Finite Group Databasebeta

← #27301034

Subgroups — #27301034

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810860C211yesmaximal, characteristic
2C2×#16810660C211yesmaximal, characteristic, complemented
3C2×#16810425C211yesmaximal, characteristic, complemented
4C2×#16810423C211yesmaximal, characteristic, complemented
5C2×#16810402C211yesmaximal, characteristic, complemented
6#20272806C211yesmaximal, characteristic, complemented
7C53×#1380C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×C8×D53C2211yescharacteristic
10C2×#13133192—21no—
11C2×#13133175—21no—
12C22×#11719908C2211yescharacteristic
13#16810860—22no—
14#16810860C414yescomplemented
15#16810660—22no—
16#16810660—22no—
17C2×#13133151C2211yescharacteristic
18C2×#13133148—21no—
19C2×#13133137—21no—
20#16810425—22no—
21#16810423—22no—
22#16810402—22no—
23C22×C16×C53C2211yescharacteristic
24C2×#107×C53C2211yescharacteristic
25C2×#107×C53C2211yescharacteristic
26C22×D212—21no—
27C22×D212—41no—
28C22×C4×D53D411yescharacteristic
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424C2×C412yes—
35C22×#88322D411yescharacteristic
36C2×C8×D53—21no—
37C2×C8×D53—21no—
38C2×C8×D53—21no—
39C2×C8×D53—21no—
40C2×C8×D53C2×C412yes—
41C2×C8×D53D412yes—
42#13133192—24no—
43#13133192—42no—
44#13133175—24no—
45#13133175—42no—
46C2×C4⋊Dic53—21no—
47C2×#11719921—41no—
48C2×#11719921—41no—
49C2×C4×Dic53—41no—
50C2×#11719908—21no—
51C2×#11719908—21no—
52C2×#11719908—21no—
53#13133151—21no—
54#13133151D412yescomplemented
55#13133148—24no—
56#13133137—24no—
57C22×C8×C53C2311yescharacteristic
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C16×C53—21no—
61C2×C16×C53—21no—
62C2×C16×C53—21no—
63#107×C53—21no—
64#107×C53—21no—
65#107×C53C2×C412yes—
66#107×C53D412yescomplemented
67C23×D53—41no—
68C23×D53D811yescharacteristic
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—41no—
73C22×Dic53D811yescharacteristic
74C2×D212—22no—
75C2×D212—41no—
76C2×D212—41no—
77C2×D212—41no—
78C2×D212—41no—
79C2×D212—44no—
80C2×D212—41no—
81C2×D212—82no—
82C2×D212—82no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×C4×D53C22⋊C412yes—
88C2×C4×D53SD1612yes—
89C2×C8×C53C22×C411yescharacteristic
90C2×C8×C53C2×D411yescharacteristic
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53#4111yescharacteristic
94C2×C8×C53#4111yescharacteristic
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99#11719921—44no—
100#11719921—44no—
101C4⋊Dic53—22no—
102C4⋊Dic53—41no—
103C4×Dic53—44no—
104C2×#88322—21no—
105C2×#88322—21no—
106C2×#88322C22⋊C412yes—
107D424—22no—
108D424—22no—
109D424—41no—
110D424—41no—
111C8×D53—22no—
112C8×D53—22no—
113C8×D53—22no—
114C8×D53—22no—
115C16×C53—41no—
116#11719908—41no—
117C23×C53C2×D811yescharacteristic
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—42no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—81no—
131C22×D53#10212yes—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53C2×C22⋊C411yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53C2×SD1611yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—42no—
147C2×Dic53—81no—
148C2×Dic53—81no—
149C2×Dic53#10212yes—
150D212—42no—
151D212—42no—
152D212—88no—
153D212—81no—
154D212—81no—
155C4×D53—22no—
156C4×D53—22no—
157C4×D53—22no—
158C4×D53—22no—
159C8×C53C4×D411yescharacteristic
160C8×C53C4×D411yescharacteristic
161C8×C53#12111yescharacteristic
162C8×C53#12111yescharacteristic
163#88322—22no—
164#12992—531nomaximal, Sylow
165C22×C53C2×#10211yescharacteristic
166C22×C53#44711yescharacteristic
167C22×C53#48011yescharacteristic
168C22×C53#50811yescharacteristic
169C22×C53#50811yescharacteristic
170C22×C53#50911yescharacteristic
171C22×C53#50911yescharacteristic
172C2×D53—22no—
173C2×D53—22no—
174C2×D53—22no—
175C2×D53—41no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—82no—
180C2×D53—82no—
181C2×D53—82no—
182C2×D53—161no—
183C2×D53—161no—
184C2×D53—161no—
185C2×D53—161no—
186C4×C53—82no—
187C4×C53—82no—
188C4×C53#38611yescharacteristic
189C4×C53#41711yescharacteristic
190C4×C53#44811yescharacteristic
191C4×C53#48111yescharacteristic
192Dic53—22no—
193Dic53—82no—
194C2×#466—531no—
195C2×#427—531no—
196C2×#357—531no—
197C2×#357—531no—
198C2×C22⋊C16—531no—
199#1380—531no—
200#1380D5311yescharacteristic, complemented
201C2×C53#187411yescharacteristic
202C2×C53#213811yescharacteristic
203C2×C53#214011yescharacteristic
204C2×C53#218611yescharacteristic
205C2×C53#218611yescharacteristic
206C2×C53#223411yescharacteristic
207C2×C53#223411yescharacteristic
208D53—42no—
209D53—162no—
210C22×D8—531no—
211C23×C8—531no—
