Finite Group Databasebeta

← #27367386

Subgroups — #27367386

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285660C211yesmaximal, characteristic, complemented
2#20285653C211yesmaximal, characteristic
3C2×#16810970C211yesmaximal, characteristic
4#20274283C214yesmaximal, complemented
5#20274159C214yesmaximal, complemented
6#20273463C211yesmaximal, characteristic, complemented
7#20273401C211yesmaximal, characteristic, complemented
8#20273384C211yesmaximal, characteristic
9C53×#2960C211yesmaximal, characteristic, complemented
10C2×#13133217C2211yescharacteristic
11C2×#13133214C2211yescharacteristic
12#16811002—21no—
13#16810997—21no—
14#16810975—21no—
15#16810971—21no—
16#16810970—21no—
17#16810970—21no—
18#16810970C2212yes—
19#16810970C2212yes—
20#16810833C2214yescomplemented
21#16810823C2214yes—
22#16810778—22no—
23#16810773—22no—
24#16810676C2211yescharacteristic
25#16810676C2211yescharacteristic
26C2×#13133139—21no—
27C2×#13133139—21no—
28#16810376C2212yescomplemented
29#16810374C2212yescomplemented
30#16810368C2214yescomplemented
31#16810366C2214yes—
32C53×#533C2211yescharacteristic
33C53×#446—21no—
34C53×#446C2212yescomplemented
35C8×Q8×C53—21no—
36C8×Q8×C53C2212yescomplemented
37C53×#424—21no—
38C53×#423C2211yescharacteristic
39C53×#405—21no—
40C2×C4×C8×C53C2211yescharacteristic
41C22×Dic106—21no—
42C22×D212—21no—
43C22×D212—21no—
44C22×D212—21no—
45C2×#11719925—24no—
46C2×#11719924—24no—
47#13133217—21no—
48#13133217—21no—
49#13133217—21no—
50#13133217C2312yes—
51#13133214—21no—
52#13133214—21no—
53#13133214—21no—
54#13133214C2312yes—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C4⋊#88322C2312yes—
58C4⋊#88322C2312yes—
59#13133140—24no—
60#13133139—22no—
61#13133139—22no—
62#13133139—22no—
63#13133139—22no—
64#13133137—24no—
65C22⋊#88322—21no—
66C22⋊#88322—21no—
67C4×#88322—21no—
68C4×#88322—21no—
69C2×M16×C53—21no—
70C22×C8×C53D411yescharacteristic
71C22×C8×C53D411yescharacteristic
72C4⋊Q8×C53D411yescharacteristic
73C4⋊Q8×C53D411yescharacteristic
74#125×C53—21no—
75C22⋊Q8×C53—21no—
76C22⋊Q8×C53D412yes—
77C4×Q8×C53—21no—
78C4×Q8×C53C2312yes—
79#117×C53—21no—
80C2×C42×C53C2311yescharacteristic
81C4⋊C8×C53—22no—
82C4⋊C8×C53—22no—
83C4⋊C8×C53—21no—
84C4⋊C8×C53C2312yes—
85#97×C53—22no—
86C4×C8×C53—22no—
87C4×C8×C53D411yescharacteristic
88C4×C8×C53D411yescharacteristic
89C4×C8×C53C2311yescharacteristic
90C4×C8×C53C2311yescharacteristic
91C23×D53—41no—
92C2×Q8×C53—21no—
93C2×Q8×C53C2×D412yes—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×Dic53—41no—
98C2×D212—22no—
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—42no—
102C2×D212—42no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—82no—
107C2×D212—82no—
108C2×Dic106—22no—
109C2×Dic106—41no—
110C2×Dic106—82no—
111M16×C53—42no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53—22no—
115C2×C8×C53C2×D411yescharacteristic
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53C2×D412yes—
125C22⋊C4×C53—22no—
126C42×C53—22no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53#4111yescharacteristic
130C42×C53C2411yescharacteristic
