Finite Group Databasebeta

← #27368048

Subgroups — #27368048

Order 13568.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic, complemented
2#20284195C211yesmaximal, characteristic
3#41×#88322C211yesmaximal, characteristic, complemented
4#20274133C214yesmaximal, complemented
5#20274126C214yesmaximal, complemented
6#20273470C211yesmaximal, characteristic, complemented
7#20273396C211yesmaximal, characteristic, complemented
8#20273384C211yesmaximal, characteristic, complemented
9C53×#3079C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C2×#13133214C2211yescharacteristic
12#16810971—21no—
13#16810971—21no—
14#16810971C2214yes—
15#16810823—21no—
16#16810823C2212yes—
17#16810818—21no—
18#16810818C2212yes—
19Q8×#88322C2212yescomplemented
20D4×#88322—22no—
21D4×#88322C2212yescomplemented
22#16810676—21no—
23#16810676C2211yescharacteristic
24#16810647C2211yescharacteristic
25#16810627—21no—
26C2×#13133137—22no—
27C2×C4×#88322—21no—
28C2×C4×#88322C2211yescharacteristic
29#16810373C2214yescomplemented
30#16810372C2214yescomplemented
31#16810366C2214yes—
32C2×C4⋊Q8×C53C2211yescharacteristic
33C4×#41×C53C2211yescharacteristic
34C53×#477C2212yescomplemented
35C53×#476C2212yescomplemented
36C53×#475—21no—
37C53×#474—21no—
38C53×#428—21no—
39C53×#423C2211yescharacteristic
40C53×#417—21no—
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44#13133242—44no—
45C2×#11719924—22no—
46C2×#11719922—22no—
47C2×#11719917—21no—
48C22×#88322—22no—
49C22×#88322D412yes—
50#13133214—21no—
51#13133214—21no—
52#13133214—21no—
53#13133214C2312yes—
54C4⋊#88322—21no—
55C4⋊#88322—22no—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58#13133137—22no—
59#13133137—22no—
60#13133137—24no—
61#13133137—24no—
62C22⋊#88322—22no—
63C22⋊#88322—21no—
64C4×#88322—21no—
65C4×#88322—22no—
66C4×#88322D411yescharacteristic
67C4×#88322D411yescharacteristic
68C4×#88322C2312yes—
69C2×#41×C53D411yescharacteristic
70C22×Q8×C53—21no—
71C2×M16×C53—21no—
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53C2312yes—
76C4×Q8×C53D411yescharacteristic
77C4×Q8×C53C2311yescharacteristic
78C4×D4×C53—21no—
79C4×D4×C53—21no—
80C4×D4×C53D411yescharacteristic
81C4×D4×C53C2311yescharacteristic
82#117×C53—21no—
83#117×C53D411yescharacteristic
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C42×C53—21no—
87C2×C42×C53C2311yescharacteristic
88#107×C53—22no—
89#106×C53—22no—
90C4⋊C8×C53—21no—
91C4⋊C8×C53C2312yes—
92#103×C53—22no—
93#102×C53—22no—
94C23×D53—41no—
95#41×C53—22no—
96#41×C53SD1614yes—
97C2×Q8×C53—22no—
98C2×Q8×C53—41no—
99C2×Q8×C53—82no—
100C2×Q8×C53C2×D411yescharacteristic
101C2×D4×C53—21no—
102C2×D4×C53C2×D411yescharacteristic
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—41no—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C2×D212—22no—
111C2×D212—22no—
112C2×D212—22no—
113C2×D212—44no—
114C2×D212—42no—
115C2×D212—42no—
116C2×D212—41no—
117C2×D212—41no—
118C2×D212—41no—
119C2×D212—82no—
120M16×C53—42no—
121C2×C8×C53—22no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53C2×D411yescharacteristic
129C4⋊C4×C53C2×D411yescharacteristic
130C22⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C22⋊C4×C53—21no—
133C42×C53—21no—
134C42×C53—21no—
135C42×C53—21no—
