Finite Group Databasebeta

← #27369282

Subgroups — #27369282

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285660C211yesmaximal, characteristic, complemented
2#20284508C211yesmaximal, characteristic
3C2×#16810904C211yesmaximal, characteristic
4#20283738C211yesmaximal, characteristic
5#20274169C214yesmaximal, complemented
6#20274079C214yesmaximal, complemented
7#20273489C211yesmaximal, characteristic, complemented
8#20273399C211yesmaximal, characteristic, complemented
9C53×#1542C211yesmaximal, characteristic, complemented
10C2×#13133217C2211yescharacteristic
11C2×C4×D212C2211yescharacteristic
12#16811002C2212yescomplemented
13#16810975C2212yescomplemented
14#16810920—22no—
15#16810910C2212yes—
16#16810904—22no—
17#16810904C2214yes—
18#16810900C2212yes—
19#16810833—22no—
20#16810778—22no—
21C8×D212—22no—
22#16810688C2214yes—
23#16810676C2211yescharacteristic
24#16810666—21no—
25C2×#13133151—21no—
26#16810647C2211yescharacteristic
27#16810627—21no—
28C2×#13133139—21no—
29C22⋊C8×D53—22no—
30#16810375C2214yescomplemented
31#16810370C2214yescomplemented
32C53×#533C2211yescharacteristic
33C53×#431C2211yescharacteristic
34C2×C4⋊C8×C53C2211yescharacteristic
35C53×#333C2212yescomplemented
36C53×#332C2212yescomplemented
37C22×Dic106—21no—
38C22×D212D411yescharacteristic
39C22×C4×D53—21no—
40C2×#11719925—24no—
41C2×#11719917—21no—
42C2×C8×D53—24no—
43#13133217—21no—
44#13133217—21no—
45#13133217—21no—
46#13133217C2312yes—
47C4×D212—21no—
48C4×D212—21no—
49C4×D212D412yes—
50C4×D212C2312yes—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53D411yescharacteristic
54C2×#11719921—21no—
55#13133151—24no—
56#13133150—22no—
57#13133148—22no—
58C4⋊#88322—21no—
59C4⋊#88322D412yes—
60C4⋊#88322C2312yes—
61#13133140—24no—
62#13133139—24no—
63#13133137—22no—
64#13133137—22no—
65C22⋊#88322D412yes—
66#13133129—24no—
67C4×#88322C2312yes—
68C22×C8×C53—21no—
69C4⋊Q8×C53C2311yescharacteristic
70C4⋊Q8×C53C2311yescharacteristic
71#125×C53—21no—
72C22⋊Q8×C53—21no—
73C22⋊Q8×C53—21no—
74C4×Q8×C53—21no—
75C4×Q8×C53—21no—
76#117×C53—21no—
77C2×C42×C53C2311yescharacteristic
78C4⋊C8×C53—21no—
79C4⋊C8×C53C2312yes—
80C4⋊C8×C53C2312yes—
81C22⋊C8×C53—21no—
82C4×C8×C53—21no—
83C23×D53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×Dic53—21no—
90C22×Dic53—41no—
91C2×D212—21no—
92C2×D212—22no—
93C2×D212Q1614yescomplemented
94C2×D212C2×D412yes—
95C2×C4×D53—22no—
96C2×C4×D53—22no—
97C2×C4×D53—82no—
98C2×Dic106—22no—
99C2×Dic106—41no—
100C2×Dic106—82no—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53—42no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C42×C53—41no—
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C42×C53C2411yescharacteristic
116#11719925—84no—
117#11719921—22no—
118#11719921—22no—
119C4⋊Dic53—21no—
120C4⋊Dic53—22no—
121C4⋊Dic53—22no—
122C4⋊Dic53—41no—
123C4⋊Dic53—41no—
124C4⋊Dic53C2×D412yes—
125#11719917—42no—
126C2×#88322—22no—
127C2×#88322C2×D414yes—
128C8×D53—84no—
129C23×C53#12111yescharacteristic
130C22×D53—22no—
131C22×D53—24no—
132C22×D53—81no—
133C22×D53—81no—
134Q8×C53—42no—
135Q8×C53—81no—
136C2×C4×C53—22no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53C2×Q1612yes—
144C2×C4×C53#13512yes—
145C2×C4×C53C22×D411yescharacteristic
146C2×C4×C53C22×D411yescharacteristic
147C2×C4×C53C2×#4111yescharacteristic
148C2×Dic53—22no—
149C2×Dic53—42no—
150C2×Dic53—81no—
151C2×Dic53—82no—
152D212—22no—
153D212—24no—
154D212C2×Q1614yes—
155C4×D53—44no—
156C4×D53—82no—
157Dic106—44no—
158Dic106—82no—
159C8×C53—42no—
160C8×C53—42no—
161#88322—24no—
162#88322—42no—
163#19387—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53#46712yes—
167C22×C53C2×#12111yescharacteristic
168C2×D53—24no—
169C2×D53—42no—
170C2×D53—42no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53#46712yes—
178C4×C53C22×Q1611yescharacteristic
179C4×C53C2×#13511yescharacteristic
180C4×C53#57612yes—
181Dic53—42no—
182Dic53—82no—
183#3209—531no—
184#3077—531no—
185C2×#474—531no—
186#2972—531no—
187#1683—534no—
188#1669—534no—
189#1571—531no—
190#1549—531no—
191#1542D5311yescharacteristic, complemented
192C2×C53—22no—
193C2×C53#304111yescharacteristic
194C2×C53C2×#46711yescharacteristic
195C2×C53#306911yescharacteristic
