Finite Group Databasebeta

← #357×#6157 #28310665

Subgroups — #357×#6157 #28310665

Order 13568.

534 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D8×#6157C211yesmaximal, characteristic
2C2×#16811535C211yesmaximal, characteristic
3C2×C16×#6157C211yesmaximal, characteristic, complemented
4#107×#6157C211yesmaximal, characteristic, complemented
5#20289773C211yesmaximal, characteristic, complemented
6#20289772C211yesmaximal, characteristic, complemented
7#357×D53C211yesmaximal, characteristic
8C2×D4×#6157—21no—
9C2×D53⋊C16C2211yescharacteristic
10D8×#6157—22no—
11D8×#6157C414yescomplemented
12C2×#13133286C2211yescharacteristic
13C2×#13133286C2211yescharacteristic
14C2×#13133279—21no—
15C2×C8×#6157C2211yescharacteristic
16C4⋊C4×#6157—21no—
17#16811535C414yescomplemented
18C16×#6157—22no—
19#16811496—21no—
20C2×D8×D53C2211yescharacteristic
21C2×C16×D53C2211yescharacteristic
22#107×D53C2211yescharacteristic
23#16810425C411yescharacteristic, complemented
24#16810424C411yescharacteristic, complemented
25#16810423C411yescharacteristic, complemented
26C53×#357C411yescharacteristic, complemented
27C23×#6157—41no—
28C2×D53⋊C8D411yescharacteristic
29D4×#6157—24no—
30D4×#6157—42no—
31C2×C4⋊#6157—21no—
32C2×C4⋊#6157—21no—
33C2×C22⋊#6157—41no—
34C2×C4×#6157—41no—
35C2×C4×#6157D411yescharacteristic
36D53⋊C16—21no—
37D53⋊C16C2×C412yescomplemented
38C2×#11719910—21no—
39#13133286—21no—
40#13133286—21no—
41#13133286C2×C412yes—
42#13133286C2×C412yescomplemented
43#13133279—24no—
44C8×#6157C2×C412yes—
45C8×#6157D412yes—
46(C2×#6157)⋊C4—41no—
47C2×D4×D53—21no—
48C2×D424C2×C411yescharacteristic
49C2×#11719922—21no—
50D8×D53—21no—
51D8×D53—21no—
52D8×D53C2×C412yes—
53D8×D53C2×C412yes—
54C2×C8×D53C2311yescharacteristic
55C4⋊C4×D53—21no—
56C2×#11719908C2×C411yescharacteristic, complemented
57C16×D53—21no—
58C16×D53—21no—
59#13133151C2×C411yescharacteristic, complemented
60#13133150—21no—
61C2×D8×C53C2×C411yescharacteristic
62C2×C16×C53C2×C411yescharacteristic, complemented
63#107×C53C2×C411yescharacteristic, complemented
64C23×D53—41no—
65C22×#6157—41no—
66C22×#6157—41no—
67C22×#6157—44no—
68C22×#6157—82no—
69C22×#6157—82no—
70C22×#6157D811yescharacteristic
71C22×#6157D811yescharacteristic
72C2×D4×C53—21no—
73C2×C22⋊D53—41no—
74D4×D53—22no—
75D4×D53—22no—
76D4×D53—41no—
77D4×D53—41no—
78C2×D212—21no—
79C2×C4×D53—41no—
80C2×C4×D53C2×D411yescharacteristic
81C22⋊#6157—44no—
82C2×#88324—21no—
83C4⋊#6157—22no—
84C4⋊#6157—22no—
85C4⋊#6157—41no—
86C4⋊#6157—41no—
87C4×#6157—82no—
88C4×#6157C22⋊C412yes—
89C4×#6157SD1612yes—
90D53⋊C8—21no—
91D53⋊C8C22⋊C412yes—
92D8×C53—21no—
93D8×C53C4212yescomplemented
94C2×C8×C53C22×C411yescharacteristic
95C4⋊C4×C53—21no—
96#11719922—22no—
97#11719922—22no—
98C4⋊Dic53—21no—
99#11719919—41no—
100C2×#88322#4111yescharacteristic
101D424—21no—
102D424C4212yescomplemented
103C8×D53C22×C411yescharacteristic
104C8×D53C22×C411yescharacteristic
105C8×D53C2×D411yescharacteristic
106C8×D53#4111yescharacteristic
107#11719910—22no—
108C16×C53—21no—
109#11719908—21no—
110C23×C53—41no—
111C22×D53—41no—
112C22×D53—42no—
113C22×D53—42no—
114C22×D53—81no—
115C22×D53—81no—
116C22×D53—81no—
