Finite Group Databasebeta

← #27367980

Subgroups — #27367980

Order 13568.

786 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810997C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic
3#41×#88322C211yesmaximal, characteristic, complemented
4#20274065C214yesmaximal, complemented
5#20274058C214yesmaximal, complemented
6#20273470C211yesmaximal, characteristic, complemented
7#20273400C211yesmaximal, characteristic, complemented
8#20273384C211yesmaximal, characteristic, complemented
9C53×#3079C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C22×#11719924—22no—
12C2×#13133214C2211yescharacteristic
13#16810997—21no—
14#16810997—21no—
15#16810997C2214yes—
16#16810823—21no—
17#16810823C2212yes—
18#16810818—21no—
19#16810818C2212yes—
20Q8×#88322C2212yescomplemented
21D4×#88322—22no—
22D4×#88322C2212yescomplemented
23#16810676—21no—
24#16810676C2211yescharacteristic
25#16810647C2211yescharacteristic
26#16810627—21no—
27C2×C4×#88322—21no—
28C2×C4×#88322C2211yescharacteristic
29#16810373C2214yescomplemented
30#16810369C2214yescomplemented
31#16810366C2214yes—
32C2×C4⋊Q8×C53C2211yescharacteristic
33C4×#41×C53C2211yescharacteristic
34C53×#477—21no—
35C53×#476—21no—
36C53×#475C2212yescomplemented
37C53×#474C2212yescomplemented
38C53×#428—21no—
39C53×#423C2211yescharacteristic
40C53×#417—21no—
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44#13133242—44no—
45C2×#11719924—24no—
46C2×#11719924—24no—
47C2×#11719924—22no—
48C2×#11719924—48no—
49C2×#11719922—22no—
50C2×#11719917—21no—
51C22×#88322—22no—
52C22×#88322D412yes—
53#13133214—21no—
54#13133214—21no—
55#13133214—21no—
56#13133214C2312yes—
57C4⋊#88322—21no—
58C4⋊#88322—22no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61#13133137—22no—
62#13133137—22no—
63C22⋊#88322—22no—
64C22⋊#88322—21no—
65C4×#88322—21no—
66C4×#88322—22no—
67C4×#88322D411yescharacteristic
68C4×#88322D411yescharacteristic
69C4×#88322C2312yes—
70C2×#41×C53D411yescharacteristic
71C22×Q8×C53—21no—
72C2×M16×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53C2312yes—
77C4×Q8×C53D411yescharacteristic
78C4×Q8×C53C2311yescharacteristic
79C4×D4×C53—21no—
80C4×D4×C53—21no—
81C4×D4×C53D411yescharacteristic
82C4×D4×C53C2311yescharacteristic
83#117×C53—21no—
84#117×C53D411yescharacteristic
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C42×C53—21no—
88C2×C42×C53C2311yescharacteristic
89#107×C53—22no—
90#106×C53—22no—
91C4⋊C8×C53—21no—
92C4⋊C8×C53C2312yes—
93#103×C53—22no—
94#102×C53—22no—
95C23×D53—41no—
96#41×C53—22no—
97#41×C53SD1614yescomplemented
98C2×Q8×C53—22no—
99C2×Q8×C53—44no—
100C2×Q8×C53—41no—
101C2×Q8×C53C2×D411yescharacteristic
102C2×D4×C53—21no—
103C2×D4×C53C2×D411yescharacteristic
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—41no—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C2×D212—22no—
112C2×D212—22no—
113C2×D212—22no—
114C2×D212—44no—
115C2×D212—42no—
116C2×D212—42no—
117C2×D212—41no—
118C2×D212—41no—
119C2×D212—41no—
120C2×D212—82no—
121M16×C53—42no—
122C2×C8×C53—22no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53—41no—
129C4⋊C4×C53C2×D411yescharacteristic
130C4⋊C4×C53C2×D411yescharacteristic
131C22⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C22⋊C4×C53—21no—
134C42×C53—21no—
135C42×C53—21no—
136C42×C53—21no—
137C42×C53C2×D411yescharacteristic
138C42×C53C2×D411yescharacteristic
139C42×C53#4111yescharacteristic
140C42×C53C2411yescharacteristic
141#11719924—48no—
142#11719924—44no—
