Finite Group Databasebeta

← #27369333

Subgroups — #27369333

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284508C211yesmaximal, characteristic
2C2×#16810910C211yesmaximal, characteristic
3#20283738C211yesmaximal, characteristic
4#20274163C214yesmaximal, complemented
5#20274111C214yesmaximal, complemented
6#20273490C211yesmaximal, characteristic, complemented
7#20273396C211yesmaximal, characteristic, complemented
8#20273372C211yesmaximal, characteristic, complemented
9C53×#3230C211yesmaximal, characteristic, complemented
10C2×C4×D212C2211yescharacteristic
11#16810920—22no—
12#16810910—22no—
13#16810910C2214yes—
14#16810910C2212yes—
15#16810900C2212yes—
16C8×D212C2214yes—
17#16810688—22no—
18#16810676C2211yescharacteristic
19#16810666—21no—
20C2×#13133148—21no—
21#16810647C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23#16810627—21no—
24C2×#13133137—21no—
25C22⋊C8×D53—22no—
26#16810376C2214yescomplemented
27#16810372C2212yescomplemented
28#16810372C2214yescomplemented
29#16810369C2212yescomplemented
30C53×#533C2211yescharacteristic
31C2×C4⋊Q8×C53C2211yescharacteristic
32C53×#500C2212yescomplemented
33C53×#487C2212yescomplemented
34C53×#484—22no—
35C53×#477—22no—
36C53×#431C2211yescharacteristic
37C22×D212D411yescharacteristic
38C22×C4×D53—21no—
39C2×#11719917—21no—
40C22×#88322—21no—
41C2×C8×D53—24no—
42C4×D212—21no—
43C4×D212—21no—
44C4×D212D412yes—
45C4×D212C2312yes—
46C2×C4⋊Dic53D411yescharacteristic
47C2×#11719921—21no—
48#13133150—22no—
49#13133148—24no—
50#13133148—22no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55C4⋊#88322C2312yes—
56#13133137—24no—
57#13133137—22no—
58#13133137—22no—
59C22⋊#88322—21no—
60#13133129—24no—
61C4×#88322—21no—
62C22×Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53C2311yescharacteristic
67C4⋊Q8×C53C2312yes—
68C4⋊Q8×C53C2311yescharacteristic
69#125×C53—21no—
70C22⋊Q8×C53—21no—
71C22⋊Q8×C53—21no—
72C4×Q8×C53—21no—
73C4×Q8×C53—21no—
74#117×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C42×C53C2311yescharacteristic
78#107×C53—24no—
79C4⋊C8×C53D412yes—
80#103×C53—24no—
81C22⋊C8×C53D412yes—
82C4×C8×C53C2312yes—
83C23×D53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—22no—
87C2×Q8×C53—41no—
88C2×Q8×C53—82no—
89C22×C4×C53—41no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—21no—
94C2×D212—21no—
95C2×D212—22no—
96C2×D212Q1614yes—
97C2×D212C2×D412yes—
98C2×C4×D53—22no—
99C2×C4×D53—22no—
100C2×C4×D53—82no—
101C2×C8×C53C2×D414yes—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C42×C53—41no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53C2411yescharacteristic
118#11719921—22no—
119#11719921—22no—
120C4⋊Dic53—21no—
121C4⋊Dic53C2×D412yes—
122#11719917—42no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—22no—
126C2×#88322—42no—
127C8×D53—84no—
128C23×C53#12111yescharacteristic
129C22×D53—22no—
130C22×D53—24no—
131C22×D53—81no—
132C22×D53—81no—
133Q8×C53—42no—
134Q8×C53—82no—
135Q8×C53—81no—
136Q8×C53—82no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—82no—
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53C2×Q1612yes—
147C2×C4×C53#13512yes—
148C2×C4×C53C22×D411yescharacteristic
149C2×C4×C53C22×D411yescharacteristic
150C2×C4×C53C2×#4111yescharacteristic
151C2×Dic53—22no—
152C2×Dic53—81no—
153D212—22no—
154D212—24no—
155D212C2×Q1614yes—
156C4×D53—44no—
157C4×D53—82no—
158C8×C53—24no—
159#88322—42no—
160#88322—42no—
161#88322—42no—
162#19393—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53#46712yes—
166C22×C53C2×#12111yescharacteristic
167C2×D53—24no—
168C2×D53—42no—
169C2×D53—42no—
170C2×D53—84no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—81no—
175C4×C53—82no—
176C4×C53—81no—
177C4×C53#46712yes—
178C4×C53C22×Q1611yescharacteristic
179C4×C53C2×#13511yescharacteristic
180C4×C53#57712yes—
181Dic53—42no—
182#3230D5311yescharacteristic, complemented
183#3077—531no—
184C2×#476—531no—
185#2972—531no—
186#1683—534no—
187#1675—534no—
188#1571—531no—
189#1548—531no—
190#1542—531no—
191C2×C53—22no—
192C2×C53#304211yescharacteristic
193C2×C53C2×#46711yescharacteristic
