Finite Group Databasebeta

← #27287965

Subgroups — #27287965

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810900C211yesmaximal, characteristic
2#20275509C211yesmaximal, characteristic
3#20274712C211yesmaximal, characteristic
4C2×#16810374C211yesmaximal, characteristic, complemented
5C2×#16810369C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1956C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810900—22no—
10#16810900—22no—
11C2×#13133150—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#16810369—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×#103×C53—21no—
24C2×#103×C53—21no—
25C2×C4×C8×C53C2211yescharacteristic
26C53×#384C2211yescharacteristic
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#13133150—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
62#103×C53—42no—
63#103×C53—42no—
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66C23×D53—21no—
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×Dic53—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212SD1612yes—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53—21no—
96C2×C8×C53D812yes—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—81no—
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110#11719921—41no—
111#11719921—41no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C2×#88322—22no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123C23×C53#12111yescharacteristic
124C22×D53—22no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131Q8×C53—42no—
132Q8×C53—42no—
133Q8×C53—81no—
134Q8×C53—81no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—42no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12311yescharacteristic
147C2×C4×C53#12311yescharacteristic
148C2×C4×C53#12411yescharacteristic
149C2×C4×C53C2×D811yescharacteristic
150C2×C4×C53C2×SD1611yescharacteristic
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×C4×C53#13711yescharacteristic
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160D212—22no—
161D212—22no—
162D212—41no—
163D212—41no—
164C4×D53—81no—
165C4×D53—81no—
166C4×D53—81no—
167C4×D53—81no—
168C8×C53—22no—
169C8×C53—41no—
170#88322—42no—
171#88322—81no—
172#88322—81no—
173#10484—531nomaximal, Sylow
174C22×C53#39411yescharacteristic
175C22×C53#45911yescharacteristic
176C22×C53#46311yescharacteristic
177C22×C53#46511yescharacteristic
178C22×C53#47011yescharacteristic
179C22×C53#47111yescharacteristic
180C22×C53#47311yescharacteristic
181C2×D53—42no—
182C2×D53—42no—
183C2×D53—42no—
184C2×D53—81no—
185C2×D53—81no—
186C2×D53—81no—
187C2×D53—81no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—82no—
191C4×C53—161no—
192C4×C53#44811yescharacteristic
193C4×C53#45211yescharacteristic
194C4×C53#45911yescharacteristic
195C4×C53#46311yescharacteristic
196C4×C53#46611yescharacteristic
197C4×C53#46911yescharacteristic
198C4×C53#48311yescharacteristic
199C4×C53#48511yescharacteristic
200Dic53—81no—
201C2×#474—531no—
202#1956D5311yescharacteristic, complemented
203#1954—531no—
204#1802—531no—
205C2×#334—531no—
206C2×#332—531no—
207#1296—531no—
208C2×C53#166511yescharacteristic
209C2×C53#167911yescharacteristic
210C2×C53#168911yescharacteristic
211C2×C53#170311yescharacteristic
212C2×C53#203511yescharacteristic
213C2×C53#205211yescharacteristic
214C2×C53#205411yescharacteristic
215D53—82no—
216C2×C4⋊Q8C2×D5311yescharacteristic
217C2×C4×D4—531no—
218#474—1062no—
219#474—1062no—
220C2×#107—1061no—
221C2×C4⋊C8—531no—
222C2×C4⋊C8—531no—
223C2×C4⋊C8—531no—
224C2×#103—21no—
225C2×#103—21no—
226C2×#102—1061no—
227C2×#102—1061no—
228C2×C22⋊C8—1061no—
229C2×C22⋊C8—1061no—
230C2×C4×C8C2×D5311yescharacteristic
231#384C2×D5311yescharacteristic
232#340—1061no—
233#334—1062no—
234#332—1062no—
235C53#1048411yescharacteristic, complemented, Sylow
