Finite Group Databasebeta

← #27302938

Subgroups — #27302938

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133160C211yesmaximal, characteristic, complemented
2C2×#16810998C211yesmaximal, characteristic
3C2×#16810425C211yesmaximal, characteristic, complemented
4C2×#16810424C211yesmaximal, characteristic, complemented
5#20275577C211yesmaximal, characteristic, complemented
6#20272792C211yesmaximal, characteristic, complemented
7C22×SD32×C53C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9C22×#11719922—21no—
10C2×#13133213—21no—
11C2×#13133160—22no—
12C2×#13133160—22no—
13C2×#13133160—22no—
14C22×#11719908C2211yescharacteristic
15#16810998—22no—
16#16810998—22no—
17C2×#13133151C2211yescharacteristic
18C2×#13133141—21no—
19C2×C8×Dic53C2211yescharacteristic
20#16810530—21no—
21#16810425—22no—
22#16810424—22no—
23C22×Q16×C53C2211yescharacteristic
24C22×D8×C53C2211yescharacteristic
25C2×SD32×C53—22no—
26C2×SD32×C53—22no—
27C2×SD32×C53—22no—
28C22×C16×C53C2211yescharacteristic
29C22×D212—21no—
30C22×C22⋊D53—42no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424—21no—
35C2×D424—21no—
36C2×D424D412yes—
37C2×#11719922—24no—
38C2×#11719922—42no—
39C2×#11719922—42no—
40C22×#88322D411yescharacteristic
41#13133213—24no—
42#13133213—42no—
43C2×C4⋊Dic53—21no—
44C2×C4×Dic53D411yescharacteristic
45#13133160—28no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48C2×#11719908D412yescomplemented
49#13133151—21no—
50#13133151—21no—
51#13133141—42no—
52C8×Dic53—21no—
53C8×Dic53—21no—
54#13133124—42no—
55C22×Q8×C53—21no—
56C22×D4×C53—21no—
57C2×Q16×C53—21no—
58C2×Q16×C53—21no—
59C2×Q16×C53—21no—
60C2×Q16×C53—21no—
61C2×Q16×C53—21no—
62C2×Q16×C53D412yescomplemented
63C2×D8×C53—21no—
64C2×D8×C53—21no—
65C2×D8×C53—21no—
66C2×D8×C53—21no—
67C2×D8×C53—21no—
68C2×D8×C53D412yes—
69C22×C8×C53C2311yescharacteristic
70SD32×C53—28no—
71C2×C16×C53—21no—
72C2×C16×C53—21no—
73C2×C16×C53D412yescomplemented
74C24×C53—41no—
75C23×D53—41no—
76C2×Q8×C53—22no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×D4×C53—22no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90C2×C22⋊D53—48no—
91C2×C22⋊D53—84no—
92C2×C22⋊D53—84no—
93C22×Dic53—41no—
94C22×Dic53D812yes—
95C2×D212—22no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101Q16×C53—22no—
102Q16×C53—22no—
103Q16×C53—41no—
104Q16×C53—41no—
105D8×C53—22no—
106D8×C53—22no—
107D8×C53—41no—
108D8×C53—41no—
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53#4111yescharacteristic
115#11719922—48no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118C4×Dic53—21no—
119C4×Dic53—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322D812yes—
123D424—22no—
124D424—22no—
125D424—41no—
126D424—41no—
127C16×C53—22no—
128#11719908—22no—
129C23×C53—42no—
130C23×C53—81no—
131C23×C53—81no—
132C23×C53—82no—
133C23×C53—82no—
134C23×C53C2×D811yescharacteristic
135C22×D53—42no—
136C22×D53—81no—
137C22×D53—81no—
138C22×D53—82no—
139C22×D53—82no—