131#11719925—84no—
132#11719924—84no—
133C4⋊Dic53—22no—
134C4⋊Dic53—22no—
135C4⋊Dic53—41no—
136C4⋊Dic53—41no—
137C2×#88322—22no—
138C2×#88322—22no—
139C23×C53#12111yescharacteristic
140C22×D53—42no—
141C22×D53—82no—
142C22×D53—82no—
143C22×D53—82no—
144Q8×C53—24no—
145Q8×C53—42no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#13512yes—
154C2×C4×C53#13712yes—
155C2×C4×C53C22×D411yescharacteristic
156C2×C4×C53C22×D411yescharacteristic
157C2×C4×C53C2×#4111yescharacteristic
158C2×Dic53—42no—
159C2×Dic53—82no—
160D212—42no—
161D212—42no—
162D212—44no—
163D212—82no—
164D212—82no—
165D212—82no—
166D212—82no—
167Dic106—44no—
168Dic106—82no—
169C8×C53—21no—
170C8×C53—21no—
171C8×C53—21no—
172C8×C53—21no—
173C8×C53—42no—
174#88322—42no—
175#88322—42no—
176#19214—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53#46412yes—
180C22×C53C2×#12111yescharacteristic
181C2×D53—82no—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—164no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—42no—
188C4×C53—42no—
189C4×C53#46412yes—
190C4×C53#47112yes—
191C4×C53C2×#13511yescharacteristic
192C4×C53C2×#13711yescharacteristic
193C4×C53#57612yes—
194Dic53—82no—
195#3209—531no—
196#3208—531no—
197C2×#486—531no—
198#2960D5311yescharacteristic, complemented
199#1702—534no—
200#1682—534no—
201#1564—531no—
202#1549—531no—
203#1545—531no—
204C2×C53—22no—
205C2×C53C2×#46411yescharacteristic
206C2×C53#305811yescharacteristic
207C2×C53C2×#47111yescharacteristic
208D53—162no—
209#533C2×D5311yescharacteristic
210C2×C4⋊Q8—531no—
211C2×#127—531no—
212#494—1061no—
213#492—1061no—
214#487—1061no—
215#486—532no—
216#486—532no—
217#486—1061no—
218#486—1061no—
219#486—1061no—
220#462—534no—
221#460—534no—
222#451—1062no—
223#450—1062no—
224#446—21no—
225#446C2×D5312yescomplemented
226C8×Q8—21no—
227C8×Q8C2×D5312yescomplemented
228#431—531no—
229#431—531no—
230#424—21no—
231#423C2×D5311yescharacteristic
232C2×#102—1061no—
233C2×#102—1061no—
234#405—21no—
235C2×C4×C8C2×D5311yescharacteristic
236#335—532no—
237#334—532no—
238#332—534no—
239#331—534no—
240C53#1921411yescharacteristic, complemented, Sylow
241C22×Q8—1061no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4—1061no—
245C2×Q16—1064no—
246C2×SD16—1064no—
247C2×M16—21no—
248C22×C8C22⋊D5311yescharacteristic
249C22×C8C22⋊D5311yescharacteristic
250C4⋊Q8—532no—
251C4⋊Q8—1061no—
252C4⋊Q8—1061no—
253C4⋊Q8—1061no—
254C4⋊Q8D21211yescharacteristic
255C4⋊Q8D21211yescharacteristic
256#127—532no—
257#127—1061no—
258#127—1061no—
259#127—1061no—
260#125—21no—
261C22⋊Q8—21no—
262C22⋊Q8D21212yes—
263C4×Q8—21no—
264C4×Q8C22×D5312yes—
265#117—21no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4—1061no—
268C2×C42C22×D5311yescharacteristic
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—21no—
272C4⋊C8—532no—
273C4⋊C8—532no—
274C4⋊C8C22×D5312yes—
275#103—1064no—
276#102—1062no—
277#102—1062no—
278#102—1062no—
279#102—1062no—
280#102—1064no—
281C22⋊C8—1061no—
282C22⋊C8—1061no—
283#97—22no—
284C4×C8—22no—
285C4×C8—1061no—
286C4×C8—1061no—
287C4×C8C22⋊D5311yescharacteristic
288C4×C8C22⋊D5311yescharacteristic
