Finite Group Databasebeta

← #27277261

Subgroups — #27277261

Order 13568.

454 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
1#20274735C212yesmaximal
2C4×#13133151C211yesmaximal, characteristic, complemented
3#20273474C211yesmaximal, characteristic, complemented
4#20272617C212yesmaximal, complemented
5C8×#41×C53C211yesmaximal, characteristic
6C2×#13133151—21no—
7#16810647C2212yes—
8C2×#13133144—22no—
9C4×C4⋊Dic53C2212yes—
10#16810386—21no—
11#16810382—22no—
12#16810376—21no—
13#16810376—21no—
14C4×#41×C53C2211yescharacteristic
15C8×Q8×C53C411yescharacteristic, complemented
16C8×Q8×C53C411yescharacteristic, complemented
17C8×D4×C53—21no—
18C8×D4×C53—21no—
19C8×D4×C53C411yescharacteristic, complemented
20C8×D4×C53C411yescharacteristic, complemented
21C53×#431—21no—
22C53×#431C2211yescharacteristic
23C2×C4×C8×C53—21no—
24C2×C4×C8×C53C2211yescharacteristic
25C2×#11719917—22no—
26C2×C4⋊Dic53—22no—
27C2×C4×Dic53—22no—
28#13133151—24no—
29C4⋊#88322—22no—
30C4⋊#88322—22no—
31#13133144—44no—
32#13133124—22no—
33C2×#41×C53D411yescharacteristic
34C22×C8×C53—21no—
35C22×C8×C53—21no—
36C22×C8×C53D411yescharacteristic
37C22×C8×C53D411yescharacteristic
38C4×Q8×C53C2×C411yescharacteristic
39C4×Q8×C53Q811yescharacteristic
40C4×D4×C53—21no—
41C4×D4×C53—21no—
42C4×D4×C53C2×C411yescharacteristic
43C4×D4×C53Q811yescharacteristic
44#117×C53—21no—
45#117×C53D411yescharacteristic
46C2×C42×C53—21no—
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—21no—
49C4⋊C8×C53—21no—
50C4⋊C8×C53C2×C411yescharacteristic
51C4⋊C8×C53C2×C411yescharacteristic
52C22⋊C8×C53—21no—
53C22⋊C8×C53—21no—
54C22⋊C8×C53—21no—
55C4×C8×C53—21no—
56C4×C8×C53—21no—
57C4×C8×C53—21no—
58C4×C8×C53C2×C411yescharacteristic
59C4×C8×C53C2×C411yescharacteristic
60C4×C8×C53D411yescharacteristic
61C4×C8×C53D411yescharacteristic
62#41×C53—21no—
63#41×C53—21no—
64#41×C53D812yes—
65#41×C53Q1612yes—
66C2×Q8×C53C4⋊C411yescharacteristic
67C2×D4×C53—21no—
68C2×D4×C53C4⋊C411yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4111yescharacteristic
75C22×Dic53—42no—
76C2×C8×C53—22no—
77C2×C8×C53—21no—
78C2×C8×C53—21no—
79C2×C8×C53—41no—
80C2×C8×C53—42no—
81C2×C8×C53—42no—
82C2×C8×C53C22⋊C411yescharacteristic
83C2×C8×C53C22⋊C411yescharacteristic
84C2×C8×C53C22⋊C411yescharacteristic
85C2×C8×C53C22⋊C411yescharacteristic
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53C4⋊C411yescharacteristic
89C4⋊C4×C53C4⋊C411yescharacteristic
90C22⋊C4×C53—21no—
91C22⋊C4×C53—21no—
92C22⋊C4×C53—21no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53C22×C411yescharacteristic
97C42×C53C2×D411yescharacteristic
98C42×C53C2×Q811yescharacteristic
99C42×C53#4111yescharacteristic
100C4⋊Dic53—24no—
101C4⋊Dic53—42no—
102C4×Dic53—42no—
103C4×Dic53—42no—
104#11719917—42no—
105#11719917—42no—
106C2×#88322—42no—
107C23×C53—21no—
108C23×C53#12111yescharacteristic
109Q8×C53—21no—
110Q8×C53#10712yescomplemented
111D4×C53—22no—
112D4×C53—22no—
113D4×C53—21no—
114D4×C53#10712yescomplemented
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53C2×C22⋊C411yescharacteristic
127C2×C4×C53C2×C4⋊C411yescharacteristic
128C2×C4×C53#11711yescharacteristic
129C2×C4×C53#12111yescharacteristic
130C2×C4×C53C22⋊Q811yescharacteristic
131C2×C4×C53C22⋊Q811yescharacteristic
132C2×C4×C53C2×D811yescharacteristic
133C2×C4×C53C2×Q1611yescharacteristic
134C2×C4×C53#13511yescharacteristic
135C2×C4×C53#13511yescharacteristic
136C2×Dic53—44no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139C8×C53—21no—
140C8×C53—21no—
141C8×C53—21no—
142C8×C53—41no—
143C8×C53—41no—
144C8×C53#10412yescomplemented
145#88322—82no—
146#8667—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—22no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53#38011yescharacteristic
153C22×C53#46611yescharacteristic
154C22×C53#46711yescharacteristic
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53C2×#10411yescharacteristic
162C4×C53#42111yescharacteristic
163C4×C53C2×#10711yescharacteristic
164C4×C53#42711yescharacteristic
165C4×C53#46611yescharacteristic
166C4×C53#46711yescharacteristic
167Dic53—84no—
168C8×#41D5311yescharacteristic
169#1806—532no—
170C4×#107—531no—
171#1567—531no—
172#1303—532no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—42no—
176C2×C53#180611yescharacteristic
177C2×C53#180811yescharacteristic
178C2×C53#181411yescharacteristic
179C4×#41C2×D5311yescharacteristic
180C8×Q8Dic5311yescharacteristic, complemented