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139C42×C53C2411yescharacteristic
140#11719924—44no—
141#11719922—44no—
142#11719917—42no—
143C2×#88322—22no—
144C2×#88322—22no—
145C2×#88322—42no—
146C2×#88322—44no—
147C2×#88322C2×D412yes—
148C2×#88322C2×D412yes—
149C23×C53—21no—
150C23×C53#12111yescharacteristic
151C22×D53—42no—
152C22×D53—82no—
153C22×D53—82no—
154C22×D53—82no—
155Q8×C53—21no—
156Q8×C53—82no—
157Q8×C53—82no—
158Q8×C53C2×SD1612yes—
159D4×C53—24no—
160D4×C53—21no—
161D4×C53C2×SD1612yes—
162C2×C4×C53—21no—
163C2×C4×C53—22no—
164C2×C4×C53—24no—
165C2×C4×C53—22no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—82no—
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53C2×SD1612yes—
176C2×C4×C53#13512yes—
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180D212—42no—
181D212—42no—
182D212—44no—
183D212—82no—
184D212—82no—
185D212—82no—
186D212—82no—
187C8×C53—42no—
188#88322—22no—
189#88322—22no—
190#88322—42no—
191#88322—42no—
192#19274—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—22no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53#46512yes—
199C22×C53C2×#12111yescharacteristic
200C2×D53—82no—
201C2×D53—82no—
202C2×D53—82no—
203C2×D53—164no—
204C4×C53—21no—
205C4×C53—22no—
206C4×C53—21no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53—82no—
210C4×C53#46512yes—
211C4×C53#47212yes—
212C4×C53C22×SD1611yescharacteristic
213C4×C53C2×#13511yescharacteristic
214C4×C53#57512yes—
215C2×#486—531no—
216#3079D5311yescharacteristic, complemented
217#3038—531no—
218C8×#41—531no—
219#1678—534no—
220#1677—534no—
221#1566—531no—
222#1548—531no—
223#1545—531no—
224C2×C53—22no—
225C2×C53—42no—
226C2×C53#304011yescharacteristic
227C2×C53C2×#46511yescharacteristic
228C2×C53C2×#47211yescharacteristic
229D53—162no—
230C2×AGL(1,ℤ/8)—1061no—
231C2×C4⋊Q8C2×D5311yescharacteristic
232C2×#127—531no—
233C4×#41C2×D5311yescharacteristic
234#486—534no—
235#486—1061no—
236#486—1061no—
237#477C2×D5312yescomplemented
238#476C2×D5312yescomplemented
239#475—21no—
240#474—21no—
241#460—532no—
242#460—1061no—
243#459—532no—
244#459—1061no—
245C8×Q8—532no—
246C8×D4—532no—
247C8×D4—1062no—
248#431—531no—
249#431—1061no—
250#428—21no—
251#423—531no—
252#423C2×D5311yescharacteristic
253#418—1061no—
254#417—21no—
255C2×#102—1062no—
256C2×C4×C8—531no—
257C2×C4×C8—1061no—
258#334—534no—
259#333—534no—
260#331—534no—
261C53#1927411yescharacteristic, complemented, Sylow
262C2×#41C22⋊D5311yescharacteristic
263C22×Q8—21no—
264C22×D4—1061no—
265C22×D4—1061no—
266C22×D4—1061no—
267AGL(1,ℤ/8)—2124no—
268C2×SD16—1062no—
269C2×D8—1062no—
270C2×M16—21no—
271C2×M16—1061no—
272C22×C8—532no—
273C22×C8—1062no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8C22×D5312yes—
278#127—532no—
279#127—1061no—
280#127—1061no—
281#127—1061no—
282C4×Q8C22⋊D5311yescharacteristic
283C4×Q8C22×D5311yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C22⋊D5311yescharacteristic
287C4×D4C22×D5311yescharacteristic
288#117—21no—
289#117C22⋊D5311yescharacteristic
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C42—21no—
293C2×C42C22×D5311yescharacteristic
294#107—22no—
295#106—22no—
296C4⋊C8—21no—
297C4⋊C8—532no—
298C4⋊C8—532no—
299C4⋊C8—1062no—
300C4⋊C8—1061no—
