Finite Group Databasebeta

← #27539428

Subgroups — #27539428

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285255C211yesmaximal, characteristic, complemented
2#20285202C211yesmaximal, characteristic
3#20276955C212yesmaximal, complemented
4#20276953C212yesmaximal, complemented
5#20276948C212yesmaximal, complemented
6#20276937C212yesmaximal, complemented
7#20276430C211yesmaximal, characteristic, complemented
8#20276427C211yesmaximal, characteristic, complemented
9C2×#16810425C211yesmaximal, characteristic, complemented
10#20276311C211yesmaximal, characteristic, complemented
11C53×#2138C211yesmaximal, characteristic, complemented
12C2×#13133242—21no—
13C22×D424C2211yescharacteristic
14#16811251—21no—
15#16811187—21no—
16C2×#13133157C2211yescharacteristic
17#16811030—22no—
18#16811027C2212yescomplemented
19#16811024C2212yescomplemented
20#16811009—22no—
21#16811008C2212yes—
22#16810998C2212yes—
23#16810666—21no—
24C2×#13133151C2211yescharacteristic
25#16810578C2211yescharacteristic
26#16810427C2212yescomplemented
27#16810425C2212yes—
28#16810425C2212yes—
29#16810424C2212yescomplemented
30#16810423C2212yes—
31#16810423C2212yes—
32#16810398C2212yescomplemented
33C16×Dic53C2212yescomplemented
34C2×#135×C53C2211yescharacteristic
35C22×C16×C53C2211yescharacteristic
36C53×#427C2211yescharacteristic
37C53×#358C2211yescharacteristic, complemented
38C53×#358C2211yescharacteristic, complemented
39C53×#357C2211yescharacteristic, complemented
40C53×#357C2211yescharacteristic, complemented
41C22×D212—21no—
42#13133242—44no—
43C2×#11719924—22no—
44C2×D424—21no—
45C2×D424—22no—
46C2×D424—22no—
47C2×D424C2312yes—
48C2×#11719922—22no—
49C2×#11719917D411yescharacteristic
50#13133217—41no—
51#13133216—22no—
52#13133213—22no—
53#13133209—41no—
54#13133208—22no—
55#13133206—22no—
56#13133194—42no—
57#13133191—41no—
58#13133191—41no—
59#13133185—42no—
60#13133176D411yescharacteristic
61C2×C4⋊Dic53—21no—
62#13133157—22no—
63C2×#11719908C2312yes—
64#13133151—21no—
65#13133151C2312yes—
66#13133150—22no—
67#13133148—22no—
68#13133127D412yes—
69C8×Dic53C2312yes—
70C2×#41×C53—21no—
71#135×C53—22no—
72#135×C53—22no—
73C2×Q16×C53C2311yescharacteristic
74C2×SD16×C53—21no—
75C2×D8×C53C2311yescharacteristic
76C22×C8×C53C2311yescharacteristic
77#117×C53—21no—
78C2×C16×C53—22no—
79C2×C16×C53C2311yescharacteristic
80C2×C16×C53C2311yescharacteristic
81#107×C53C2311yescharacteristic
82#107×C53C2311yescharacteristic
83#106×C53—21no—
84C23×D53—41no—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C2×C22⋊D53—42no—
93C2×C22⋊D53—42no—
94C22×Dic53—41no—
95C2×D212—22no—
96C2×D212—42no—
97C2×D212—42no—
98C2×D212—41no—
99C2×Dic106—42no—
100Q16×C53—21no—
101Q16×C53—21no—
102SD16×C53—42no—
103D8×C53—21no—
104D8×C53—21no—
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4112yes—
107C2×C8×C53#4112yes—
108C2×C8×C53C2411yescharacteristic
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C22⋊C4×C53—41no—
112C42×C53—41no—
113C22⋊Dic53D812yes—
114#11719924—44no—
115#11719922—44no—
116#11719921—42no—
117#11719921—42no—
118C4⋊Dic53—22no—
119C4⋊Dic53—41no—
120C4⋊Dic53—42no—
121C4⋊Dic53D811yescharacteristic