236C22×Q8—21no—
237C22×D4—531no—
238C23×C4—1061no—
239C22×C8—1061no—
240C22×C8—1061no—
241C22×C8—1061no—
242C22×C8C22⋊D5311yescharacteristic
243C22×C8C22⋊D5311yescharacteristic
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8D21212yes—
248C4×D4—1061no—
249C4×D4—1061no—
250C4×D4—1061no—
251C4×D4—1061no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—531no—
258C2×C22⋊C4—1061no—
259C2×C42C22×D5311yescharacteristic
260#107—1064no—
261C4⋊C8—532no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267#103—42no—
268#103—42no—
269#102—1064no—
270#102—2122no—
271C22⋊C8—2122no—
272C22⋊C8—2122no—
273C4×C8—21no—
274C4×C8—21no—
275C24—1061no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—532no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—1061no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4D4×D5311yescharacteristic
298C22×C4#1171994711yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—1062no—
306C2×C8—2121no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8—2121no—
310C2×C8—2121no—
311C2×C8—2121no—
312C2×C8—2121no—
313C2×C8—2121no—
314C2×C8#1171992212yes—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—81no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C42C2×D21211yescharacteristic
329C42#1171994511yescharacteristic
330C42D4×D5311yescharacteristic
331C42#1171994711yescharacteristic
332C23—1062no—
333C23—2121no—
334C23—2121no—
335C23—2121no—
336C23—2121no—
337C23—2121no—
338C23—2121no—
339C23#1313318811yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—81no—
343Q8—81no—
344D4—1062no—
345D4—1062no—
346D4—2121no—
347D4—2121no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—4241no—
359C2×C4—4241no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4#1313318711yescharacteristic
364C2×C4#1313318911yescharacteristic
365C2×C4#1313319111yescharacteristic
366C2×C4#1313319911yescharacteristic
367C2×C4#1313320111yescharacteristic
368C2×C4#1313320311yescharacteristic
369C2×C4C2×#1171992211yescharacteristic
370C2×C4SD16×D5311yescharacteristic
371C2×C4#1313323711yescharacteristic
372C2×C4#1313323811yescharacteristic
373C2×C4#1313324211yescharacteristic
374C2×C4#1313324311yescharacteristic
375C2×C4#1313324911yescharacteristic
376C2×C4#1313324911yescharacteristic
377C8—22no—
378C8—41no—
379C8—4241no—
380C8—4241no—
381C8—2122no—
382C22—4241no—
383C22—4241no—
384C22—4241no—
385C22—4241no—
386C22—2122no—
387C22—2122no—
388C22—2122no—
389C22#1681053911yescharacteristic
390C22#1681081511yescharacteristic
391C22#1681083511yescharacteristic
392C22#1681084911yescharacteristic
393C22#1681087811yescharacteristic
394C22#1681088311yescharacteristic
395C22#1681089511yescharacteristic
396C4—21no—
397C4—21no—
398C4—82no—
399C4—161no—
400C4—4241no—
401C4#1681076311yescharacteristic
402C4#1681078811yescharacteristic
403C4#1681081811yescharacteristic
404C4#1681083811yescharacteristic
405C4#1681085711yescharacteristic
406C4#1681087411yescharacteristic
407C4#1681095211yescharacteristic
408C4#1681096211yescharacteristic
409C2—4242no—
410C2#2027405311yescharacteristic
411C2#2027413511yescharacteristic
412C2#2027420111yescharacteristic
413C2#2027428711yescharacteristic
414C2#2027591011yescharacteristic
415C2#2027601511yescharacteristic
416C2#2027602511yescharacteristic
subgroupG/H
centerC23#13133188
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1956C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenter#1956D53
hypocenterC53#10484
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
4. C2×#16810374C2semidirect
5. C2×#16810369C2semidirect
6. #20272598C2semidirect
7. C53×#1956C2semidirect
202. #1956D53semidirect
235. C53#10484semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
#202755092
#202747122
C2×#168103742
C2×#168103692
#202725982
C53×#19562
#1048453