301C4⋊C8C22×D5312yes—
302#103—22no—
303#102—22no—
304#102—1064no—
305#102—1064no—
306#102—1062no—
307#102—1062no—
308C22⋊C8—1062no—
309C22⋊C8—1061no—
310C4×C8—531no—
311C4×C8—531no—
312C4×C8—532no—
313C4×C8—1062no—
314C4×C8—1061no—
315C24—2121no—
316#41—22no—
317#41#1171992414yes—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×Q8C2×C22⋊D5311yescharacteristic
322C2×D4—21no—
323C2×D4—1062no—
324C2×D4—1062no—
325C2×D4—1062no—
326C2×D4—4242no—
327C2×D4—2124no—
328C2×D4—2121no—
329C2×D4—2121no—
330C2×D4—2121no—
331C2×D4—2122no—
332C2×D4—2122no—
333C2×D4C2×C22⋊D5311yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4D4×D5311yescharacteristic
339C22×C4#1171994711yescharacteristic
340C22×C4C2×C22⋊D5311yescharacteristic
341SD16—2124no—
342D8—2124no—
343M16—42no—
344M16—2122no—
345C2×C8—22no—
346C2×C8—532no—
347C2×C8—532no—
348C2×C8—1062no—
349C2×C8—1062no—
350C2×C8—2122no—
351C2×C8—2124no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4C2×C22⋊D5311yescharacteristic
359C4⋊C4C2×C22⋊D5311yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42D4×D5311yescharacteristic
367C42#1171994711yescharacteristic
368C42C2×C22⋊D5311yescharacteristic
369C42C23×D5311yescharacteristic
370C23—21no—
371C23—4242no—
372C23—4242no—
373C23—4242no—
374C23—2122no—
375C23#1313318911yescharacteristic
376Q8—21no—
377Q8—82no—
378Q8—82no—
379Q8C2×#1171992412yes—
380D4—24no—
381D4—21no—
382D4—4242no—
383D4—4242no—
384D4—4242no—
385D4—4242no—
386D4—2122no—
387D4—2122no—
388D4—2124no—
389D4C2×#1171992412yes—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4#1313318911yescharacteristic
401C2×C4#1313318911yescharacteristic
402C2×C4#1313318911yescharacteristic
403C2×C4C2×#1171992412yes—
404C2×C4#1313323712yes—
405C2×C4C2×D4×D5311yescharacteristic
406C2×C4C22×C22⋊D5311yescharacteristic
407C2×C4C2×#1171994711yescharacteristic
408C8—42no—
409C8—1062no—
410C8—1062no—
411C8—2122no—
412C8—2122no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—41no—
417C22—41no—
418C22—8484no—
419C22—4242no—
420C22—4242no—
421C22—4242no—
422C22#1681085412yes—
423C22C2×#1313318911yescharacteristic
424C4—21no—
425C4—21no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4#1681085412yes—
431C4#1681089112yes—
432C4C22×#1171992411yescharacteristic
433C4C2×#1313323711yescharacteristic
434C4#1681142212yes—
435C2—22no—
436C2—42no—
437C2—8482no—
438C2#2028421111yescharacteristic
439C2C2×#1681085411yescharacteristic
440C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3079C2
socleC22×C53C2×#121
prime radical#20284195C2
hypercenter#3079D53
hypocenterC53#19274
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×#16810971C2semidirect
3. #41×#88322C2semidirect
4. #20274133C2semidirect
5. #20274126C2semidirect
6. #20273470C2semidirect
7. #20273396C2semidirect
8. #20273384C2semidirect
9. C53×#3079C2semidirect
19. Q8×#88322C22semidirect
21. D4×#88322C22semidirect
29. #16810373C22semidirect
30. #16810372C22semidirect
34. C53×#477C22semidirect
35. C53×#476C22semidirect
216. #3079D53semidirect
237. #477C2×D53semidirect
238. #476C2×D53semidirect
261. C53#19274semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
#202841952
#41×#883222
#202741332
#202741262
#202734702
#202733962
#202733842
C53×#30792
#1927453