143#11719924—48no—
144#11719922—44no—
145#11719917—42no—
146C2×#88322—24no—
147C2×#88322—22no—
148C2×#88322—22no—
149C2×#88322—44no—
150C2×#88322C2×D412yes—
151C2×#88322C2×D412yes—
152C23×C53—21no—
153C23×C53#12111yescharacteristic
154C22×D53—42no—
155C22×D53—82no—
156C22×D53—82no—
157C22×D53—82no—
158Q8×C53—21no—
159Q8×C53—44no—
160Q8×C53—82no—
161Q8×C53C2×SD1612yescomplemented
162D4×C53—24no—
163D4×C53—21no—
164D4×C53C2×SD1612yescomplemented
165C2×C4×C53—21no—
166C2×C4×C53—22no—
167C2×C4×C53—24no—
168C2×C4×C53—22no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—82no—
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53C2×SD1612yes—
179C2×C4×C53#13512yes—
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C2×#4111yescharacteristic
183D212—42no—
184D212—42no—
185D212—44no—
186D212—82no—
187D212—82no—
188D212—82no—
189D212—82no—
190C8×C53—42no—
191#88322—22no—
192#88322—42no—
193#88322—42no—
194#88322AGL(1,ℤ/8)14yescomplemented
195#19266—531nomaximal, Sylow
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—22no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53#46512yes—
202C22×C53C2×#12111yescharacteristic
203C2×D53—82no—
204C2×D53—82no—
205C2×D53—82no—
206C2×D53—164no—
207C4×C53—21no—
208C4×C53—22no—
209C4×C53—21no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53—82no—
213C4×C53#46512yes—
214C4×C53#47112yes—
215C4×C53C22×SD1611yescharacteristic
216C4×C53C2×#13511yescharacteristic
217C4×C53C2×AGL(1,ℤ/8)12yes—
218C2×#492—531no—
219#3079D5311yescharacteristic, complemented
220#3038—531no—
221C8×#41—531no—
222#1666—534no—
223#1665—534no—
224#1566—531no—
225#1549—531no—
226#1545—531no—
227C2×C53—22no—
228C2×C53—42no—
229C2×C53#303811yescharacteristic
230C2×C53C2×#46511yescharacteristic
231C2×C53C2×#47111yescharacteristic
232D53—162no—
233C2×AGL(1,ℤ/8)—1061no—
234C22×SD16—1062no—
235C2×C4⋊Q8C2×D5311yescharacteristic
236C2×#127—531no—
237C4×#41C2×D5311yescharacteristic
238#492—534no—
239#492—1061no—
240#492—1061no—
241#477—21no—
242#476—21no—
243#475C2×D5312yescomplemented
244#474C2×D5312yescomplemented
245#460—532no—
246#460—1061no—
247#459—532no—
248#459—1061no—
249C8×Q8—532no—
250C8×D4—532no—
251C8×D4—1062no—
252#431—531no—
253#431—1061no—
254#428—21no—
255#423—531no—
256#423C2×D5311yescharacteristic
257#418—1061no—
258#417—21no—
259C2×C4×C8—531no—
260C2×C4×C8—1061no—
261#334—534no—
262#332—534no—
263#331—534no—
264C53#1926611yescharacteristic, complemented, Sylow
265C2×#41C22⋊D5311yescharacteristic
266C22×Q8—21no—
267C22×D4—1061no—
268C22×D4—1061no—
269C22×D4—1061no—
270AGL(1,ℤ/8)—2124no—
271C2×SD16—1064no—
272C2×SD16—1064no—
273C2×SD16—1062no—
274C2×SD16—2128no—
275C2×D8—1062no—
276C2×M16—21no—
277C2×M16—1061no—
278C22×C8—532no—
279C22×C8—1062no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8C22×D5312yes—
284#127—532no—
285#127—1061no—
286#127—1061no—
287#127—1061no—
288C4×Q8C22⋊D5311yescharacteristic
289C4×Q8C22×D5311yescharacteristic
290C4×D4—21no—
291C4×D4—21no—
292C4×D4C22⋊D5311yescharacteristic
293C4×D4C22×D5311yescharacteristic
294#117—21no—
295#117C22⋊D5311yescharacteristic
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C42—21no—
299C2×C42C22×D5311yescharacteristic
300#107—22no—
301#106—22no—
302C4⋊C8—21no—
303C4⋊C8—532no—
304C4⋊C8—532no—
305C4⋊C8—1062no—
306C4⋊C8—1061no—
307C4⋊C8C22×D5312yes—
308#103—22no—
309#102—22no—
310#102—1062no—
311#102—1062no—
312C22⋊C8—1062no—
313C22⋊C8—1061no—
314C4×C8—531no—
315C4×C8—531no—
