Finite Group Databasebeta

← #27277135

Subgroups — #27277135

Order 13568.

542 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810688C211yesmaximal, characteristic
2#20274871C211yesmaximal, characteristic, complemented
3#20274716C211yesmaximal, characteristic
4#20274712C211yesmaximal, characteristic
5C2×#16810373C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C22×C8×D53—21no—
9C2×C4×D212C2211yescharacteristic
10#16810688—22no—
11#16810688C414yescomplemented
12C2×#13133148—21no—
13C2×#13133148—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133139—21no—
16C2×#13133137—21no—
17C2×#13133129—21no—
18C2×C4×#88322C2211yescharacteristic
19#16810493C2211yescharacteristic
20#16810386—21no—
21#16810385—21no—
22#16810373—22no—
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C22×D212D411yescharacteristic
27C22×C4×D53—21no—
28C22×#88322—21no—
29C22×#88322D411yescharacteristic
30C22×#88322D411yescharacteristic
31C2×C8×D53—24no—
32C2×C8×D53—42no—
33C2×C8×D53—42no—
34C4×D212—21no—
35C4×D212—21no—
36C4×D212C2×C412yes—
37C4×D212Q812yes—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×#11719921—21no—
43#13133148—42no—
44#13133148—42no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47#13133139—42no—
48#13133137—42no—
49#13133129—24no—
50C4×#88322C2×C412yes—
51C4×#88322D412yes—
52C22×C8×C53—21no—
53C22×C8×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C42×C53C2311yescharacteristic
58C4⋊C8×C53—21no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53C2×C412yes—
62C23×D53—21no—
63C22×C4×C53—41no—
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53#4111yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—41no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212C4⋊C412yes—
73C2×D212SD1612yes—
74C2×D212SD1612yes—
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C8×C53—22no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C4⋊C4×C53—41no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—81no—
92C42×C53C22×C411yescharacteristic
93C42×C53C2×D411yescharacteristic
94C42×C53C2×Q811yescharacteristic
95C42×C53#4111yescharacteristic
96#11719921—22no—
97#11719921—22no—
98C4⋊Dic53—21no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53—41no—
102C4⋊Dic53—41no—
103C4⋊Dic53—81no—
104C4⋊Dic53C4⋊C412yes—
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322—41no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322C22⋊C412yes—
111C2×#88322C22⋊C412yes—
112C2×#88322D812yes—
113C2×#88322SD1612yes—
114C8×D53—48no—
115C23×C53#12111yescharacteristic
116C22×D53—22no—
117C22×D53—22no—
118C22×D53—22no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53—81no—
128C2×C4×C53C2×C22⋊C411yescharacteristic
129C2×C4×C53C2×C4⋊C411yescharacteristic
130C2×C4×C53#11711yescharacteristic
131C2×C4×C53#12111yescharacteristic
132C2×C4×C53C22⋊Q811yescharacteristic
133C2×C4×C53C22⋊Q811yescharacteristic
134C2×C4×C53C2×D811yescharacteristic
135C2×C4×C53C2×SD1611yescharacteristic
136C2×C4×C53C2×SD1611yescharacteristic
137C2×C4×C53C2×SD1611yescharacteristic
138C2×C4×C53#13511yescharacteristic
139C2×C4×C53#13511yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53#13711yescharacteristic
142C2×Dic53—22no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—81no—
146C2×Dic53—81no—
147C2×Dic53—81no—
148D212—22no—
149D212—22no—
150D212—22no—
151D212#10614yescomplemented
152C4×D53—42no—
153C4×D53—42no—
154C4×D53—42no—
155C4×D53—42no—
156C8×C53—42no—
157C8×C53—81no—
158#88322—22no—
159#88322—81no—
160#88322#10214yescomplemented
161#8642—531nomaximal, Sylow
162C22×C53#38011yescharacteristic
163C22×C53#45911yescharacteristic
164C22×C53#46111yescharacteristic
165C22×C53#46511yescharacteristic
166C22×C53#46511yescharacteristic
167C22×C53#47111yescharacteristic
168C22×C53#47111yescharacteristic
169C2×D53—24no—
170C2×D53—42no—
171C2×D53—42no—
172C2×D53—42no—
173C2×D53—42no—
174C2×D53—42no—
175C2×D53—42no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—161no—
179C4×C53C2×#10211yescharacteristic
180C4×C53#41711yescharacteristic
181C4×C53C2×#10611yescharacteristic
182C4×C53#42711yescharacteristic
183C4×C53#46511yescharacteristic
184C4×C53#46511yescharacteristic
185C4×C53#47411yescharacteristic
186C4×C53#47611yescharacteristic
187Dic53—42no—
188Dic53—161no—
189C2×C8×D4—531no—
190#1844—531no—
191#1802—531no—
192#1802—531no—
193C2×#334—531no—
194#1296—531no—
195#1296D5311yescharacteristic, complemented
196C2×C53#166511yescharacteristic
197C2×C53#166511yescharacteristic
198C2×C53#167111yescharacteristic
199C2×C53#167111yescharacteristic
