Finite Group Databasebeta

← #27367389

Subgroups — #27367389

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#20285663C211yesmaximal, characteristic, complemented
2#20285658C211yesmaximal, characteristic
3C2×#16810971C211yesmaximal, characteristic
4#20283645C211yesmaximal, characteristic, complemented
5#20274286C214yesmaximal, complemented
6#20274162C214yesmaximal, complemented
7#20273461C211yesmaximal, characteristic, complemented
8#20273384C211yesmaximal, characteristic
9C53×#1549C211yesmaximal, characteristic, complemented
10#16811180C2211yescharacteristic
11C2×#13133214C2211yescharacteristic
12#16811002—21no—
13#16810996—21no—
14#16810973—21no—
15#16810971—21no—
16#16810971—21no—
17#16810971C2212yes—
18#16810971C2212yes—
19#16810970—21no—
20#16810830C2214yescomplemented
21#16810824C2214yes—
22#16810779—22no—
23#16810776—22no—
24#16810750—21no—
25#16810750C2212yescomplemented
26#16810747—21no—
27#16810747C2212yescomplemented
28#16810676C2211yescharacteristic
29#16810651—21no—
30C2×#13133137—21no—
31C2×#13133137—21no—
32#16810581—21no—
33C2×C4×#88322C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810366C2214yes—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C53×#431C2211yescharacteristic
39C53×#423C2211yescharacteristic
40C53×#332C2214yescomplemented
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44C2×#11719925—24no—
45C2×#11719913—24no—
46C22×#88322D411yescharacteristic
47C22×#88322D411yescharacteristic
48#13133217D411yescharacteristic
49#13133217D411yescharacteristic
50#13133214—21no—
51#13133214—21no—
52#13133214—21no—
53#13133214C2312yes—
54#13133209—21no—
55#13133196—21no—
56#13133196D412yes—
57C4×Dic106—21no—
58C4×Dic106C2312yes—
59#13133176—21no—
60C4⋊#88322C2312yes—
61#13133146—22no—
62#13133146—22no—
63#13133141—24no—
64#13133139—24no—
65#13133137—22no—
66#13133137—22no—
67#13133137—22no—
68#13133137—22no—
69C22⋊#88322—21no—
70#13133127—22no—
71C4×#88322—21no—
72C4×#88322D411yescharacteristic
73C4×#88322D411yescharacteristic
74C4×#88322C2311yescharacteristic
75C4×#88322C2311yescharacteristic
76C8×Dic53—22no—
77C22×Q8×C53—21no—
78C2×M16×C53—21no—
79C4⋊Q8×C53—21no—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53C2312yes—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C42×C53C2311yescharacteristic
86C4⋊C8×C53—21no—
87C4⋊C8×C53C2312yes—
88C4⋊C8×C53C2312yes—
89C22⋊C8×C53—21no—
90C4×C8×C53—21no—
91C23×D53—41no—
92C2×Q8×C53—22no—
93C2×Q8×C53—41no—
94C2×Q8×C53—82no—
95C22×C4×C53—41no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—22no—
102C2×D212—42no—
103C2×D212—42no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—82no—
108C2×D212—82no—
109C2×Dic106—21no—
110C2×Dic106C2×D412yes—
111M16×C53—42no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53C2411yescharacteristic
122C22⋊Dic53—22no—
123#11719925—84no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53C2×D412yes—
128#11719919—22no—
129#11719919—22no—
130C4×Dic53—22no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—22no—
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136C2×#88322C2×D411yescharacteristic
137C2×#88322C2×D411yescharacteristic
138#11719913—84no—
139C23×C53#12111yescharacteristic
140C22×D53—42no—
141C22×D53—82no—
142C22×D53—82no—
143C22×D53—82no—
144Q8×C53—44no—
145Q8×C53—82no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—42no—
149C2×C4×C53—82no—
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—22no—
159C2×Dic53—22no—
160D212—42no—
161D212—42no—
162D212—44no—
163D212—82no—
164D212—82no—
165D212—82no—
166D212—82no—
167Dic106—24no—
168Dic106—42no—
169C8×C53—42no—
170C8×C53—42no—
171#88322—21no—
172#88322—21no—
173#88322—21no—
174#88322—21no—
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—82no—
188C4×C53#46412yes—
189C4×C53#47112yes—
190C4×C53C2×#13511yescharacteristic
191C4×C53C2×#13711yescharacteristic
192C4×C53#57612yes—
193Dic53—42no—
194Dic53—42no—
195#3209—531no—
196#3208—531no—
197C2×#486—531no—
198#2960—531no—
199#1702—534no—
200#1682—534no—
201#1564—531no—
202#1549D5311yescharacteristic, complemented
203#1545—531no—
204C2×C53—22no—
205C2×C53C2×#46411yescharacteristic