316C4×C8—532no—
317C4×C8—1062no—
318C4×C8—1061no—
319C24—2121no—
320#41—22no—
321#41#1171992414yescomplemented
322C2×Q8—22no—
323C2×Q8—44no—
324C2×Q8—41no—
325C2×Q8C2×C22⋊D5311yescharacteristic
326C2×D4—21no—
327C2×D4—1062no—
328C2×D4—1062no—
329C2×D4—1062no—
330C2×D4—4242no—
331C2×D4—2124no—
332C2×D4—2121no—
333C2×D4—2121no—
334C2×D4—2121no—
335C2×D4—2122no—
336C2×D4—2122no—
337C2×D4C2×C22⋊D5311yescharacteristic
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—41no—
342C22×C4D4×D5311yescharacteristic
343C22×C4#1171994711yescharacteristic
344C22×C4C2×C22⋊D5311yescharacteristic
345SD16—2128no—
346SD16—2128no—
347SD16—2124no—
348D8—2124no—
349M16—42no—
350M16—2122no—
351C2×C8—22no—
352C2×C8—532no—
353C2×C8—532no—
354C2×C8—1064no—
355C2×C8—1062no—
356C2×C8—1062no—
357C2×C8—2124no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4C2×C22⋊D5311yescharacteristic
365C4⋊C4C2×C22⋊D5311yescharacteristic
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42D4×D5311yescharacteristic
373C42#1171994711yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C42C23×D5311yescharacteristic
376C23—21no—
377C23—4242no—
378C23—4242no—
379C23—4242no—
380C23—2122no—
381C23#1313318911yescharacteristic
382Q8—21no—
383Q8—44no—
384Q8—82no—
385Q8C2×#1171992412yescomplemented
386D4—24no—
387D4—21no—
388D4—4242no—
389D4—4242no—
390D4—4242no—
391D4—4242no—
392D4—2122no—
393D4—2122no—
394D4—2124no—
395D4C2×#1171992412yescomplemented
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—82no—
406C2×C4#1313318911yescharacteristic
407C2×C4#1313318911yescharacteristic
408C2×C4#1313318911yescharacteristic
409C2×C4C2×#1171992412yes—
410C2×C4#1313323712yes—
411C2×C4C2×D4×D5311yescharacteristic
412C2×C4C22×C22⋊D5311yescharacteristic
413C2×C4C2×#1171994711yescharacteristic
414C8—42no—
415C8—534no—
416C8—1062no—
417C8—2122no—
418C8—2122no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—8484no—
425C22—4242no—
426C22—4242no—
427C22—4242no—
428C22#1681085412yes—
429C22C2×#1313318911yescharacteristic
430C4—21no—
431C4—21no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—82no—
436C4#1681085412yes—
437C4#1681088812yes—
438C4C22×#1171992411yescharacteristic
439C4C2×#1313323711yescharacteristic
440C4AGL(1,ℤ/8)×D5312yes—
441C2—22no—
442C2—42no—
443C2—8482no—
444C2#2028419311yescharacteristic
445C2C2×#1681085411yescharacteristic
446C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3079C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenter#3079D53
hypocenterC53#19266
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3079

Complements

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

NH ≅ G/Ntype
1. C2×#16810997C2semidirect
3. #41×#88322C2semidirect
4. #20274065C2semidirect
5. #20274058C2semidirect
6. #20273470C2semidirect
7. #20273400C2semidirect
8. #20273384C2semidirect
9. C53×#3079C2semidirect
20. Q8×#88322C22semidirect
22. D4×#88322C22semidirect
29. #16810373C22semidirect
30. #16810369C22semidirect
36. C53×#475C22semidirect
37. C53×#474C22semidirect
97. #41×C53SD16semidirect
161. Q8×C53C2×SD16semidirect
164. D4×C53C2×SD16semidirect
194. #88322AGL(1,ℤ/8)semidirect
219. #3079D53semidirect
243. #475C2×D53semidirect
244. #474C2×D53semidirect
264. C53#19266semidirect
321. #41#11719924semidirect
385. Q8C2×#11719924semidirect
395. D4C2×#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109972
#202841952
#41×#883222
#202740652
#202740582
#202734702
#202734002
#202733842
C53×#30792
#1926653