200C2×C53#180211yescharacteristic
201C2×C53#180711yescharacteristic
202C2×C53#181511yescharacteristic
203D53—44no—
204C23×C8—1061no—
205C2×C4×D4—531no—
206C8×D4—534no—
207C8×D4—1062no—
208C2×#106—1061no—
209C2×#106—1061no—
210C2×C4⋊C8—531no—
211C2×C4⋊C8C2×D5311yescharacteristic
212C2×C4⋊C8C2×D5311yescharacteristic
213C2×#102—1061no—
214C2×#102—1061no—
215C2×C22⋊C8—1061no—
216C2×C4×C8—531no—
217#384—531no—
218#384C2×D5311yescharacteristic
219#340—1061no—
220#340—1061no—
221#334—1062no—
222C53#864211yescharacteristic, complemented, Sylow
223C22×D4—531no—
224C23×C4—1061no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—531no—
228C22×C8—531no—
229C22×C8—1064no—
230C22×C8—1061no—
231C22×C8—2122no—
232C22×C8—2122no—
233C4×D4—532no—
234C4×D4—532no—
235C4×D4—1061no—
236C4×D4—1061no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C42C22×D5311yescharacteristic
246#106—2122no—
247#106—2122no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—1061no—
252C4⋊C8—1061no—
253C4⋊C8C4×D5312yes—
254#102—2122no—
255#102—2122no—
256C22⋊C8—1064no—
257C4×C8—532no—
258C4×C8—532no—
259C24—1061no—
260C2×D4—532no—
261C2×D4—532no—
262C2×D4—532no—
263C2×D4—1061no—
264C2×D4—1061no—
265C2×D4—1061no—
266C22×C4—41no—
267C22×C4—1061no—
268C22×C4—1062no—
269C22×C4—1062no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4#1171994511yescharacteristic
276C22×C4D4×D5311yescharacteristic
277C22×C4D4×D5311yescharacteristic
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—532no—
285C2×C8—532no—
286C2×C8—532no—
287C2×C8—532no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—2128no—
291C2×C8—2121no—
292C2×C8—2121no—
293C2×C8—2121no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—81no—
299C4⋊C4—532no—
300C4⋊C4—1061no—
301C4⋊C4—4241no—
302C4⋊C4—2121no—
303C4⋊C4—2121no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—1062no—
307C22⋊C4—1062no—
308C42C2×C4×D5311yescharacteristic
309C42#1171994511yescharacteristic
310C42D4×D5311yescharacteristic
311C42Q8×D5311yescharacteristic
312C23—1062no—
313C23—1062no—
314C23—1062no—
315C23—2121no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23#1313319011yescharacteristic
320D4—534no—
321D4—1062no—
322D4—1062no—
323D4—1062no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—1062no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4—4241no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4C22⋊C4×D5311yescharacteristic
340C2×C4C4⋊C4×D5311yescharacteristic
341C2×C4#1313317411yescharacteristic
342C2×C4#1313319011yescharacteristic
343C2×C4#1313319511yescharacteristic
344C2×C4#1313319511yescharacteristic
345C2×C4D8×D5311yescharacteristic
346C2×C4SD16×D5311yescharacteristic
347C2×C4SD16×D5311yescharacteristic
348C2×C4SD16×D5311yescharacteristic
349C2×C4#1313323611yescharacteristic
350C2×C4#1313324011yescharacteristic
351C2×C4#1313324411yescharacteristic
352C2×C4#1313324911yescharacteristic
353C8—42no—
354C8—81no—
355C8—534no—
356C8—1062no—
357C8—4241no—
358C22—1064no—
359C22—2122no—
360C22—2122no—
361C22—2122no—
362C22—2122no—
363C22—2122no—
364C22—2122no—
365C22#1681048211yescharacteristic
366C22#1681081711yescharacteristic
367C22#1681082711yescharacteristic
368C22#1681085111yescharacteristic
369C22#1681085511yescharacteristic
370C22#1681088511yescharacteristic
371C22#1681088911yescharacteristic
372C4—21no—
373C4—21no—
374C4—161no—
375C4—8481no—
376C4—2122no—
377C4#102×D5311yescharacteristic
378C4#1681062311yescharacteristic
379C4#106×D5311yescharacteristic
380C4#1681066011yescharacteristic
381C4#1681085111yescharacteristic
382C4#1681085511yescharacteristic
383C4#1681090211yescharacteristic
384C4#1681091211yescharacteristic
385C2—2124no—
386C2#2027405511yescharacteristic
387C2#2027405911yescharacteristic
388C2#2027408911yescharacteristic
389C2#2027409311yescharacteristic
390C2#2027471411yescharacteristic
391C2#2027473911yescharacteristic
392C2#2027477311yescharacteristic
subgroupG/H
centerC23#13133190
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#121
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#8642
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #20274871C2semidirect
5. C2×#16810373C2semidirect
6. #20272599C2semidirect
7. C53×#1296C2semidirect
11. #16810688C4semidirect
151. D212#106semidirect
160. #88322#102semidirect
195. #1296D53semidirect
222. C53#8642semidirect

Maximal subgroups

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

subgroupindex
C2×#168106882
#202748712
#202747162
#202747122
C2×#168103732
#202725992
C53×#12962
#864253