Finite Group Databasebeta

← #28310305

Subgroups — #28310305

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D53⋊M32C211yesmaximal, characteristic
2C22×#13133285C211yesmaximal, characteristic
3#20289786C214yesmaximal, complemented
4#427×D53C211yesmaximal, characteristic
5C22×C4⋊#6157—21no—
6C2×#13133292C2212yes—
7D53⋊M32—22no—
8D53⋊M32C414yescomplemented
9C2×#13133291—21no—
10C2×D53⋊C16—21no—
11C2×#13133285—22no—
12C2×#13133285—21no—
13C2×#13133285C414yescomplemented
14C2×#13133285C2212yes—
15C22×C8×D53C2211yescharacteristic
16#117×D53—21no—
17#16810661C411yescharacteristic, complemented
18#16810660C2212yes—
19#16810659—21no—
20#16810658—21no—
21#107×D53C412yescomplemented
22#106×D53—21no—
23C53×#427C411yescharacteristic, complemented
24C23×#6157—41no—
25C2×C4⋊#6157—24no—
26C2×C4⋊#6157—22no—
27C2×C4⋊#6157—41no—
28C2×C4⋊#6157—42no—
29#13133292—22no—
30#13133292C2×C414yescomplemented
31#13133291—24no—
32D53⋊C16—24no—
33C2×#11719910—22no—
34#13133285—24no—
35#13133285—22no—
36#13133285C2×C414yescomplemented
37C22×C4×D53D411yescharacteristic
38C22×#88322D411yescharacteristic
39C2×C8×D53C2×C411yescharacteristic
40C2×C8×D53C2×C412yes—
41C2×C8×D53C2×C412yes—
42C2×C8×D53D411yescharacteristic
43C2×C8×D53D412yes—
44C2×C8×D53D411yescharacteristic
45C2×C8×D53Q812yes—
46C2×C8×D53C2311yescharacteristic
47#13133176—21no—
48#13133175—22no—
49#13133174—41no—
50#13133173—41no—
51C4⋊C4×D53—22no—
52C22⋊C4×D53—41no—
53C42×D53—41no—
54#13133151C2×C412yescomplemented
55#13133150—22no—
56#13133149—21no—
57#13133148—22no—
58C22×C8×C53C2×C411yescharacteristic
59#117×C53—21no—
60#107×C53C2×C412yescomplemented
61#106×C53—21no—
62C23×D53D811yescharacteristic
63C22×#6157—44no—
64C22×#6157—42no—
65C22×#6157—84no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×Dic53SD1611yescharacteristic
68C2×C4×D53—42no—
69C2×C4×D53C22⋊C411yescharacteristic
70C2×C4×D53C22⋊C412yes—
71C2×C4×D53C4⋊C412yes—
72C2×C4×D53D811yescharacteristic
73C2×C4×D53SD1612yes—
74C2×C4×D53SD1611yescharacteristic
75C2×C4×D53Q1612yes—
76C2×C4×D53C2×D411yescharacteristic
77C4⋊#6157—24no—
78C4⋊#6157—44no—
79C4⋊#6157—42no—
80C2×C8×C53C4212yes—
81C2×C8×C53C22⋊C411yescharacteristic
82C2×C8×C53C4⋊C412yes—
83C2×C8×C53C22×C411yescharacteristic
84C4⋊C4×C53—22no—
85C22⋊C4×C53—41no—
86C42×C53—41no—
87C22⋊Dic53—41no—
88#11719921—42no—
89C4⋊Dic53—22no—
90#11719919—42no—
91C4×Dic53—41no—
92C4×Dic53—42no—
93C2×#88322C22⋊C411yescharacteristic
94C2×#88322C22⋊C412yes—
95C2×#88322C4⋊C412yes—
96C2×#88322C2×D411yescharacteristic
97C8×D53C4212yes—
98C8×D53C22⋊C412yes—
99C8×D53C22⋊C412yes—
100C8×D53C4⋊C412yes—
101C8×D53C4⋊C412yes—
102C8×D53C4⋊C412yes—
103C8×D53C22×C411yescharacteristic
104C8×D53C22×C411yescharacteristic
105C8×D53C2×D411yescharacteristic
106C8×D53C2×Q811yescharacteristic