117C22×D53—81no—
118C22×D53C2×D811yescharacteristic
119C2×#6157—21no—
120C2×#6157—21no—
121C2×#6157—42no—
122C2×#6157—42no—
123C2×#6157—81no—
124C2×#6157—81no—
125C2×#6157—81no—
126C2×#6157—81no—
127C2×#6157—88no—
128C2×#6157#10212yes—
129C2×#6157#10212yes—
130C2×#6157D1612yes—
131C2×#6157SD3212yes—
132D4×C53—22no—
133D4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C4×D411yescharacteristic
136C22⋊D53—42no—
137C22⋊D53—42no—
138C2×Dic53—41no—
139C2×Dic53#13511yescharacteristic
140D212—22no—
141D212—41no—
142C4×D53—81no—
143C4×D53—81no—
144C4×D53C2×C22⋊C411yescharacteristic
145C4×D53C4×D411yescharacteristic
146C4×D53C2×SD1611yescharacteristic
147C4×D53#13511yescharacteristic
148#88324—22no—
149C8×C53C2×C4211yescharacteristic
150C8×C53C4×D411yescharacteristic
151#88322#11711yescharacteristic
152#88322C4×D411yescharacteristic
153C4×#357—531nomaximal, Sylow
154C22×C53—42no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53C4×D811yescharacteristic
158C2×D53—42no—
159C2×D53—81no—
160C2×D53—81no—
161C2×D53—82no—
162C2×D53—82no—
163C2×D53—82no—
164C2×D53—82no—
165C2×D53C2×#10211yescharacteristic
166C2×D53C4×D811yescharacteristic
167C2×D53C2×D1611yescharacteristic
168C2×D53C2×SD3211yescharacteristic
169C2×D53#50811yescharacteristic
170C2×D53#50811yescharacteristic
171#6157—22no—
172#6157—82no—
173#6157—82no—
174#6157#35714yesdirect product factor
175C4×C53—81no—
176C4×C53C4×C22⋊C411yescharacteristic
177C4×C53C4×SD1611yescharacteristic
178Dic53—81no—
179Dic53#41611yescharacteristic
180Dic53C4×SD1611yescharacteristic
181C2×C4×D8—531no—
182C2×#357—531no—
183C2×#357—531no—
184C2×C4×C16—531no—
185C4×#107—531no—
186#1380—531no—
187#1380—531no—
188C2×C53—82no—
189C2×C53C4×#10211yescharacteristic
190C2×C53C4×D1611yescharacteristic
191C2×C53C4×SD3211yescharacteristic
192D53—82no—
193D53C2×#35711yescharacteristic
194D53#213811yescharacteristic
195D53C4×D1611yescharacteristic
196D53C4×SD3211yescharacteristic
197C22×D8—531no—
198C2×C4×D4—1061no—
199C22×C16—531no—
200C22×C16—531no—
201C4×D8—534no—
202C4×D8—1062no—
203C2×#107—531no—
204C2×#107—531no—
205C2×#107—531no—
206C2×#102—1061no—
207C2×C4×C8—531no—
208C4×C4⋊C4—1061no—
209#357—534no—
210#357—531no—
211#357—531no—
212#357—531no—
213#357#615711yescharacteristic, direct product factor
214C4×C16—1062no—
215#340—1061no—
216C53C4×#35711yescharacteristic, complemented, Sylow
217C22×D4—1061no—
218C23×C4—2121no—
219C2×D8—532no—
220C2×D8—532no—
221C2×D8—531no—
222C2×D8—1061no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8C2×#615711yescharacteristic
226C22×C8—531no—
227C22×C8—531no—
228C4×D4—1064no—
229C4×D4—2122no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C22⋊C4—2121no—
234C2×C42—531no—
235C2×C42—2121no—
236C2×C16—532no—
237C2×C16—531no—
238C2×C16—1061no—
239C2×C16—1061no—
240C2×C16—1061no—
241C2×C16—1061no—
242C2×C16C2×#615711yescharacteristic, complemented
243#107—532no—
244#107—532no—
245#107—531no—
246#107—1061no—
247#107—1061no—
248#107—1061no—
249#107C2×#615711yescharacteristic, complemented
250#102—1064no—
251C4×C8—532no—
252C4×C8—532no—
253#95—2121no—
254C24—2121no—
255C2×D4—21no—
256C2×D4—1062no—
257C2×D4—1062no—
