#135×C53C4semidirect
15. C53×#467C4semidirect
17. C53×#466C4semidirect
21. C53×#427C4semidirect
182. C2×#135Dic53semidirect
186. #467Dic53semidirect
188. #466Dic53semidirect
192. #427Dic53semidirect
199. C53#11890semidirect

Maximal subgroups

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

subgroupindex
#202763262
#202747442
#202727272
#202727242
C53×#30502
#1189053