Finite Group Databasebeta

← #27367234

Subgroups — #27367234

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285630C211yesmaximal, characteristic
2C2×#16810971C211yesmaximal, characteristic, complemented
3#20283427C211yesmaximal, characteristic, complemented
4#20274250C214yesmaximal, complemented
5#20274126C214yesmaximal, complemented
6#20273461C211yesmaximal, characteristic, complemented
7#20273396C211yesmaximal, characteristic, complemented
8#20273382C211yesmaximal, characteristic, complemented
9C53×#3227C211yesmaximal, characteristic, complemented
10C2×#13133224—21no—
11C2×#13133214C2211yescharacteristic
12#16810999—21no—
13#16810971—21no—
14#16810971—21no—
15#16810971—21no—
16#16810971C2212yes—
17#16810971C2212yes—
18#16810818C2214yes—
19#16810756—22no—
20#16810701—21no—
21#16810701C2212yescomplemented
22D4×#88322—21no—
23D4×#88322C2212yescomplemented
24#16810676C2211yescharacteristic
25#16810647C2211yescharacteristic
26C2×#13133137—21no—
27C2×#13133137—21no—
28#16810590—21no—
29C2×C4×#88322C2211yescharacteristic
30#16810375C2212yescomplemented
31#16810373C2212yescomplemented
32#16810372C2214yescomplemented
33#16810366C2214yes—
34C53×#532C2211yescharacteristic
35C2×C4⋊Q8×C53C2211yescharacteristic
36C53×#498—21no—
37C53×#486—21no—
38C53×#481—22no—
39C53×#476C2214yescomplemented
40C53×#431C2211yescharacteristic
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44C2×#11719922—24no—
45#13133224—44no—
46C2×#11719917—22no—
47C2×#11719917—21no—
48C22×#88322—22no—
49C22×#88322D411yescharacteristic
50C22×#88322D411yescharacteristic
51#13133214—21no—
52#13133214—21no—
53#13133214—21no—
54#13133214C2312yes—
55C4⋊#88322—21no—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58#13133137—24no—
59#13133137—22no—
60#13133137—22no—
61#13133137—22no—
62#13133137—22no—
63C22⋊#88322—22no—
64C22⋊#88322—22no—
65C22⋊#88322—21no—
66C4×#88322—21no—
67C4×#88322D411yescharacteristic
68C4×#88322D411yescharacteristic
69C4×#88322C2311yescharacteristic
70C4×#88322C2311yescharacteristic
71C2×#41×C53—21no—
72C22×Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53—21no—
76C4⋊Q8×C53D411yescharacteristic
77C4⋊Q8×C53C2312yes—
78#127×C53D411yescharacteristic
79#124×C53—21no—
80#121×C53—21no—
81#121×C53D412yes—
82C4×D4×C53—21no—
83C4×D4×C53C2312yes—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C42×C53C2311yescharacteristic
87#106×C53—24no—
88C4⋊C8×C53C2312yes—
89#102×C53—24no—
90C22⋊C8×C53—21no—
91C4×C8×C53—21no—
92C23×D53—41no—
93#41×C53—42no—
94#41×C53—42no—
95C2×Q8×C53—21no—
96C2×Q8×C53—22no—
97C2×Q8×C53—41no—
98C2×Q8×C53—82no—
99C2×D4×C53—22no—
100C2×D4×C53—21no—
101C2×D4×C53—21no—
102C2×D4×C53C2×D412yes—
103C22×C4×C53—22no—
104C22×C4×C53—41no—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C2×D212—22no—
109C2×D212—22no—
110C2×D212—22no—
111C2×D212—42no—
112C2×D212—42no—
113C2×D212—41no—
114C2×D212—41no—
115C2×D212—41no—
116C2×D212—82no—
117C2×D212—82no—
118C2×C8×C53—22no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53C2×D412yes—
125C22⋊C4×C53—22no—
126C22⋊C4×C53—22no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53#4111yescharacteristic
130C42×C53C2411yescharacteristic
131#11719922—84no—
132#11719917—44no—
133#11719917—42no—
134C2×#88322—22no—
135C2×#88322—22no—
136C2×#88322—22no—
137C2×#88322—22no—
138C2×#88322—44no—
139C2×#88322C2×D411yescharacteristic
140C2×#88322C2×D411yescharacteristic
141C2×#88322C2×D411yescharacteristic
142C2×#88322C2×D411yescharacteristic
143C23×C53—22no—
144C23×C53#12111yescharacteristic