194C2×C53#306811yescharacteristic
195D53—44no—
196#533C2×D5311yescharacteristic
197C2×C4⋊Q8C2×D5311yescharacteristic
198C2×C4×D4—531no—
199#500C2×D5312yescomplemented
200#487C2×D5312yescomplemented
201#484—22no—
202#478—1062no—
203#477—22no—
204#476—532no—
205#476—534no—
206#476—1062no—
207#474—532no—
208C8×D4—534no—
209C8×D4—1062no—
210#431—531no—
211#431C2×D5311yescharacteristic
212#428—1061no—
213C2×#106—1061no—
214#423—531no—
215C2×C4⋊C8—531no—
216#418—1061no—
217C2×#102—1061no—
218C2×C22⋊C8—1062no—
219#335—534no—
220#333—532no—
221#333—534no—
222#332—532no—
223C53#1939311yescharacteristic, complemented, Sylow
224C22×Q8—21no—
225C22×D4—531no—
226C23×C4—1061no—
227C2×M16—1061no—
228C22×C8—1061no—
229C22×C8—1064no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22×D5312yes—
234C4⋊Q8C22×D5311yescharacteristic
235C4⋊Q8C22×D5311yescharacteristic
236#125—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×D4—532no—
242C4×D4—532no—
243C4×D4—1061no—
244C4×D4—1061no—
245#117—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—531no—
249C2×C22⋊C4—1061no—
250C2×C42C22×D5311yescharacteristic
251#107—24no—
252#107—1062no—
253#106—1064no—
254#106—1062no—
255C4⋊C8—532no—
256C4⋊C8—532no—
257C4⋊C8—532no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260C4⋊C8C22⋊D5312yes—
261#103—24no—
262#102—1064no—
263#102—1062no—
264#102—1062no—
265C22⋊C8—1061no—
266C22⋊C8—1064no—
267C22⋊C8C22⋊D5312yes—
268C4×C8—1061no—
269C4×C8C22×D5312yes—
270C24—1061no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×D4—534no—
277C2×D4—532no—
278C2×D4—1061no—
279C2×D4—1062no—
280C22×C4—41no—
281C22×C4—1061no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—4242no—
285C22×C4D4×D5311yescharacteristic
286C22×C4#1171994711yescharacteristic
287C22×C4C2×C22⋊D5311yescharacteristic
288M16—2122no—
289C2×C8—1062no—
290C2×C8—1062no—
291C2×C8—1062no—
292C2×C8—4244no—
293C2×C8—2122no—
294C2×C8C2×C22⋊D5314yes—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—532no—
306C4⋊C4—1061no—
307C22⋊C4—41no—
308C22⋊C4—1062no—
309C22⋊C4—1062no—
310C42—41no—
311C42D4×D5311yescharacteristic
312C42#1171994711yescharacteristic
313C42C2×C22⋊D5311yescharacteristic
314C42C23×D5311yescharacteristic
315C23—1064no—
316C23—1062no—
317C23—4241no—
318C23—4241no—
319C23#1313318811yescharacteristic
320Q8—42no—
321Q8—81no—
322Q8—82no—
323Q8—82no—
324D4—534no—
325D4—1064no—
326D4—1062no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—1062no—
334C2×C4—4242no—
335C2×C4—4241no—
336C2×C4—2124no—
337C2×C4#1313318811yescharacteristic
338C2×C4#1313318811yescharacteristic
339C2×C4#1313318811yescharacteristic
340C2×C4Q16×D5312yes—
341C2×C4#1313324012yes—
342C2×C4C2×D4×D5311yescharacteristic
343C2×C4C22×C22⋊D5311yescharacteristic
344C2×C4C2×#1171994711yescharacteristic
345C8—24no—
346C8—2122no—
347C8—2122no—
348C8—2122no—
349C22—21no—
350C22—21no—
351C22—1064no—
352C22—4244no—
353C22—2122no—
354C22—2122no—
355C22#1681086112yes—
356C22C2×#1313318811yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—82no—
361C4—81no—
362C4—81no—
363C4—2122no—
364C4#1681086112yes—
365C4C2×Q16×D5311yescharacteristic
366C4C2×#1313324011yescharacteristic
367C4#1681143112yes—
368C2—22no—
369C2—2124no—
370C2#2028422311yescharacteristic
371C2C2×#1681086111yescharacteristic
372C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3230C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3230D53
hypocenterC53#19393
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
4. #20274163C2semidirect
5. #20274111C2semidirect
6. #20273490C2semidirect
7. #20273396C2semidirect
8. #20273372C2semidirect
9. C53×#3230C2semidirect
26. #16810376C22semidirect
27. #16810372C22semidirect
28. #16810372C22semidirect
29. #16810369C22semidirect
32. C53×#500C22semidirect
33. C53×#487C22semidirect
182. #3230D53semidirect
199. #500C2×D53semidirect
200. #487C2×D53semidirect
223. C53#19393semidirect

Maximal subgroups

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

subgroupindex
#202845082
C2×#168109102
#202837382
#202741632
#202741112
#202734902
#202733962
#202733722
C53×#32302
#1939353