181C8×Q8Dic5311yescharacteristic, complemented
182C8×D4—21no—
183C8×D4—21no—
184C8×D4Dic5311yescharacteristic, complemented
185C8×D4Dic5311yescharacteristic, complemented
186#431—21no—
187#431C2×D5311yescharacteristic
188C2×#107—1061no—
189#423—532no—
190C2×#104—1062no—
191C2×C4×C8—21no—
192C2×C4×C8C2×D5311yescharacteristic
193C4×C4⋊C4—532no—
194#340—1061no—
195#339—1062no—
196#335—1061no—
197#335—1061no—
198C53#866711yescharacteristic, complemented, Sylow
199C2×#41D21211yescharacteristic
200C2×M16—1062no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8C22⋊D5311yescharacteristic
204C22×C8C22⋊D5311yescharacteristic
205C4×Q8Dic10611yescharacteristic
206C4×Q8C2×Dic5311yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4Dic10611yescharacteristic
210C4×D4C2×Dic5311yescharacteristic
211#117—21no—
212#117D21211yescharacteristic
213C2×C4⋊C4—1062no—
214C2×C42—21no—
215C2×C42—1062no—
216C2×C42C22×D5311yescharacteristic
217#107—1064no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—1062no—
221C4⋊C8—1062no—
222C4⋊C8C2×Dic5311yescharacteristic
223C4⋊C8C2×Dic5311yescharacteristic
224#104—2124no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8C2×Dic5311yescharacteristic
232C4×C8C2×Dic5311yescharacteristic
233C4×C8C22⋊D5311yescharacteristic
234C4×C8C22⋊D5311yescharacteristic
235#95—1062no—
236#41—21no—
237#41—21no—
238#41D42412yes—
239#41Dic21212yes—
240C2×Q8C4⋊Dic5311yescharacteristic
241C2×D4—21no—
242C2×D4C4⋊Dic5311yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—2122no—
247C22×C4C2×D21211yescharacteristic
248C22×C4#1171994511yescharacteristic
249C22×C4C2×C22⋊D5311yescharacteristic
250M16—2122no—
251M16—2122no—
252C2×C8—22no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—2122no—
259C2×C8C22⋊Dic5311yescharacteristic
260C2×C8C22⋊Dic5311yescharacteristic
261C2×C8C22⋊Dic5311yescharacteristic
262C2×C8C22⋊Dic5311yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—1064no—
266C4⋊C4—2122no—
267C4⋊C4C4⋊Dic5311yescharacteristic
268C4⋊C4C4⋊Dic5311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—2122no—
276C42—2122no—
277C42C2×Dic10611yescharacteristic
278C42#1171994511yescharacteristic
279C42C22×Dic5311yescharacteristic
280C42C2×C22⋊D5311yescharacteristic
281C23—21no—
282C23#1313319111yescharacteristic
283Q8—21no—
284Q8#1313315112yescomplemented
285D4—22no—
286D4—22no—
287D4—21no—
288D4#1313315112yescomplemented
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—4242no—
301C2×C4—4242no—
302C2×C4—2124no—
303C2×C4C2×C22⋊Dic5311yescharacteristic
304C2×C4C2×C4⋊Dic5311yescharacteristic
305C2×C4#1313317611yescharacteristic
306C2×C4#1313319111yescharacteristic
307C2×C4#1313319611yescharacteristic
308C2×C4#1313319611yescharacteristic
309C2×C4C2×D42411yescharacteristic
310C2×C4C2×Dic21211yescharacteristic
311C2×C4#1313323911yescharacteristic
312C2×C4#1313323911yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—41no—
317C8—41no—
318C8—4242no—
319C8#1313314412yescomplemented
320C22—21no—
321C22—21no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22#1681048311yescharacteristic
326C22#1681085911yescharacteristic
327C22#1681086411yescharacteristic
328C4—21no—
329C4—22no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—4244no—
335C4C2×#1313314411yescharacteristic
336C4#1681064011yescharacteristic
337C4C2×#1313315111yescharacteristic
338C4#1681066111yescharacteristic
339C4#1681085911yescharacteristic
340C4#1681086411yescharacteristic
341C2—21no—
342C2—21no—
343C2—42no—
344C2#2027473511yescharacteristic
345C2#2027474411yescharacteristic
346C2#2027476911yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC8×#41×C53C2
socleC22×C53#380
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenterC8×#41D53
hypocenterC53#8667
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. C4×#13133151C2semidirect
3. #20273474C2semidirect
4. #20272617C2semidirect
15. C8×Q8×C53C4semidirect
16. C8×Q8×C53C4semidirect
19. C8×D4×C53C4semidirect
20. C8×D4×C53C4semidirect
110. Q8×C53#107semidirect
114. D4×C53#107semidirect
144. C8×C53#104semidirect
180. C8×Q8Dic53semidirect
181. C8×Q8Dic53semidirect
184. C8×D4Dic53semidirect
185. C8×D4Dic53semidirect
198. C53#8667semidirect
284. Q8#13133151semidirect
288. D4#13133151semidirect
319. C8#13133144semidirect

Maximal subgroups

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

subgroupindex
#202747352
C4×#131331512
#202734742
#202726172
C8×#41×C532
#866753