107#11719910—24no—
108C23×C53#10211yescharacteristic
109C22×D53—24no—
110C22×D53#10212yes—
111C22×D53#10211yescharacteristic
112C22×D53#10712yes—
113C22×D53C2×D811yescharacteristic
114C2×#6157—44no—
115C2×#6157—84no—
116C2×#6157—88no—
117C2×C4×C53—42no—
118C2×C4×C53#9512yes—
119C2×C4×C53#10211yescharacteristic
120C2×C4×C53#10312yes—
121C2×C4×C53C2×C22⋊C411yescharacteristic
122C2×Dic53—42no—
123C2×Dic53#10211yescharacteristic
124C2×Dic53#10312yes—
125C2×Dic53#10612yes—
126C2×Dic53C2×SD1611yescharacteristic
127C4×D53—84no—
128C4×D53#9512yes—
129C4×D53#10212yes—
130C4×D53#10312yes—
131C4×D53#10312yes—
132C4×D53#10612yes—
133C4×D53#10712yes—
134C4×D53C2×C22⋊C411yescharacteristic
135C4×D53C2×C4⋊C411yescharacteristic
136C4×D53C2×SD1611yescharacteristic
137C4×D53C2×Q1611yescharacteristic
138C8×C53#9512yes—
139C8×C53C2×C4211yescharacteristic
140C8×C53C2×C4⋊C411yescharacteristic
141#88322#9512yes—
142#88322C2×C22⋊C411yescharacteristic
143#88322C2×C4⋊C411yescharacteristic
144C2×#1386—531nomaximal, Sylow
145C22×C53—22no—
146C22×C53#34012yes—
147C22×C53C2×#10211yescharacteristic
148C2×D53—22no—
149C2×D53—24no—
150C2×D53—24no—
151C2×D53#34012yes—
152C2×D53#36014yes—
153C2×D53C2×#10211yescharacteristic
154C2×D53C2×#10711yescharacteristic
155#6157—84no—
156C4×C53—82no—
157C4×C53#34012yes—
158C4×C53C2×#9511yescharacteristic
159C4×C53C2×#10311yescharacteristic
160Dic53—82no—
161Dic53#34012yes—
162Dic53C2×#10311yescharacteristic
163Dic53C2×#10611yescharacteristic
164C22×M32—531no—
165C2×#427—531no—
166C22×#106—531no—
167#1386—534no—
168C2×C53—22no—
169C2×C53C2×#34011yescharacteristic
170C2×C53C2×#36012yes—
171D53—21no—
172D53—21no—
173D53#138614yescomplemented
174C23×C8—531no—
175C2×#117—1061no—
176C22×C4⋊C4—1061no—
177C2×M32—532no—
178C2×M32—534no—
179C2×M32—1061no—
180C2×M32—1062no—
181C22×C16—1061no—
182#427—531no—
183#427—532no—
184#427—1061no—
185#427—1061no—
186#427#615711yescharacteristic, complemented
187C2×#107—532no—
188C2×#106—532no—
189C2×#106—534no—
190C2×#106—1061no—
191C2×#106—1062no—
192C2×#106—1061no—
193C53C2×#138611yescharacteristic, complemented, Sylow
194C23×C4—531no—
195C23×C4—2121no—
196C22×C8—531no—
197C22×C8—532no—
198C22×C8—532no—
199C22×C8—532no—
200C22×C8—532no—
201C22×C8—531no—
202C22×C8—531no—
203C22×C8—531no—
204C22×C8—531no—
205C22×C8C2×#615711yescharacteristic
206#117—21no—
207#117—1062no—
208#117—1061no—
209#117—2121no—
210#117—2121no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1064no—
213C2×C4⋊C4—1062no—
214C2×C4⋊C4—2121no—
215C2×C4⋊C4—2122no—
216C2×C22⋊C4—2121no—
217C2×C42—2121no—
218M32—534no—
219M32—1064no—
220M32—1062no—
221C2×C16—1062no—
222C2×C16—1064no—
223#107—532no—
224#107—1062no—
225#107C2×#615712yescomplemented
226#106—21no—
227#106—534no—
228#106—1062no—
229#106—1064no—
