Finite Group Databasebeta

← #27287937

Subgroups — #27287937

Order 13568.

474 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×#16810910C211yesmaximal, characteristic
2#20275509C211yesmaximal, characteristic
3#20274712C211yesmaximal, characteristic
4C2×#16810374C211yesmaximal, characteristic, complemented
5C2×#16810372C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1990C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810910—22no—
10#16810910—22no—
11C2×#13133148—21no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133137—21no—
16C2×#13133137—21no—
17C2×#13133129—21no—
18C2×#13133129—21no—
19#16810385—21no—
20#16810374—22no—
21#16810372—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×C8×C53C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C53×#340—21no—
26C53×#340—21no—
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31C22×#88322—21no—
32C4×D212—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C2×C4⋊Dic53D411yescharacteristic
37C2×#11719921—21no—
38#13133148—24no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322D412yes—
45#13133137—24no—
46#13133137—42no—
47#13133129—42no—
48#13133129—42no—
49C22×Q8×C53—21no—
50C22×C8×C53D411yescharacteristic
51C22×C8×C53D411yescharacteristic
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53D412yes—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4×C8×C53—21no—
63C4×C8×C53—21no—
64C23×D53—21no—
65C2×Q8×C53—22no—
66C2×Q8×C53—41no—
67C2×Q8×C53—81no—
68C2×Q8×C53—81no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4111yescharacteristic
74C22×Dic53—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212SD1612yes—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53SD1612yes—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—81no—
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C42×C53#4111yescharacteristic
106#11719921—41no—
107#11719921—41no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C2×#88322—22no—
111C2×#88322—41no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C23×C53#12111yescharacteristic
120C22×D53—22no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—41no—
127Q8×C53—82no—
128Q8×C53—82no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—42no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53#12311yescharacteristic
142C2×C4×C53#12411yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53C2×SD1611yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53#13511yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C2×C4×C53#13711yescharacteristic
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154D212—22no—
155D212—22no—
156D212—41no—
157D212—41no—
158C4×D53—81no—
159C4×D53—81no—
160C4×D53—81no—
161C4×D53—81no—
162C8×C53—22no—
163C8×C53—41no—
164#88322—42no—
165#88322—81no—
166#88322—81no—
167#10480—531nomaximal, Sylow
168C22×C53#39411yescharacteristic
169C22×C53#46111yescharacteristic
170C22×C53#46311yescharacteristic
171C22×C53#46511yescharacteristic
172C22×C53#47011yescharacteristic
173C22×C53#47111yescharacteristic
174C22×C53#47211yescharacteristic
175C2×D53—42no—
176C2×D53—42no—
177C2×D53—42no—
178C2×D53—81no—
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—82no—
185C4×C53—161no—
186C4×C53#44811yescharacteristic
187C4×C53#45211yescharacteristic
188C4×C53#46011yescharacteristic
189C4×C53#46411yescharacteristic
190C4×C53#46511yescharacteristic
191C4×C53#46811yescharacteristic
192C4×C53#48311yescharacteristic
193C4×C53#48511yescharacteristic
194Dic53—81no—
195C2×#476—531no—
196#1990D5311yescharacteristic, complemented
197#1954—531no—
198#1802—531no—
199C2×#334—531no—
200C2×#333—531no—
201#1296—531no—
202C2×C53#167111yescharacteristic
203C2×C53#167711yescharacteristic
204C2×C53#169511yescharacteristic
205C2×C53#170311yescharacteristic
206C2×C53#203511yescharacteristic
207C2×C53#205111yescharacteristic
208C2×C53#205511yescharacteristic
209D53—82no—
210C2×C4⋊Q8C2×D5311yescharacteristic
211C2×C4×D4—531no—
212#476—1062no—
213#476—1062no—
214C2×#106—1061no—
215C2×C4⋊C8—531no—
216C2×C4⋊C8—531no—
217C2×C4⋊C8—531no—
218C2×#102—1061no—
