Finite Group Databasebeta

← #27369635

Subgroups — #27369635

Order 13568.

714 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#20285653C211yesmaximal, characteristic, complemented
2#20284508C211yesmaximal, characteristic, complemented
3C2×#16810819C211yesmaximal, characteristic
4#20283755C211yesmaximal, characteristic, complemented
5#20274251C214yesmaximal, complemented
6#20274183C214yesmaximal, complemented
7#20273489C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic, complemented
9C53×#1542C211yesmaximal, characteristic, complemented
10C22×D424—21no—
11C2×#13133214C2211yescharacteristic
12C2×C4×D212C2211yescharacteristic
13#16810997C2212yescomplemented
14#16810971C2212yescomplemented
15#16810920—22no—
16#16810910C2212yescomplemented
17#16810900C2212yescomplemented
18#16810823—22no—
19#16810819—22no—
20#16810819C2214yes—
21#16810773—22no—
22#16810702—22no—
23#16810700C2214yescomplemented
24#16810676C2211yescharacteristic
25#16810666—21no—
26#16810647C2211yescharacteristic
27#16810627—21no—
28C2×#13133139—21no—
29#16810585—22no—
30#16810375C2214yescomplemented
31#16810366C2214yescomplemented
32C53×#533C2211yescharacteristic
33C53×#431C2211yescharacteristic
34C2×C4⋊C8×C53C2211yescharacteristic
35C53×#333C2212yescomplemented
36C53×#332C2212yescomplemented
37C22×D212—21no—
38C22×D212—21no—
39C22×D212—21no—
40C22×D212D411yescharacteristic
41C22×C4×D53—21no—
42C2×#11719924—24no—
43C2×D424—24no—
44C2×D424—44no—
45C2×#11719917—21no—
46C2×#11719912—24no—
47#13133214—21no—
48#13133214—21no—
49#13133214—21no—
50#13133214C2312yes—
51C4×D212—21no—
52C4×D212—21no—
53C4×D212D412yes—
54C4×D212C2312yes—
55C2×C4⋊Dic53D411yescharacteristic
56C2×#11719921—21no—
57#13133150—22no—
58#13133148—22no—
59C4⋊#88322—21no—
60C4⋊#88322D412yes—
61C4⋊#88322C2312yes—
62#13133139—24no—
63#13133137—24no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322D412yes—
67#13133129—24no—
68C4×#88322C2312yes—
69C22×C8×C53—21no—
70C4⋊Q8×C53C2311yescharacteristic
71C4⋊Q8×C53C2311yescharacteristic
72#125×C53—21no—
73C22⋊Q8×C53—21no—
74C22⋊Q8×C53—21no—
75C4×Q8×C53—21no—
76C4×Q8×C53—21no—
77#117×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—21no—
80C4⋊C8×C53C2312yes—
81C4⋊C8×C53C2312yes—
82C22⋊C8×C53—21no—
83C4×C8×C53—21no—
84C23×D53—21no—
85C23×D53—41no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C22×Dic53—21no—
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—21no—
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—22no—
98C2×D212—42no—
99C2×D212—42no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—82no—
104C2×D212—82no—
105C2×D212C2×D412yes—
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—82no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—42no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C42×C53—41no—
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C42×C53C2411yescharacteristic
124#11719924—84no—
125#11719921—22no—
126#11719921—22no—
127C4⋊Dic53—21no—
128C4⋊Dic53C2×D412yes—
129#11719917—42no—
130C2×#88322—22no—
131C2×#88322C2×D414yes—
132D424—48no—
133#11719912—84no—
134C23×C53#12111yescharacteristic
135C22×D53—22no—
136C22×D53—42no—
137C22×D53—42no—
138C22×D53—81no—
139C22×D53—81no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—82no—
143Q8×C53—42no—
144Q8×C53—81no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)12yes—
153C2×C4×C53#13712yes—
154C2×C4×C53C22×D411yescharacteristic
155C2×C4×C53C22×D411yescharacteristic
156C2×C4×C53C2×#4111yescharacteristic
157C2×Dic53—22no—
158C2×Dic53—81no—
159D212—22no—
160D212—22no—
161D212—42no—
162D212—42no—
163D212—44no—
164D212—42no—
165D212—82no—
166D212—82no—
167D212—82no—
168D212—82no—
169C4×D53—44no—
170C4×D53—82no—
171C8×C53—42no—
172C8×C53—42no—
173#88322—24no—
174#88322—42no—
175#19420—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53#46812yes—
179C22×C53C2×#12111yescharacteristic
180C2×D53—42no—
181C2×D53—42no—
182C2×D53—42no—
183C2×D53—84no—
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—82no—
187C2×D53—164no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—81no—
192C4×C53—81no—
193C4×C53#46812yes—
194C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
195C4×C53C2×#13711yescharacteristic
196C4×C53#57612yes—
197Dic53—42no—
198#3208—531no—
199#3077—531no—
200C2×#459—531no—
201#2975—531no—
202#1697—534no—
203#1685—534no—
204#1571—531no—