140Q8×C53—42no—
141Q8×C53—42no—
142Q8×C53—81no—
143Q8×C53—81no—
144D4×C53—42no—
145D4×C53—42no—
146D4×C53—81no—
147D4×C53—81no—
148C2×C4×C53—42no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12711yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C22⋊D53—816no—
158C2×Dic53—22no—
159C2×Dic53—22no—
160C2×Dic53—81no—
161C2×Dic53—82no—
162C2×Dic53SD3214yescomplemented
163D212—42no—
164D212—42no—
165D212—81no—
166D212—81no—
167C8×C53#12011yescharacteristic
168C8×C53#12011yescharacteristic
169C8×C53#12111yescharacteristic
170C8×C53#12111yescharacteristic
171#88322—22no—
172#13328—531nomaximal, Sylow
173C22×C53—84no—
174C22×C53—82no—
175C22×C53—162no—
176C22×C53—161no—
177C22×C53—161no—
178C22×C53#44711yescharacteristic
179C22×C53#45911yescharacteristic
180C22×C53#49311yescharacteristic
181C22×C53C2×SD3212yes—
182C22×C53#50912yes—
183C2×D53—84no—
184C2×D53—82no—
185C2×D53—162no—
186C2×D53—161no—
187C2×D53—161no—
188C4×C53—82no—
189C4×C53#39211yescharacteristic
190C4×C53#44711yescharacteristic
191C4×C53#45911yescharacteristic
192C4×C53#49611yescharacteristic
193Dic53—24no—
194Dic53—161no—
195C22×SD32—531no—
196C22×SD32D5311yescharacteristic, complemented
197C2×#493—531no—
198C2×#357—531no—
199C2×#357—531no—
200#1969—531no—
201#1374—531no—
202C2×C53—162no—
203C2×C53#201111yescharacteristic
204C2×C53#218712yes—
205C2×C53#220712yes—
206C2×C53#224811yescharacteristic
207C2×C53#225011yescharacteristic
208D53—162no—
209C22×Q16C2×D5311yescharacteristic
210C22×D8—531no—
211C22×D8—1061no—
212C22×D8C2×D5311yescharacteristic
213C2×#127—1061no—
214C2×SD32—22no—
215C2×SD32—22no—
216C2×SD32—22no—
217C2×SD32—1062no—
218C2×SD32—1062no—
219C2×SD32—1062no—
220C22×C16—531no—
221C22×C16C2×D5311yescharacteristic
222#493—1062no—
223#493—1062no—
224C2×#107—531no—
225C2×#103—1061no—
226C2×C4×C8—531no—
227#391—1061no—
228#357—1062no—
229#357—1062no—
230C53#1332811yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×D4—21no—
233C22×D4—1061no—
234C22×D4—2122no—
235C2×Q16—21no—
236C2×Q16—21no—
237C2×Q16—21no—
238C2×Q16—21no—
239C2×Q16—21no—
240C2×Q16C22⋊D5312yescomplemented
241C2×D8—21no—
242C2×D8—21no—
243C2×D8—21no—
244C2×D8—21no—
245C2×D8—21no—
246C2×D8—532no—
247C2×D8—1064no—
248C2×D8—1061no—
249C2×D8—1061no—
250C2×D8—1061no—
251C2×D8—1061no—
252C2×D8—1061no—
253C2×D8—2122no—
254C2×D8—2122no—
255C2×D8C22⋊D5312yes—
256C22×C8—531no—
257C22×C8C22×D5311yescharacteristic
258#127—1064no—
259#127—2122no—
260C2×C4⋊C4—1061no—
261C2×C42—531no—
262SD32—28no—
263SD32—1068no—
264C2×C16—21no—
265C2×C16—21no—
266C2×C16—532no—
267C2×C16—1061no—
268C2×C16—1061no—
269C2×C16C22⋊D5312yescomplemented
270#107—1061no—
271#107—1061no—
272#103—2122no—
273C4×C8—1061no—
274C4×C8—1061no—
275#95—2122no—
276C24—41no—
277C24—2121no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—1062no—
291C2×D4—4244no—
292C2×D4—4244no—
293C2×D4—2128no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2121no—
299C22×C4—41no—
300C22×C4—532no—