219C2×#102—1061no—
220C2×C22⋊C8—1061no—
221C2×C22⋊C8—1061no—
222C2×C4×C8C2×D5311yescharacteristic
223#384C2×D5311yescharacteristic
224#340—21no—
225#340—21no—
226#340—1061no—
227#334—1062no—
228#333—1062no—
229C53#1048011yescharacteristic, complemented, Sylow
230C22×Q8—21no—
231C22×D4—531no—
232C23×C4—1061no—
233C22×C8—1061no—
234C22×C8—1061no—
235C22×C8—1061no—
236C22×C8C22⋊D5311yescharacteristic
237C22×C8C22⋊D5311yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8D21212yes—
242C4×D4—1061no—
243C4×D4—1061no—
244C4×D4—1061no—
245C4×D4—1061no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—531no—
252C2×C22⋊C4—1061no—
253C2×C42C22×D5311yescharacteristic
254#106—1064no—
255C4⋊C8—532no—
256C4⋊C8—1061no—
257C4⋊C8—1061no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260C4⋊C8—1061no—
261#102—1064no—
262#102—2122no—
263C22⋊C8—2122no—
264C22⋊C8—2122no—
265C4×C8—21no—
266C4×C8—21no—
267C24—1061no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—532no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—1061no—
281C22×C4—2121no—
282C22×C4—2121no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4D4×D5311yescharacteristic
288C22×C4#1171994711yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—1062no—
296C2×C8—2121no—
297C2×C8—2121no—
298C2×C8—2121no—
299C2×C8—2121no—
300C2×C8—2121no—
301C2×C8—2121no—
302C2×C8—2121no—
303C2×C8—2121no—
304C2×C8#1171992412yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4—1061no—
315C4⋊C4—1061no—
316C22⋊C4—2121no—
317C22⋊C4—2121no—
318C42C2×D21211yescharacteristic
319C42#1171994511yescharacteristic
320C42D4×D5311yescharacteristic
321C42#1171994711yescharacteristic
322C23—1062no—
323C23—2121no—
324C23—2121no—
325C23—2121no—
326C23—2121no—
327C23—2121no—
328C23—2121no—
329C23#1313318811yescharacteristic
330Q8—82no—
331Q8—82no—
332D4—1062no—
333D4—1062no—
334D4—2121no—
335D4—2121no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—4241no—
345C2×C4—4241no—
346C2×C4—4241no—
347C2×C4—4241no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4#1313318711yescharacteristic
352C2×C4#1313318911yescharacteristic
353C2×C4#1313319111yescharacteristic
354C2×C4#1313319911yescharacteristic
355C2×C4#1313320111yescharacteristic
356C2×C4#1313320311yescharacteristic
357C2×C4SD16×D5311yescharacteristic
358C2×C4C2×#1171992411yescharacteristic
359C2×C4#1313323711yescharacteristic
360C2×C4#1313323811yescharacteristic
361C2×C4#1313324211yescharacteristic
362C2×C4#1313324811yescharacteristic
363C2×C4#1313324911yescharacteristic
364C2×C4#1313324911yescharacteristic
365C8—22no—
366C8—41no—
367C8—4241no—
368C8—4241no—
369C8—2122no—
370C22—4241no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22—2122no—
375C22—2122no—
376C22—2122no—
377C22#1681053911yescharacteristic
378C22#1681082511yescharacteristic
379C22#1681083511yescharacteristic
380C22#1681084911yescharacteristic
381C22#1681087811yescharacteristic
382C22#1681088311yescharacteristic
383C22#1681089011yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—161no—
388C4—4241no—
389C4#1681076311yescharacteristic
390C4#1681078811yescharacteristic
391C4#1681082311yescharacteristic
392C4#1681084511yescharacteristic
393C4#1681085411yescharacteristic
394C4#1681087111yescharacteristic
395C4#1681095211yescharacteristic
396C4#1681096211yescharacteristic
397C2—4242no—
398C2#2027408711yescharacteristic
399C2#2027412111yescharacteristic
400C2#2027423511yescharacteristic
401C2#2027428711yescharacteristic
402C2#2027591011yescharacteristic
403C2#2027600811yescharacteristic
404C2#2027603011yescharacteristic
subgroupG/H
centerC23#13133188
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1990C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenter#1990D53
hypocenterC53#10480
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#16810374C2semidirect
5. C2×#16810372C2semidirect
6. #20272598C2semidirect
7. C53×#1990C2semidirect
196. #1990D53semidirect
229. C53#10480semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
#202755092
#202747122
C2×#168103742
C2×#168103722
#202725982
C53×#19902
#1048053