196D53—44no—
197#533C2×D5311yescharacteristic
198C2×C4⋊Q8—531no—
199C2×C4×D4—531no—
200#494—532no—
201#487—532no—
202#478—1062no—
203#476—532no—
204#474—532no—
205#474—534no—
206#474—1062no—
207#462—1062no—
208#451—1062no—
209C8×D4—534no—
210C8×D4—1062no—
211#431—531no—
212#431C2×D5311yescharacteristic
213#428—1061no—
214C2×#107—1061no—
215#423—531no—
216C2×C4⋊C8C2×D5311yescharacteristic
217#418—1061no—
218C2×#102—1061no—
219C2×C22⋊C8—1062no—
220#335—534no—
221#333C2×D5312yescomplemented
222#332—534no—
223#332C2×D5312yescomplemented
224C53#1938711yescharacteristic, complemented, Sylow
225C22×Q8—1061no—
226C22×D4—531no—
227C23×C4—1061no—
228C2×Q16—1064no—
229C2×M16—1061no—
230C22×C8—21no—
231C22×C8—1064no—
232C4⋊Q8—532no—
233C4⋊Q8—1061no—
234C4⋊Q8—1061no—
235C4⋊Q8—1061no—
236C4⋊Q8C22×D5311yescharacteristic
237C4⋊Q8C22×D5311yescharacteristic
238#125—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—532no—
244C4×D4—532no—
245C4×D4—1061no—
246C4×D4—1061no—
247#117—21no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C22⋊C4—1061no—
252C2×C42C22×D5311yescharacteristic
253#107—1062no—
254#107—1064no—
255#106—1062no—
256C4⋊C8—21no—
257C4⋊C8—532no—
258C4⋊C8—532no—
259C4⋊C8—1061no—
260C4⋊C8C22×D5312yes—
261C4⋊C8C22×D5312yes—
262#103—1064no—
263#102—1062no—
264#102—1062no—
265#102—1064no—
266C22⋊C8—21no—
267C22⋊C8—532no—
268C22⋊C8—1064no—
269C4×C8—21no—
270C4×C8—532no—
271C24—1061no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—1062no—
275C2×Q8—4242no—
276C2×Q8—2121no—
277C2×D4—534no—
278C2×D4—532no—
279C2×D4—1061no—
280C2×D4—1062no—
281C22×C4—1061no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—4242no—
285C22×C4—2121no—
286C22×C4#1171994511yescharacteristic
287C22×C4D4×D5311yescharacteristic
288C22×C4D4×D5311yescharacteristic
289Q16—4244no—
290M16—2122no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—534no—
295C2×C8—1062no—
296C2×C8—4244no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—532no—
304C4⋊C4—1061no—
305C4⋊C4—1062no—
306C4⋊C4—1062no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C22⋊C4—41no—
310C22⋊C4—1062no—
311C22⋊C4—1062no—
312C42—41no—
313C42#1171994511yescharacteristic
314C42D4×D5311yescharacteristic
315C42D4×D5311yescharacteristic
316C42C23×D5311yescharacteristic
317C23—1064no—
318C23—1062no—
319C23—4241no—
320C23—4241no—
321C23#1313319011yescharacteristic
322Q8—42no—
323Q8—81no—
324Q8—4242no—
325Q8—2124no—
326D4—534no—
327D4—1064no—
328D4—1062no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—1062no—
334C2×C4—4242no—
335C2×C4—4242no—
336C2×C4—4241no—
337C2×C4—2122no—
338C2×C4—2124no—
339C2×C4#1313319011yescharacteristic
340C2×C4#1313319011yescharacteristic
341C2×C4#1313319011yescharacteristic
342C2×C4Q16×D5312yes—
343C2×C4#1313323812yes—
344C2×C4C2×D4×D5311yescharacteristic
345C2×C4C2×D4×D5311yescharacteristic
346C2×C4C2×#1171994511yescharacteristic
347C8—42no—
348C8—42no—
349C8—1064no—
350C8—2122no—
351C22—21no—
352C22—21no—
353C22—1064no—
354C22—4244no—
355C22—2122no—
356C22—2122no—
357C22#1681086312yes—
358C22C2×#1313319011yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—81no—
364C4—4242no—
365C4—2122no—
366C4#1681086312yes—
367C4C2×Q16×D5311yescharacteristic
368C4C2×#1313323811yescharacteristic
369C4#1681142512yes—
370C2—22no—
371C2—2124no—
372C2#2028422111yescharacteristic
373C2C2×#1681086311yescharacteristic
374C2#2028445511yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1542C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#1542D53
hypocenterC53#19387
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #20285660C2semidirect
5. #20274169C2semidirect
6. #20274079C2semidirect
7. #20273489C2semidirect
8. #20273399C2semidirect
9. C53×#1542C2semidirect
12. #16811002C22semidirect
13. #16810975C22semidirect
30. #16810375C22semidirect
31. #16810370C22semidirect
35. C53×#333C22semidirect
36. C53×#332C22semidirect
93. C2×D212Q16semidirect
191. #1542D53semidirect
221. #333C2×D53semidirect
223. #332C2×D53semidirect
224. C53#19387semidirect

Maximal subgroups

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

subgroupindex
#202856602
#202845082
C2×#168109042
#202837382
#202741692
#202740792
#202734892
#202733992
C53×#15422
#1938753