Finite Group Databasebeta

← #27288796

Subgroups — #27288796

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275508C211yesmaximal, characteristic
2#20274874C211yesmaximal, characteristic, complemented
3#20274716C211yesmaximal, characteristic
4#20274712C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133150—21no—
10C2×#13133150—21no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133139—21no—
13C2×#13133137—21no—
14C2×#13133129—21no—
15C2×#13133129—21no—
16C2×C4×#88322C2211yescharacteristic
17#16810493C2211yescharacteristic
18#16810386—21no—
19#16810385—21no—
20C2×C4⋊C8×C53C2211yescharacteristic
21C2×C4⋊C8×C53C2211yescharacteristic
22C53×#384C2211yescharacteristic
23C22×D212D411yescharacteristic
24C22×C4×D53—21no—
25C22×#88322—21no—
26C22×#88322D411yescharacteristic
27C22×#88322D411yescharacteristic
28C4×D212—21no—
29C4×D212—21no—
30C4×D212—21no—
31C4×D212—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×#11719921—21no—
37#13133150—42no—
38#13133150—42no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41#13133139—42no—
42#13133137—42no—
43#13133129—42no—
44#13133129—42no—
45C4×#88322—21no—
46C4×#88322—21no—
47C22×C8×C53—21no—
48C22×C8×C53—21no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C42×C53C2311yescharacteristic
53C4⋊C8×C53—21no—
54C4⋊C8×C53—21no—
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57C23×D53—21no—
58C22×C4×C53—41no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53#4111yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—41no—
64C2×D212—21no—
65C2×D212—21no—
66C2×D212—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212D812yes—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C8×C53—41no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C2×C8×C53—41no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—81no—
87C42×C53#4111yescharacteristic
88C42×C53#4111yescharacteristic
89C42×C53#4111yescharacteristic
90C42×C53#4111yescharacteristic
91#11719921—41no—
92#11719921—41no—
93C4⋊Dic53—21no—
94C4⋊Dic53—21no—
95C4⋊Dic53—41no—
96C4⋊Dic53—41no—
97C4⋊Dic53—41no—
98C4⋊Dic53—41no—
99C4⋊Dic53—81no—
100C2×#88322—21no—
101C2×#88322—21no—
102C2×#88322—21no—
103C2×#88322—21no—
104C2×#88322—41no—
105C2×#88322—41no—
106C2×#88322—41no—
107C2×#88322D812yes—
108C2×#88322SD1612yes—
109C23×C53#12111yescharacteristic
110C22×D53—22no—
111C22×D53—41no—
112C22×D53—41no—
113C22×D53—41no—
114C22×D53—41no—
115C22×D53—41no—
116C22×D53—41no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—81no—
120C2×C4×C53—81no—
121C2×C4×C53—81no—
122C2×C4×C53#12311yescharacteristic
123C2×C4×C53#12311yescharacteristic
124C2×C4×C53#12311yescharacteristic
125C2×C4×C53#12411yescharacteristic
126C2×C4×C53#12611yescharacteristic
127C2×C4×C53#12611yescharacteristic
128C2×C4×C53C2×D811yescharacteristic
129C2×C4×C53C2×D811yescharacteristic
130C2×C4×C53C2×SD1611yescharacteristic
131C2×C4×C53#13511yescharacteristic
132C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C53#13711yescharacteristic
134C2×C4×C53#13711yescharacteristic
135C2×C4×C53#13711yescharacteristic
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142D212—22no—
143D212—22no—
144D212—41no—
145D212—41no—
146C4×D53—81no—
147C4×D53—81no—
148C4×D53—81no—
149C4×D53—81no—
150C8×C53—81no—
151C8×C53—81no—
152#88322—22no—
153#88322—22no—
154#88322—81no—
155#10615—531nomaximal, Sylow
156C22×C53#39711yescharacteristic
157C22×C53#45911yescharacteristic
158C22×C53#46111yescharacteristic
159C22×C53#46611yescharacteristic
160C22×C53#47011yescharacteristic
161C22×C53#47211yescharacteristic
162C22×C53#47311yescharacteristic
163C2×D53—42no—
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—81no—
167C2×D53—81no—
168C2×D53—81no—
169C2×D53—81no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—161no—
173C4×C53#48011yescharacteristic
174C4×C53#48011yescharacteristic
175C4×C53#48011yescharacteristic
176C4×C53#48111yescharacteristic
177C4×C53#48511yescharacteristic
178C4×C53#48511yescharacteristic
179C4×C53#48611yescharacteristic
180C4×C53#48911yescharacteristic
181Dic53—81no—
182Dic53—161no—
183#1954—531no—
184#1845—531no—
185#1802—531no—
186#1802—531no—
187#1296—531no—
188#1296D5311yescharacteristic, complemented
189#1277—531no—
190C2×C53#166711yescharacteristic
191C2×C53#167311yescharacteristic
192C2×C53#168911yescharacteristic