206C2×C53#305811yescharacteristic
207C2×C53C2×#47111yescharacteristic
208D53—162no—
209#533—531no—
210C2×C4⋊Q8C2×D5311yescharacteristic
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—532no—
225#446—1061no—
226C8×Q8—532no—
227C8×Q8—1061no—
228#431—531no—
229#431C2×D5311yescharacteristic
230#424—1061no—
231#423C2×D5311yescharacteristic
232C2×#102—1061no—
233C2×#102—1061no—
234#405—1061no—
235C2×C4×C8—531no—
236#335—532no—
237#334—532no—
238#332C2×D5314yescomplemented
239#331—534no—
240C53#1921411yescharacteristic, complemented, Sylow
241C22×Q8—21no—
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4—1061no—
245C2×Q16—1064no—
246C2×SD16—1064no—
247C2×M16—21no—
248C22×C8—531no—
249C22×C8—531no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—531no—
254C4⋊Q8—531no—
255C4⋊Q8C22×D5312yes—
256#127—532no—
257#127—1061no—
258#127—1061no—
259#127—1061no—
260#125—1061no—
261C22⋊Q8—532no—
262C22⋊Q8—1061no—
263C4×Q8—532no—
264C4×Q8—1061no—
265#117—1061no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C42C22×D5311yescharacteristic
269C4⋊C8—21no—
270C4⋊C8—532no—
271C4⋊C8—1062no—
272C4⋊C8—1062no—
273C4⋊C8C22×D5312yes—
274C4⋊C8C22×D5312yes—
275#103—1064no—
276#102—1062no—
277#102—1062no—
278#102—1062no—
279#102—1062no—
280#102—1064no—
281C22⋊C8—21no—
282C22⋊C8—1061no—
283#97—1062no—
284C4×C8—21no—
285C4×C8—531no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—531no—
289C4×C8—1062no—
290C4×C8—1061no—
291C24—2121no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8—532no—
296C2×Q8—1061no—
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—41no—
308C22×C4#1171994511yescharacteristic
309C22×C4D4×D5311yescharacteristic
310C22×C4D4×D5311yescharacteristic
311Q16—4244no—
312SD16—4244no—
313M16—42no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—531no—
317C2×C8—531no—
318C2×C8—531no—
319C2×C8—531no—
320C2×C8—1062no—
321C2×C8—1062no—
322C2×C8—1062no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—532no—
328C4⋊C4—1061no—
329C4⋊C4—1061no—
330C4⋊C4—1062no—
331C4⋊C4—1061no—
332C4⋊C4—1062no—
333C22⋊C4—1062no—
334C42—1062no—
335C42#1171994511yescharacteristic
336C42D4×D5311yescharacteristic
337C42D4×D5311yescharacteristic
338C42C23×D5311yescharacteristic
339C23—4242no—
340C23—4242no—
341C23—4242no—
342C23—2122no—
343C23#1313319011yescharacteristic
344Q8—44no—
345Q8—82no—
346Q8—1064no—
347Q8—2122no—
348D4—4242no—
349D4—4242no—
350D4—4242no—
351D4—4242no—
352D4—2122no—
353D4—2122no—
354D4—2124no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—1062no—
360C2×C4—1062no—
361C2×C4#1313319011yescharacteristic
362C2×C4#1313319011yescharacteristic
363C2×C4#1313319011yescharacteristic
364C2×C4#1313323812yes—
365C2×C4#1313324912yes—
366C2×C4C2×D4×D5311yescharacteristic
367C2×C4C2×D4×D5311yescharacteristic
368C2×C4C2×#1171994511yescharacteristic
369C8—42no—
370C8—42no—
371C8—1061no—
372C8—1061no—
373C8—1061no—
374C8—1061no—
375C8—2122no—
376C22—21no—
377C22—21no—
378C22—8484no—
379C22—4242no—
380C22—4242no—
381C22—4242no—
382C22#1681084812yes—
383C22C2×#1313319011yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—2122no—
388C4—2122no—
389C4#1681084812yes—
390C4#1681088912yes—
391C4C2×#1313323811yescharacteristic
392C4C2×#1313324911yescharacteristic
393C4#1681142712yes—
394C2—22no—
395C2—8482no—
396C2C2×#1681084811yescharacteristic
397C2#2028436211yescharacteristic
398C2C2×#1681088911yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1549C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#1549D53
hypocenterC53#19214
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. #20285663C2semidirect
4. #20283645C2semidirect
5. #20274286C2semidirect
6. #20274162C2semidirect
7. #20273461C2semidirect
9. C53×#1549C2semidirect
20. #16810830C22semidirect
25. #16810750C22semidirect
27. #16810747C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
40. C53×#332C22semidirect
202. #1549D53semidirect
238. #332C2×D53semidirect
240. C53#19214semidirect

Maximal subgroups

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

subgroupindex
#202856632
#202856582
C2×#168109712
#202836452
#202742862
#202741622
#202734612
#202733842
C53×#15492
#1921453