Finite Group Databasebeta

← #27369248

Subgroups — #27369248

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284508C211yesmaximal, characteristic, complemented
2C2×#16810815C211yesmaximal, characteristic
3#20283738C211yesmaximal, characteristic, complemented
4#20274121C214yesmaximal, complemented
5#20274053C214yesmaximal, complemented
6#20273486C211yesmaximal, characteristic, complemented
7#20273383C211yesmaximal, characteristic, complemented
8#20273372C211yesmaximal, characteristic, complemented
9C53×#3208C211yesmaximal, characteristic, complemented
10C22×#11719922—21no—
11C2×C4×D212C2211yescharacteristic
12#16810920—22no—
13#16810910C2212yescomplemented
14#16810900C2212yescomplemented
15#16810815—22no—
16#16810815C2214yes—
17C8×D212C2214yescomplemented
18#16810688—22no—
19#16810676C2211yescharacteristic
20#16810666—21no—
21#16810647C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23#16810627—21no—
24C2×#13133138—21no—
25C22⋊C8×D53—22no—
26#16810374C2214yescomplemented
27#16810372C2212yescomplemented
28#16810369C2212yescomplemented
29#16810367C2214yescomplemented
30C53×#533C2211yescharacteristic
31C2×#127×C53C2211yescharacteristic
32C53×#492C2212yescomplemented
33C53×#486C2212yescomplemented
34C53×#460—22no—
35C53×#450—22no—
36C53×#431C2211yescharacteristic
37C22×D212D411yescharacteristic
38C22×C4×D53—21no—
39C2×#11719922—24no—
40C2×#11719922—44no—
41C2×#11719917—21no—
42C22×#88322—21no—
43C2×C8×D53—24no—
44C4×D212—21no—
45C4×D212—21no—
46C4×D212D412yes—
47C4×D212C2312yes—
48C2×C4⋊Dic53D411yescharacteristic
49C2×#11719921—21no—
50#13133150—22no—
51#13133148—22no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322C2312yes—
55C4⋊#88322C2312yes—
56C4⋊#88322C2312yes—
57#13133138—24no—
58#13133137—22no—
59#13133137—22no—
60C22⋊#88322—21no—
61#13133129—24no—
62C4×#88322—21no—
63C22×D4×C53—21no—
64C22×D4×C53—21no—
65C22×D4×C53—21no—
66C2×SD16×C53—24no—
67C4⋊Q8×C53C2311yescharacteristic
68C4⋊Q8×C53C2311yescharacteristic
69#127×C53—21no—
70#127×C53—21no—
71#127×C53—21no—
72#127×C53C2312yes—
73#125×C53—21no—
74C22⋊Q8×C53—21no—
75C22⋊Q8×C53—21no—
76C4×Q8×C53—21no—
77C4×Q8×C53—21no—
78#117×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53D412yes—
81#102×C53—24no—
82C22⋊C8×C53D412yes—
83C4×C8×C53C2312yes—
84C24×C53—41no—
85C23×D53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×D4×C53—22no—
89C2×D4×C53—22no—
90C2×D4×C53—22no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—42no—
95C2×D4×C53—42no—
96C2×D4×C53—82no—
97C2×D4×C53—82no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53#4111yescharacteristic
101C22×Dic53—21no—
102C2×D212—21no—
103C2×D212—22no—
104C2×D212SD1614yescomplemented
105C2×D212C2×D412yes—
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—82no—
109SD16×C53—84no—
110C2×C8×C53C2×D414yes—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C42×C53—41no—
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C42×C53C2411yescharacteristic
123#11719922—48no—
124#11719921—22no—
125#11719921—22no—
126C4⋊Dic53—21no—
127C4⋊Dic53C2×D412yes—
128#11719917—42no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322—42no—
133C8×D53—84no—
134C23×C53—42no—
135C23×C53—82no—
136C23×C53—82no—
137C23×C53—82no—
138C23×C53#12111yescharacteristic
139C22×D53—22no—
140C22×D53—24no—
141C22×D53—81no—
142C22×D53—81no—
143Q8×C53—42no—
144Q8×C53—81no—
145D4×C53—42no—
146D4×C53—42no—
147D4×C53—44no—
148D4×C53—82no—
149D4×C53—82no—
150D4×C53—82no—
151D4×C53—82no—
152C2×C4×C53—22no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53C2×SD1612yes—
160C2×C4×C53#13512yes—
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C22×D411yescharacteristic
163C2×C4×C53C2×#4111yescharacteristic
164C2×Dic53—22no—
165C2×Dic53—81no—
166D212—22no—
167D212—24no—
168D212C2×SD1614yes—
169C4×D53—44no—
170C4×D53—82no—
171C8×C53—24no—
172#88322—42no—
173#88322—42no—
174#88322—42no—
175#19384—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—82no—
179C22×C53—82no—
180C22×C53—82no—
181C22×C53—164no—
182C22×C53#46512yes—
183C22×C53C2×#12111yescharacteristic
184C2×D53—24no—
185C2×D53—42no—
186C2×D53—42no—
187C2×D53—84no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—81no—
192C4×C53—81no—
193C4×C53#46512yes—
194C4×C53C22×SD1611yescharacteristic
195C4×C53C2×#13511yescharacteristic
196C4×C53#57612yes—
197Dic53—42no—
198#3208D5311yescharacteristic, complemented
199#3077—531no—
200C2×#459—531no—
201#2972—531no—
202#1677—534no—
203#1665—534no—
204#1570—531no—
205#1545—531no—
206#1542—531no—
207C2×C53—22no—