289C4×C8C22×D5311yescharacteristic
290C4×C8C22×D5311yescharacteristic
291C24—2121no—
292C2×Q8—21no—
293C2×Q8—1062no—
294C2×Q8—4242no—
295C2×Q8—2121no—
296C2×Q8C2×D21212yes—
297C2×D4—1062no—
298C2×D4—1062no—
299C2×D4—1062no—
300C2×D4—4242no—
301C2×D4—4242no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4—2122no—
306C2×D4—2122no—
307C22×C4—2121no—
308C22×C4C2×D21211yescharacteristic
309C22×C4#1171994511yescharacteristic
310C22×C4C2×C22⋊D5311yescharacteristic
311Q16—4244no—
312SD16—4244no—
313M16—42no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—1062no—
318C2×C8—1062no—
319C2×C8C2×C22⋊D5311yescharacteristic
320C2×C8C2×C22⋊D5311yescharacteristic
321C2×C8C2×C22⋊D5311yescharacteristic
322C2×C8C2×C22⋊D5311yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—1062no—
329C4⋊C4—1062no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C4⋊C4C2×D21212yes—
333C22⋊C4—22no—
334C42—22no—
335C42C2×D21211yescharacteristic
336C42#1171994511yescharacteristic
337C42C2×C22⋊D5311yescharacteristic
338C42C23×D5311yescharacteristic
339C23—4242no—
340C23—4242no—
341C23—4242no—
342C23—2122no—
343C23#1313319111yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—4242no—
347Q8—2124no—
348D4—4242no—
349D4—4242no—
350D4—4242no—
351D4—4242no—
352D4—2122no—
353D4—2122no—
354D4—2124no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—4242no—
360C2×C4—2122no—
361C2×C4#1313319111yescharacteristic
362C2×C4#1313319111yescharacteristic
363C2×C4#1313319111yescharacteristic
364C2×C4#1313323712yes—
365C2×C4#1313324812yes—
366C2×C4C22×C22⋊D5311yescharacteristic
367C2×C4C22×D21211yescharacteristic
368C2×C4C2×#1171994511yescharacteristic
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—42no—
374C8—2122no—
375C8—2122no—
376C22—21no—
377C22—21no—
378C22—8484no—
379C22—4242no—
380C22—4242no—
381C22—4242no—
382C22#1681084512yes—
383C22C2×#1313319111yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—4242no—
389C4#1681084512yes—
390C4#1681088612yes—
391C4C2×#1313323711yescharacteristic
392C4C2×#1313324811yescharacteristic
393C4#1681142812yes—
394C2—22no—
395C2—8482no—
396C2C2×#1681084511yescharacteristic
397C2#2028436311yescharacteristic
398C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2960C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#2960D53
hypocenterC53#19214
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960

Complements

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

NH ≅ G/Ntype
1. #20285660C2semidirect
4. #20274283C2semidirect
5. #20274159C2semidirect
6. #20273463C2semidirect
7. #20273401C2semidirect
9. C53×#2960C2semidirect
20. #16810833C22semidirect
28. #16810376C22semidirect
29. #16810374C22semidirect
30. #16810368C22semidirect
34. C53×#446C22semidirect
36. C8×Q8×C53C22semidirect
198. #2960D53semidirect
225. #446C2×D53semidirect
227. C8×Q8C2×D53semidirect
240. C53#19214semidirect

Maximal subgroups

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

subgroupindex
#202856602
#202856532
C2×#168109702
#202742832
#202741592
#202734632
#202734012
#202733842
C53×#29602
#1921453