258C2×D4—1061no—
259C2×D4—2121no—
260C2×D4—2121no—
261C2×D4—2121no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—4242no—
266C22×C4—4242no—
267C22×C4—2121no—
268C22×C4—2124no—
269C22×C4—2121no—
270C22×C4—2121no—
271D8—21no—
272D8—532no—
273D8—1062no—
274D8—1062no—
275D8—1061no—
276D8C4×#615712yescomplemented
277C2×C8—531no—
278C2×C8—531no—
279C2×C8—531no—
280C2×C8—531no—
281C2×C8—531no—
282C2×C8—532no—
283C2×C8—1061no—
284C2×C8—1061no—
285C2×C8C22×#615711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—1062no—
288C4⋊C4—1062no—
289C4⋊C4—1061no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C22⋊C4—2124no—
294C42—532no—
295C42—532no—
296C42—4242no—
297C16—21no—
298C16—1062no—
299C16—1061no—
300C23—41no—
301C23—531no—
302C23—4241no—
303C23—4241no—
304C23—4241no—
305C23—4241no—
306C23—2122no—
307C23—2122no—
308C23—2121no—
309D4—22no—
310D4—41no—
311D4—1062no—
312D4—2122no—
313D4—2122no—
314D4—2121no—
315C2×C4—41no—
316C2×C4—532no—
317C2×C4—532no—
318C2×C4—532no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—531no—
322C2×C4—531no—
323C2×C4—531no—
324C2×C4—532no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—4241no—
328C2×C4—4241no—
329C2×C4—4248no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2121no—
337C2×C4D4×#615711yescharacteristic
338C8—531no—
339C8—531no—
340C8—1062no—
341C8C2×C4×#615711yescharacteristic
342C8D4×#615711yescharacteristic
343C22—42no—
344C22—81no—
345C22—81no—
346C22—531no—
347C22—531no—
348C22—531no—
349C22—531no—
350C22—531no—
351C22—531no—
352C22—4242no—
353C22—4242no—
354C22—4242no—
355C22—4242no—
356C22—4241no—
357C22—4241no—
358C22—2122no—
359C22D8×#615711yescharacteristic
360C4—81no—
361C4—534no—
362C4—531no—
363C4—531no—
364C4—1062no—
365C4—4242no—
366C4—4242no—
367C4—4241no—
368C4C22⋊C4×#615711yescharacteristic
369C4SD16×#615711yescharacteristic
370C2—82no—
371C2—531no—
372C2—531no—
373C2—531no—
374C2—531no—
375C2—4242no—
376C2#102×#615711yescharacteristic
377C2D16×#615711yescharacteristic
378C2SD32×#615711yescharacteristic
subgroupG/H
centerC22D8×#6157
commutatorC8×C53C2×C42
FrattiniC2×C8C22×#6157
FittingC53×#357C4
socleC22×C53C4×D8
prime radicalC2×D8×D53C22
cosocleC2×C8×D53C23
hypercenter#357#6157
hypocenterC53C4×#357
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ #357

Complements

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

NH ≅ G/Ntype
3. C2×C16×#6157C2semidirect
4. #107×#6157C2semidirect
5. #20289773C2semidirect
6. #20289772C2semidirect
11. D8×#6157C4semidirect
17. #16811535C4semidirect
23. #16810425C4semidirect
24. #16810424C4semidirect
25. #16810423C4semidirect
26. C53×#357C4semidirect
37. D53⋊C16C2×C4semidirect
42. #13133286C2×C4semidirect
56. C2×#11719908C2×C4semidirect
59. #13133151C2×C4semidirect
62. C2×C16×C53C2×C4semidirect
63. #107×C53C2×C4semidirect
93. D8×C53C42semidirect
102. D424C42semidirect
174. #6157#357direct (N × H)
213. #357#6157direct (N × H)
216. C53C4×#357semidirect
242. C2×C16C2×#6157semidirect
249. #107C2×#6157semidirect
276. D8C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×D8×#61572
C2×#168115352
C2×C16×#61572
#107×#61572
#202897732
#202897722
#357×D532
C4×#35753