208C2×C53—162no—
209C2×C53#304111yescharacteristic
210C2×C53C2×#46511yescharacteristic
211C2×C53#306811yescharacteristic
212D53—44no—
213C22×D8—1061no—
214#533C2×D5311yescharacteristic
215C2×#127C2×D5311yescharacteristic
216C2×C4×D4—531no—
217#492C2×D5312yescomplemented
218#486C2×D5312yescomplemented
219#478—1062no—
220#476—532no—
221#474—532no—
222#460—22no—
223#459—534no—
224#459—1062no—
225#450—22no—
226C8×D4—534no—
227C8×D4—1062no—
228#431—531no—
229#431C2×D5311yescharacteristic
230#428—1061no—
231#423—531no—
232C2×C4⋊C8—531no—
233#418—1061no—
234C2×#102—1061no—
235C2×C22⋊C8—1062no—
236#334—534no—
237#333—532no—
238#332—532no—
239#331—534no—
240C53#1938411yescharacteristic, complemented, Sylow
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—531no—
245C23×C4—1061no—
246C2×SD16—24no—
247C2×D8—1064no—
248C2×D8—2124no—
249C2×M16—1061no—
250C22×C8—1061no—
251C22×C8—1064no—
252C4⋊Q8C22×D5311yescharacteristic
253C4⋊Q8C22×D5311yescharacteristic
254#127—21no—
255#127—21no—
256#127—21no—
257#127C22×D5312yes—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—532no—
264C4×D4—532no—
265C4×D4—1061no—
266C4×D4—1061no—
267#117—21no—
268C2×C4⋊C4—531no—
269C2×C22⋊C4—1061no—
270C2×C42C22×D5311yescharacteristic
271#107—1062no—
272#106—1062no—
273C4⋊C8—532no—
274C4⋊C8—532no—
275C4⋊C8—532no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278C4⋊C8C22⋊D5312yes—
279#102—24no—
280#102—1062no—
281#102—1062no—
282#102—1064no—
283C22⋊C8—1061no—
284C22⋊C8—1064no—
285C22⋊C8C22⋊D5312yes—
286C4×C8—1061no—
287C4×C8C22×D5312yes—
288C24—41no—
289C24—1061no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—534no—
303C2×D4—532no—
304C2×D4—1061no—
305C2×D4—1062no—
306C22×C4—1061no—
307C22×C4—1062no—
308C22×C4—1062no—
309C22×C4—4242no—
310C22×C4D4×D5311yescharacteristic
311C22×C4#1171994711yescharacteristic
312C22×C4C2×C22⋊D5311yescharacteristic
313SD16—84no—
314D8—2128no—
315M16—2122no—
316C2×C8—1062no—
317C2×C8—1062no—
318C2×C8—1062no—
319C2×C8—4244no—
320C2×C8—2122no—
321C2×C8C2×C22⋊D5314yes—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—532no—
329C4⋊C4—1061no—
330C22⋊C4—41no—
331C22⋊C4—1062no—
332C22⋊C4—1062no—
333C42—41no—
334C42D4×D5311yescharacteristic
335C42#1171994711yescharacteristic
336C42C2×C22⋊D5311yescharacteristic
337C42C23×D5311yescharacteristic
338C23—42no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—1064no—
343C23—1062no—
344C23—4241no—
345C23—4241no—
346C23#1313318811yescharacteristic
347Q8—42no—
348Q8—81no—
349D4—42no—
350D4—42no—
351D4—44no—
352D4—82no—
353D4—82no—
354D4—82no—
355D4—82no—
356D4—534no—
357D4—1064no—
358D4—1062no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—1062no—
364C2×C4—4242no—
365C2×C4—4241no—
366C2×C4—2124no—
367C2×C4#1313318811yescharacteristic
368C2×C4#1313318811yescharacteristic
369C2×C4#1313318811yescharacteristic
370C2×C4SD16×D5312yes—
371C2×C4#1313323812yes—
372C2×C4C2×D4×D5311yescharacteristic
373C2×C4C22×C22⋊D5311yescharacteristic
374C2×C4C2×#1171994711yescharacteristic
375C8—24no—
376C8—2122no—
377C8—2122no—
378C8—2122no—
379C22—21no—
380C22—21no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—164no—
385C22—1064no—
386C22—4244no—
387C22—2122no—
388C22—2122no—
389C22#1681084912yes—
390C22C2×#1313318811yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—81no—
395C4—81no—
396C4—2122no—
397C4#1681084912yes—
398C4C2×SD16×D5311yescharacteristic
399C4C2×#1313323811yescharacteristic
400C4#1681142612yes—
401C2—22no—
402C2—162no—
403C2—2124no—
404C2#2028421411yescharacteristic
405C2C2×#1681084911yescharacteristic
406C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3208C2
socleC22×C53C2×#121
prime radicalC2×#16810815C2
hypercenter#3208D53
hypocenterC53#19384
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #20284508C2semidirect
3. #20283738C2semidirect
4. #20274121C2semidirect
5. #20274053C2semidirect
6. #20273486C2semidirect
7. #20273383C2semidirect
8. #20273372C2semidirect
9. C53×#3208C2semidirect
13. #16810910C22semidirect
14. #16810900C22semidirect
17. C8×D212C22semidirect
26. #16810374C22semidirect
27. #16810372C22semidirect
28. #16810369C22semidirect
29. #16810367C22semidirect
32. C53×#492C22semidirect
33. C53×#486C22semidirect
104. C2×D212SD16semidirect
198. #3208D53semidirect
217. #492C2×D53semidirect
218. #486C2×D53semidirect
240. C53#19384semidirect

Maximal subgroups

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

subgroupindex
#202845082
C2×#168108152
#202837382
#202741212
#202740532
#202734862
#202733832
#202733722
C53×#32082
#1938453