301C22×C4—2121no—
302C22×C4D4×D5311yescharacteristic
303Q16—22no—
304Q16—22no—
305Q16—41no—
306Q16—41no—
307D8—22no—
308D8—22no—
309D8—41no—
310D8—41no—
311D8—1062no—
312D8—1062no—
313D8—2128no—
314D8—2121no—
315D8—2121no—
316C2×C8—532no—
317C2×C8—1061no—
318C2×C8—1061no—
319C2×C8D4×D5311yescharacteristic
320C2×C8D4×D5311yescharacteristic
321C2×C8#1171994711yescharacteristic
322C2×C8C2×C22⋊D5311yescharacteristic
323C2×C8C2×C22⋊D5311yescharacteristic
324C2×C8C2×C22⋊D5311yescharacteristic
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C42—1061no—
328C42—1061no—
329C16—22no—
330C16—1062no—
331C23—42no—
332C23—82no—
333C23—82no—
334C23—81no—
335C23—81no—
336C23—4242no—
337C23—4242no—
338C23—4241no—
339C23—4241no—
340C23—2122no—
341C23D8×D5311yescharacteristic
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346D4—42no—
347D4—42no—
348D4—81no—
349D4—81no—
350D4—42416no—
351D4—4241no—
352D4—4241no—
353D4—2122no—
354D4—2122no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—534no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4—4242no—
362C2×C4—4241no—
363C2×C4#1313318511yescharacteristic
364C2×C4#1313318911yescharacteristic
365C2×C4#1313321311yescharacteristic
366C2×C4D8×D5311yescharacteristic
367C2×C4#1313324111yescharacteristic
368C2×C4#1313324111yescharacteristic
369C8—1062no—
370C8#1313318511yescharacteristic
371C8#1313318611yescharacteristic
372C8#1313318811yescharacteristic
373C8#1313318911yescharacteristic
374C22—82no—
375C22—84no—
376C22—162no—
377C22—161no—
378C22—161no—
379C22—8482no—
380C22—8481no—
381C22—8481no—
382C22—4242no—
383C22—4244no—
384C22#1681075511yescharacteristic
385C22#1681081611yescharacteristic
386C22#1681099811yescharacteristic
387C22SD32×D5312yes—
388C22#1681106012yes—
389C4—82no—
390C4—8481no—
391C4—1064no—
392C4#1681053111yescharacteristic
393C4#1681075511yescharacteristic
394C4#1681081611yescharacteristic
395C4#1681101111yescharacteristic
396C2—162no—
397C2—8482no—
398C2#2027578411yescharacteristic
399C2#2027664212yes—
400C2#2027675312yes—
401C2#2027696911yescharacteristic
402C2#2027698011yescharacteristic
subgroupG/H
centerC23D8×D53
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC22×SD32×C53C2
socleC23×C53C2×D8
prime radicalC2×#16810998C2
hypercenterC22×SD32D53
hypocenterC53#13328
LCSG ▷ C22×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×SD32

Complements

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

NH ≅ G/Ntype
1. C22×#13133160C2semidirect
3. C2×#16810425C2semidirect
4. C2×#16810424C2semidirect
5. #20275577C2semidirect
6. #20272792C2semidirect
7. C22×SD32×C53C2semidirect
48. C2×#11719908D4semidirect
62. C2×Q16×C53D4semidirect
73. C2×C16×C53D4semidirect
162. C2×Dic53SD32semidirect
196. C22×SD32D53semidirect
230. C53#13328semidirect
240. C2×Q16C22⋊D53semidirect
269. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131331602
C2×#168109982
C2×#168104252
C2×#168104242
#202755772
#202727922
C22×SD32×C532
#1332853