230#106—1062no—
231#106—1061no—
232C24—531no—
233C22×C4—531no—
234C22×C4—532no—
235C22×C4—532no—
236C22×C4—531no—
237C22×C4—531no—
238C22×C4—532no—
239C22×C4—532no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—4244no—
243C22×C4—2122no—
244C22×C4—2122no—
245C22×C4—2124no—
246C22×C4C22⋊#615711yescharacteristic
247C2×C8—532no—
248C2×C8—532no—
249C2×C8—531no—
250C2×C8—531no—
251C2×C8—532no—
252C2×C8—532no—
253C2×C8—532no—
254C2×C8—532no—
255C2×C8—531no—
256C2×C8—531no—
257C2×C8—532no—
258C2×C8—531no—
259C2×C8—532no—
260C2×C8—531no—
261C2×C8C4×#615712yes—
262C2×C8C4⋊#615712yes—
263C2×C8C22⋊#615711yescharacteristic
264C2×C8C22×#615711yescharacteristic
265C4⋊C4—22no—
266C4⋊C4—1064no—
267C4⋊C4—1062no—
268C4⋊C4—2122no—
269C4⋊C4—2122no—
270C4⋊C4—2124no—
271C22⋊C4—41no—
272C22⋊C4—2122no—
273C22⋊C4—2121no—
274C42—41no—
275C42—2122no—
276C42—2121no—
277C16—1064no—
278C23—531no—
279C23—531no—
280C23—532no—
281C23—532no—
282C23—1064no—
283C23#1313327911yescharacteristic
284C2×C4—42no—
285C2×C4—531no—
286C2×C4—531no—
287C2×C4—532no—
288C2×C4—532no—
289C2×C4—532no—
290C2×C4—531no—
291C2×C4—531no—
292C2×C4—532no—
293C2×C4—531no—
294C2×C4—531no—
295C2×C4—532no—
296C2×C4—532no—
297C2×C4—532no—
298C2×C4—532no—
299C2×C4—4244no—
300C2×C4—4248no—
301C2×C4—4244no—
302C2×C4—2124no—
303C2×C4—2122no—
304C2×C4(C2×#6157)⋊C412yes—
305C2×C4#1313327911yescharacteristic
306C2×C4#1313328012yes—
307C2×C4C2×C22⋊#615711yescharacteristic
308C8—532no—
309C8—531no—
310C8—531no—
311C8(C2×#6157)⋊C412yes—
312C8C2×C4×#615711yescharacteristic
313C8C2×C4⋊#615711yescharacteristic
314C22—22no—
315C22—532no—
316C22—534no—
317C22—531no—
318C22—531no—
319C22—1064no—
320C22—1064no—
321C22—1062no—
322C22#1681149612yes—
323C22C2×#1313327911yescharacteristic
324C4—82no—
325C4—532no—
326C4—531no—
327C4—531no—
328C4—4244no—
329C4—4242no—
330C4#1681149612yes—
331C4C2×(C2×#6157)⋊C411yescharacteristic
332C4C2×#1313328011yescharacteristic
333C2—22no—
334C2—534no—
335C2—1061no—
336C2—1061no—
337C2C2×#1681149611yescharacteristic
338C2D53⋊#36012yes—
subgroupG/H
centerC22C2×#13133279
commutatorC8×C53C2×C42
FrattiniC2×C8C22×#6157
FittingC53×#427C4
socleC22×C53C2×#102
prime radicalC23×D53D8
cosocleC2×C8×D53C23
hypercenter#427#6157
hypocenterC53C2×#1386
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #20289786C2semidirect
8. D53⋊M32C4semidirect
13. C2×#13133285C4semidirect
17. #16810661C4semidirect
21. #107×D53C4semidirect
23. C53×#427C4semidirect
30. #13133292C2×C4semidirect
36. #13133285C2×C4semidirect
54. #13133151C2×C4semidirect
60. #107×C53C2×C4semidirect
173. D53#1386semidirect
186. #427#6157semidirect
193. C53C2×#1386semidirect
225. #107C2×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×D53⋊M322
C22×#131332852
#202897862
#427×D532
C2×#138653