212C2×#121—1061no—
213C2×#117—1061no—
214C22×C16—531no—
215C22×C16C2×D5311yescharacteristic
216#466—534no—
217#466—1062no—
218#427—1062no—
219#427—1062no—
220C2×#107—531no—
221C2×#107C2×D5311yescharacteristic
222C2×#107C2×D5311yescharacteristic
223C2×#106—1061no—
224C2×#102—1061no—
225#357—1062no—
226#357—1062no—
227C22⋊C16—1062no—
228C53#1299211yescharacteristic, complemented, Sylow
229C22×D4—1061no—
230C22×D4—2121no—
231C23×C4—531no—
232C2×D8—532no—
233C2×D8—1061no—
234C2×D8—1061no—
235C2×D8—1061no—
236C2×D8—1061no—
237C2×D8—1061no—
238C22×C8—531no—
239C22×C8—532no—
240C22×C8—532no—
241C22×C8—1061no—
242C22×C8—1061no—
243C22×C8—1061no—
244C22×C8—1061no—
245C22×C8C22×D5311yescharacteristic
246#121—1064no—
247#121—2122no—
248#117—1064no—
249#117—2122no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—1061no—
253C2×C22⋊C4—2121no—
254C2×C22⋊C4—2121no—
255C2×C42—2121no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—1061no—
260C2×C16—1061no—
261C2×C16—1061no—
262#107—21no—
263#107—21no—
264#107—532no—
265#107—1061no—
266#107C4×D5312yes—
267#107D21212yescomplemented
268#106—1064no—
269#102—1064no—
270C24—531no—
271C24—2121no—
272C2×D4—1062no—
273C2×D4—4242no—
274C2×D4—4242no—
275C2×D4—2124no—
276C2×D4—2121no—
277C2×D4—2121no—
278C2×D4—2121no—
279C2×D4—2121no—
280C2×D4—2121no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—531no—
284C22×C4—532no—
285C22×C4—532no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—2121no—
291C22×C4D4×D5311yescharacteristic
292D8—1062no—
293D8—1062no—
294D8—2121no—
295D8—2121no—
296C2×C8—532no—
297C2×C8—1061no—
298C2×C8—1061no—
299C2×C8—1062no—
300C2×C8—1062no—
301C2×C8—1062no—
302C2×C8—1062no—
303C2×C8C2×C4×D5311yescharacteristic
304C2×C8C2×D21211yescharacteristic
305C2×C8#1171994511yescharacteristic
306C2×C8D4×D5311yescharacteristic
307C2×C8D4×D5311yescharacteristic
308C2×C8#1171994911yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—1062no—
314C4⋊C4—2121no—
315C22⋊C4—2124no—
316C22⋊C4—2124no—
317C42—2124no—
318C16—41no—
319C16—2121no—
320C23—532no—
321C23—1061no—
322C23—1061no—
323C23—1061no—
324C23—1061no—
325C23—1061no—
326C23—1061no—
327C23—4241no—
328C23—4241no—
329C23—4241no—
330C23—4241no—
331C23—4241no—
332C23—4241no—
333C23—2122no—
334C23D8×D5311yescharacteristic
335D4—4248no—
336D4—4241no—
337D4—4241no—
338D4—2122no—
339D4—2122no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—532no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1062no—
350C2×C4—1062no—
351C2×C4—1062no—
352C2×C4—1062no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—2122no—
356C2×C4C22⋊C4×D5311yescharacteristic
357C2×C4#1313318711yescharacteristic
358C2×C4#1313320211yescharacteristic
359C2×C4SD16×D5311yescharacteristic
360C2×C4#1313324411yescharacteristic
361C2×C4#1313324911yescharacteristic
362C8—1062no—
363C8#1313317911yescharacteristic
364C8C4×D21211yescharacteristic
365C8#1313319011yescharacteristic
366C8#1313319211yescharacteristic
367C22—8481no—
368C22—8481no—
369C22—8481no—
370C22—8481no—
371C22—1062no—
372C22—1062no—
373C22—1062no—
374C22—4242no—
375C22—4242no—
376C22—4242no—
377C22—2121no—
378C22—2121no—
379C22—2121no—
380C22—2121no—
381C22#102×D5311yescharacteristic
382C22#1681075811yescharacteristic
383C22#1681093811yescharacteristic
384C22#1681105411yescharacteristic
385C22#1681105611yescharacteristic
386C22#1681105711yescharacteristic
387C22#1681106011yescharacteristic
388C4—82no—
389C4—82no—
390C4—1062no—
391C4—4242no—
392C4#1681050111yescharacteristic
393C4#1681062311yescharacteristic
394C4#1681076311yescharacteristic
395C4#1681094311yescharacteristic
396C2—8482no—
397C2—2122no—
398C2#2027503111yescharacteristic
399C2#2027642111yescharacteristic
400C2#2027642911yescharacteristic
401C2#2027663911yescharacteristic
402C2#2027664111yescharacteristic
403C2#2027690811yescharacteristic
404C2#2027690911yescharacteristic
subgroupG/H
centerC23D8×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1380C2
socleC23×C53C2×D8
prime radicalC2×#16810860C2
cosocleC22×C8×C53C23
hypercenter#1380D53
hypocenterC53#12992
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#16810660C2semidirect
3. C2×#16810425C2semidirect
4. C2×#16810423C2semidirect
5. C2×#16810402C2semidirect
6. #20272806C2semidirect
7. C53×#1380C2semidirect
14. #16810860C4semidirect
54. #13133151D4semidirect
66. #107×C53D4semidirect
200. #1380D53semidirect
228. C53#12992semidirect
267. #107D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168108602
C2×#168106602
C2×#168104252
C2×#168104232
C2×#168104022
#202728062
C53×#13802
#1299253