122C4⋊Dic53D811yescharacteristic
123#11719919—42no—
124#11719919—42no—
125C4×Dic53C2×D412yes—
126#11719917SD1614yes—
127C2×#88322C2×D412yes—
128D424—22no—
129D424—22no—
130D424—42no—
131C16×C53—21no—
132C16×C53—21no—
133#11719908—22no—
134C23×C53—41no—
135C23×C53C2×D811yescharacteristic
136C22×D53—42no—
137C22×D53—82no—
138C22×D53—161no—
139C22×D53—161no—
140Q8×C53—41no—
141Q8×C53—41no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—82no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—82no—
149C2×C4×C53#12412yes—
150C2×C4×C53C2×D811yescharacteristic
151C2×C4×C53C2×SD1612yes—
152C2×C4×C53C22×D411yescharacteristic
153C22⋊D53—162no—
154C22⋊D53—162no—
155C2×Dic53—42no—
156C2×Dic53—161no—
157C2×Dic53C2×D812yes—
158C2×Dic53C2×D812yes—
159D212—42no—
160D212—42no—
161D212—82no—
162Dic106—162no—
163C8×C53#12412yes—
164C8×C53C2×#4111yescharacteristic
165C8×C53C2×#4111yescharacteristic
166#88322C2×SD1614yes—
167#33702—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53#48612yes—
174C22×C53C22×D811yescharacteristic
175C2×D53—82no—
176C2×D53—82no—
177C2×D53—82no—
178C2×D53—164no—
179C4×C53—81no—
180C4×C53—81no—
181C4×C53—81no—
182C4×C53#48612yes—
183C4×C53C2×#12411yescharacteristic
184C4×C53C22×SD1611yescharacteristic
185Dic53—22no—
186Dic53—22no—
187Dic53—82no—
188#3164—531no—
189#3155—531no—
190#2244—532no—
191#2243—532no—
192#2242—532no—
193#2240—532no—
194#2140—531no—
195#2139—531no—
196#2138D5311yescharacteristic, complemented
197C2×#357—531no—
198#2108—531no—
199C2×C53—22no—
200C2×C53—161no—
201C2×C53C2×#48611yescharacteristic
202C2×C53#341511yescharacteristic
203C2×C53#341611yescharacteristic
204D53—162no—
205C2×AGL(1,ℤ/8)—1061no—
206C2×#135C2×D5311yescharacteristic
207C22×D8—531no—
208#544—1061no—
209#535—1061no—
210C2×M32—531no—
211C22×C16C2×D5311yescharacteristic
212#501—1062no—
213#500—532no—
214#499—532no—
215#496—1062no—
216#495—532no—
217#493—532no—
218#428—1061no—
219#427C2×D5311yescharacteristic
220C2×#107—531no—
221#405—531no—
222#358—532no—
223#358C2×D5311yescharacteristic, complemented
224#358C2×D5311yescharacteristic, complemented
225#357—532no—
226#357—532no—
227#357—532no—
228#357—532no—
229#357—532no—
230#357C2×D5311yescharacteristic, complemented
231#357C2×D5311yescharacteristic, complemented
232#346—532no—
233C4×C16—532no—
234C53#3370211yescharacteristic, complemented, Sylow
235C2×#41—21no—
236C22×D4—1061no—
237AGL(1,ℤ/8)—2124no—
238#135—22no—
239#135—22no—
240C2×Q16C22×D5311yescharacteristic
241C2×SD16—21no—
242C2×SD16—1062no—
243C2×D8—532no—
244C2×D8—1062no—
245C2×D8—1062no—
246C2×D8—1061no—
247C2×D8—1062no—
248C2×D8C22×D5311yescharacteristic
249C2×M16—531no—
250C22×C8C22×D5311yescharacteristic
251C4⋊Q8—1062no—
252C4⋊Q8—2121no—
253#127—1062no—
254#125—1062no—
255#125—2121no—
256#124—1062no—
257C22⋊Q8—2122no—
258#121—2121no—
259#121—2121no—
260#120—2122no—
261#117—21no—
262#117—531no—
263C2×C4⋊C4—1061no—
264M32—1062no—
265C2×C16—22no—
266C2×C16—532no—
267C2×C16C22×D5311yescharacteristic
268C2×C16C22×D5311yescharacteristic
269#107—532no—
270#107—1062no—
271#107—1061no—
272#107C22×D5311yescharacteristic