193C2×C53#169511yescharacteristic
194C2×C53#207111yescharacteristic
195C2×C53#207511yescharacteristic
196C2×C53#207811yescharacteristic
197D53—82no—
198C2×C4×D4—531no—
199C2×#107—1061no—
200C2×#107—1061no—
201C2×C4⋊C8—531no—
202C2×C4⋊C8C2×D5311yescharacteristic
203C2×C4⋊C8C2×D5311yescharacteristic
204C2×#102—1061no—
205C2×#102—1061no—
206C2×C22⋊C8—1061no—
207C2×C22⋊C8—1061no—
208C2×C4×C8—531no—
209#384—531no—
210#384C2×D5311yescharacteristic
211#340—1061no—
212#340—1061no—
213C53#1061511yescharacteristic, complemented, Sylow
214C22×D4—531no—
215C23×C4—1061no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—531no—
219C22×C8—531no—
220C22×C8—1061no—
221C4×D4—1061no—
222C4×D4—1061no—
223C4×D4—1061no—
224C4×D4—1061no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C22⋊C4—1061no—
233C2×C42C22×D5311yescharacteristic
234#107—2122no—
235#107—2122no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—1061no—
241C4⋊C8—1061no—
242#102—2122no—
243#102—2122no—
244C22⋊C8—2122no—
245C22⋊C8—2122no—
246C4×C8—1061no—
247C4×C8—1061no—
248C24—1061no—
249C2×D4—532no—
250C2×D4—1061no—
251C2×D4—1061no—
252C2×D4—1061no—
253C2×D4—1061no—
254C2×D4—1061no—
255C22×C4—41no—
256C22×C4—1061no—
257C22×C4—2121no—
258C22×C4—2121no—
259C22×C4—2121no—
260C22×C4—2121no—
261C22×C4—2121no—
262C22×C4—2121no—
263C22×C4—2121no—
264C22×C4#1171994511yescharacteristic
265C22×C4D4×D5311yescharacteristic
266C22×C4D4×D5311yescharacteristic
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—532no—
274C2×C8—532no—
275C2×C8—1061no—
276C2×C8—1061no—
277C2×C8—1061no—
278C2×C8—1061no—
279C2×C8—2121no—
280C2×C8—2121no—
281C2×C8—2121no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—4241no—
290C4⋊C4—2121no—
291C4⋊C4—2121no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C22⋊C4—2121no—
295C22⋊C4—2121no—
296C42#1171994511yescharacteristic
297C42#1171994511yescharacteristic
298C42#1171994711yescharacteristic
299C42#1171994911yescharacteristic
300C23—1062no—
301C23—2121no—
302C23—2121no—
303C23—2121no—
304C23—2121no—
305C23—2121no—
306C23—2121no—
307C23#1313319011yescharacteristic
308D4—1062no—
309D4—1062no—
310D4—2121no—
311D4—2121no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—4241no—
318C2×C4—4241no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—4241no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4#1313319911yescharacteristic
328C2×C4#1313320211yescharacteristic
329C2×C4#1313320211yescharacteristic
330C2×C4#1313320311yescharacteristic
331C2×C4#1313321111yescharacteristic
332C2×C4#1313321211yescharacteristic
333C2×C4D8×D5311yescharacteristic
334C2×C4D8×D5311yescharacteristic
335C2×C4SD16×D5311yescharacteristic
336C2×C4#1313323811yescharacteristic
337C2×C4#1313324411yescharacteristic
338C2×C4#1313324611yescharacteristic
339C2×C4#1313324911yescharacteristic
340C2×C4#1313324911yescharacteristic
341C8—81no—
342C8—81no—
343C8—1062no—
344C8—1062no—
345C8—4241no—
346C22—4241no—
347C22—4241no—
348C22—4241no—
349C22—4241no—
350C22—2122no—
351C22—2122no—
352C22—2122no—
353C22#1681055211yescharacteristic
354C22#1681081711yescharacteristic
355C22#1681082711yescharacteristic
356C22#1681085811yescharacteristic
357C22#1681088011yescharacteristic
358C22#1681089211yescharacteristic
359C22#1681089711yescharacteristic
360C4—21no—
361C4—21no—
362C4—161no—
363C4—8481no—
364C4—4241no—
365C4#1681093511yescharacteristic
366C4#1681093811yescharacteristic
367C4#1681093811yescharacteristic
368C4#1681094311yescharacteristic
369C4#1681096211yescharacteristic
370C4#1681096711yescharacteristic
371C4#1681096811yescharacteristic
372C4#1681098411yescharacteristic
373C2—4242no—
374C2#2027406911yescharacteristic
375C2#2027410311yescharacteristic
376C2#2027420311yescharacteristic
377C2#2027423711yescharacteristic
378C2#2027612211yescharacteristic
379C2#2027614211yescharacteristic
380C2#2027615911yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenter#1296D53
hypocenterC53#10615
LCSG ▷ C2×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. #20274874C2semidirect
5. #20272599C2semidirect
6. #20272549C2semidirect
7. C53×#1296C2semidirect
188. #1296D53semidirect
213. C53#10615semidirect

Maximal subgroups

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

subgroupindex
#202755082
#202748742
#202747162
#202747122
#202725992
#202725492
C53×#12962
#1061553