205#1545—531no—
206#1542D5311yescharacteristic, complemented
207C2×C53—22no—
208C2×C53#304111yescharacteristic
209C2×C53C2×#46811yescharacteristic
210C2×C53#306811yescharacteristic
211D53—82no—
212D53—162no—
213C22×D8—1061no—
214#533C2×D5311yescharacteristic
215C2×#127—531no—
216C2×C4×D4—531no—
217#492—532no—
218#486—532no—
219#478—1062no—
220#476—532no—
221#474—532no—
222#460—1062no—
223#459—534no—
224#459—1062no—
225#450—1062no—
226#436—534no—
227#436—1062no—
228#431—531no—
229#431C2×D5311yescharacteristic
230#428—1061no—
231#423—531no—
232C2×C4⋊C8C2×D5311yescharacteristic
233#418—1061no—
234C2×#102—1061no—
235#407—1062no—
236#335—534no—
237#333C2×D5312yescomplemented
238#332C2×D5312yescomplemented
239#331—534no—
240C53#1942011yescharacteristic, complemented, Sylow
241C22×D4—531no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4—1061no—
245C23×C4—1061no—
246C2×SD16—1064no—
247C2×D8—1064no—
248C2×D8—2124no—
249C2×M16—1064no—
250C2×M16—1061no—
251C22×C8—21no—
252C4⋊Q8C22×D5311yescharacteristic
253C4⋊Q8C22×D5311yescharacteristic
254#127—532no—
255#127—1061no—
256#127—1061no—
257#127—1061no—
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—21no—
274C4⋊C8—532no—
275C4⋊C8—532no—
276C4⋊C8—1061no—
277C4⋊C8C22×D5312yes—
278C4⋊C8C22×D5312yes—
279#102—1064no—
280#102—1062no—
281#102—1062no—
282#102—1064no—
283C22⋊C8—21no—
284C22⋊C8—532no—
285C22⋊C8—1064no—
286C4×C8—21no—
287C4×C8—532no—
288C24—1061no—
289C24—2121no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×D4—532no—
293C2×D4—1061no—
294C2×D4—1062no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—1062no—
299C2×D4—4242no—
300C2×D4—4242no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2122no—
305C2×D4—2122no—
306C22×C4—1061no—
307C22×C4—1062no—
308C22×C4—1062no—
309C22×C4—4242no—
310C22×C4#1171994511yescharacteristic
311C22×C4D4×D5311yescharacteristic
312C22×C4D4×D5311yescharacteristic
313SD16—4244no—
314D8—2128no—
315M16—4244no—
316M16—2122no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—534no—
321C2×C8—1062no—
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—
334C42#1171994511yescharacteristic
335C42D4×D5311yescharacteristic
336C42D4×D5311yescharacteristic
337C42C23×D5311yescharacteristic
338C23—1062no—
339C23—4242no—
340C23—4242no—
341C23—4241no—
342C23—4241no—
343C23—4242no—
344C23—2122no—
345C23—2122no—
346C23#1313319011yescharacteristic
347Q8—42no—
348Q8—81no—
349D4—1062no—
350D4—1062no—
351D4—4242no—
352D4—4242no—
353D4—4242no—
354D4—4242no—
355D4—2122no—
356D4—2122no—
357D4—2122no—
358D4—2124no—
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#1313319011yescharacteristic
368C2×C4#1313319011yescharacteristic
369C2×C4#1313319011yescharacteristic
370C2×C4#1313324412yes—
371C2×C4#1313324912yes—
372C2×C4C2×D4×D5311yescharacteristic
373C2×C4C2×D4×D5311yescharacteristic
374C2×C4C2×#1171994511yescharacteristic
375C8—42no—
376C8—42no—
377C8—1064no—
378C8—2122no—
379C22—21no—
380C22—21no—
381C22—8484no—
382C22—4244no—
383C22—4242no—
384C22—4242no—
385C22—4242no—
386C22—2122no—
387C22—2122no—
388C22—2122no—
389C22#1681087212yes—
390C22C2×#1313319011yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—81no—
395C4—81no—
396C4—2122no—
397C4#1681087212yes—
398C4C2×#1313324411yescharacteristic
399C4C2×#1313324911yescharacteristic
400C4#1681142512yes—
401C2—22no—
402C2—8482no—
403C2—4242no—
404C2#2028422111yescharacteristic
405C2C2×#1681087211yescharacteristic
406C2#2028444611yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1542C2
socleC22×C53C2×#121
prime radicalC2×#16810819C2
hypercenter#1542D53
hypocenterC53#19420
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. #20285653C2semidirect
2. #20284508C2semidirect
4. #20283755C2semidirect
5. #20274251C2semidirect
6. #20274183C2semidirect
7. #20273489C2semidirect
8. #20273382C2semidirect
9. C53×#1542C2semidirect
13. #16810997C22semidirect
14. #16810971C22semidirect
16. #16810910C22semidirect
17. #16810900C22semidirect
23. #16810700C22semidirect
30. #16810375C22semidirect
31. #16810366C22semidirect
35. C53×#333C22semidirect
36. C53×#332C22semidirect
206. #1542D53semidirect
237. #333C2×D53semidirect
238. #332C2×D53semidirect
240. C53#19420semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202845082
C2×#168108192
#202837552
#202742512
#202741832
#202734892
#202733822
C53×#15422
#1942053