273#107C22×D5311yescharacteristic
274#106—21no—
275#106—1062no—
276#97—532no—
277C4×C8—532no—
278C24—2121no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—2122no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—1062no—
286C2×D4—2122no—
287C2×D4—2122no—
288C2×D4—2122no—
289C2×D4—2121no—
290C2×D4—2122no—
291C22×C4—41no—
292C22×C4—2121no—
293C22×C4D4×D5311yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—42no—
297SD16—2124no—
298D8—21no—
299D8—21no—
300D8—1062no—
301D8—1062no—
302D8—2124no—
303D8—2122no—
304M16—534no—
305C2×C8—532no—
306C2×C8#1171994512yes—
307C2×C8D4×D5311yescharacteristic
308C2×C8#1171994712yes—
309C2×C8C23×D5311yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—531no—
313C4⋊C4—531no—
314C4⋊C4—1062no—
315C4⋊C4—2122no—
316C4⋊C4—2122no—
317C4⋊C4—2122no—
318C4⋊C4—2121no—
319C22⋊C4—41no—
320C22⋊C4—532no—
321C22⋊C4—2122no—
322C22⋊C4—2122no—
323C42—41no—
324C42—532no—
325C16—21no—
326C16—21no—
327C16—1062no—
328C23—41no—
329C23—8481no—
330C23—8481no—
331C23—4242no—
332C23—2122no—
333C23D8×D5311yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—8482no—
337D4—41no—
338D4—41no—
339D4—82no—
340D4—8482no—
341D4—8482no—
342D4—4242no—
343D4—2122no—
344D4—2122no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—532no—
350C2×C4—532no—
351C2×C4—8481no—
352C2×C4—2122no—
353C2×C4#1313320312yes—
354C2×C4D8×D5311yescharacteristic
355C2×C4SD16×D5312yes—
356C2×C4C2×D4×D5311yescharacteristic
357C8—534no—
358C8#1313320312yes—
359C8C2×#1171994711yescharacteristic
360C8C2×#1171994511yescharacteristic
361C22—21no—
362C22—21no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—8484no—
367C22—4242no—
368C22—4242no—
369C22—4242no—
370C22#1681096812yes—
371C22C2×D8×D5311yescharacteristic
372C4—81no—
373C4—81no—
374C4—81no—
375C4—1062no—
376C4—1062no—
377C4—4242no—
378C4#1681096812yes—
379C4C2×#1313320311yescharacteristic
380C4C2×SD16×D5311yescharacteristic
381C2—22no—
382C2—161no—
383C2—8482no—
384C2C2×#1681096811yescharacteristic
385C2#2028785311yescharacteristic
386C2#2028785811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2138C2
socleC22×C53C22×D8
prime radical#20285202C2
hypercenter#2138D53
hypocenterC53#33702
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #20285255C2semidirect
3. #20276955C2semidirect
4. #20276953C2semidirect
5. #20276948C2semidirect
6. #20276937C2semidirect
7. #20276430C2semidirect
8. #20276427C2semidirect
9. C2×#16810425C2semidirect
10. #20276311C2semidirect
11. C53×#2138C2semidirect
18. #16811027C22semidirect
19. #16811024C22semidirect
26. #16810427C22semidirect
29. #16810424C22semidirect
32. #16810398C22semidirect
33. C16×Dic53C22semidirect
37. C53×#358C22semidirect
38. C53×#358C22semidirect
39. C53×#357C22semidirect
40. C53×#357C22semidirect
196. #2138D53semidirect
223. #358C2×D53semidirect
224. #358C2×D53semidirect
230. #357C2×D53semidirect
231. #357C2×D53semidirect
234. C53#33702semidirect

Maximal subgroups

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

subgroupindex
#202852552
#202852022
#202769552
#202769532
#202769482
#202769372
#202764302
#202764272
C2×#168104252
#202763112
C53×#21382
#3370253