145C22×D53—42no—
146C22×D53—82no—
147C22×D53—82no—
148C22×D53—82no—
149Q8×C53—42no—
150Q8×C53—82no—
151Q8×C53—82no—
152D4×C53—24no—
153D4×C53—44no—
154D4×C53—42no—
155D4×C53—42no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—22no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—82no—
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#13512yes—
167C2×C4×C53AGL(1,ℤ/8)12yes—
168C2×C4×C53C22×D411yescharacteristic
169C2×C4×C53C22×D411yescharacteristic
170C2×C4×C53C2×#4111yescharacteristic
171D212—42no—
172D212—42no—
173D212—44no—
174D212—82no—
175D212—82no—
176D212—82no—
177D212—82no—
178C8×C53—42no—
179#88322—21no—
180#88322—21no—
181#88322—21no—
182#88322—21no—
183#88322—42no—
184#88322—42no—
185#19196—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—42no—
189C22×C53—42no—
190C22×C53—42no—
191C22×C53#46312yes—
192C22×C53C2×#12111yescharacteristic
193C2×D53—82no—
194C2×D53—82no—
195C2×D53—82no—
196C2×D53—164no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—42no—
200C4×C53—82no—
201C4×C53#46312yes—
202C4×C53#47212yes—
203C4×C53C2×#13511yescharacteristic
204C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C53#57712yes—
206#3227D5311yescharacteristic, complemented
207#3205—531no—
208C2×#486—531no—
209#2931—531no—
210#1697—534no—
211#1677—534no—
212#1564—531no—
213#1548—531no—
214#1545—531no—
215C2×C53—22no—
216C2×C53—82no—
217C2×C53C2×#46311yescharacteristic
218C2×C53#305711yescharacteristic
219C2×C53C2×#47211yescharacteristic
220D53—162no—
221C2×#131—1061no—
222#532C2×D5311yescharacteristic
223C2×C4⋊Q8C2×D5311yescharacteristic
224C2×#127—531no—
225#498—21no—
226#493—1061no—
227#486—21no—
228#486—532no—
229#486—532no—
230#486—1061no—
231#486—1061no—
232#486—1061no—
233#481—22no—
234#476C2×D5314yescomplemented
235#459—534no—
236#447—1062no—
237#436—532no—
238#436—1061no—
239C8×D4—532no—
240C8×D4—1061no—
241#431—531no—
242#431C2×D5311yescharacteristic
243#423—531no—
244C2×#102—1061no—
245C2×#102—1061no—
246#408—1061no—
247C2×C4×C8—531no—
248#335—532no—
249#334—532no—
250#333—534no—
251#331—534no—
252C53#1919611yescharacteristic, complemented, Sylow
253C2×#41—21no—
254C22×Q8—21no—
255C22×D4—1061no—
256C22×D4—1061no—
257C22×D4—1061no—
258C2×D8—1064no—
259#131—2124no—
260C2×M16—1062no—
261C2×M16—1061no—
262C22×C8—531no—
263C22×C8—531no—
264C22×C8—1062no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8C22⋊D5311yescharacteristic
269C4⋊Q8C22×D5312yes—
270#127—532no—
271#127—1061no—
272#127—1061no—
273#127—1061no—
274#127C22⋊D5311yescharacteristic
275#124—21no—
276#121—21no—
277#121C22⋊D5312yes—
278C4×D4—21no—
279C4×D4C22×D5312yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C42C22×D5311yescharacteristic
283#106—24no—
284C4⋊C8—532no—
285C4⋊C8—532no—
286C4⋊C8—1061no—
287C4⋊C8C22×D5312yes—
288#102—24no—
289#102—1062no—
290#102—1062no—
291#102—1062no—
292#102—1062no—
293#102—1064no—
294C22⋊C8—21no—
295C22⋊C8—1062no—
296C22⋊C8—1062no—
297C22⋊C8—1061no—
298C4×C8—21no—
299C4×C8—531no—
300C4×C8—531no—
301C4×C8—531no—
302C4×C8—531no—
303C4×C8—1061no—
304C24—2121no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×Q8—22no—
309C2×Q8—41no—
310C2×Q8—82no—
311C2×D4—22no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—1062no—
317C2×D4—4242no—
318C2×D4—4242no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2122no—
323C2×D4—2122no—
324C2×D4C2×C22⋊D5312yes—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4D4×D5311yescharacteristic
328C22×C4#1171994711yescharacteristic
329C22×C4C2×C22⋊D5311yescharacteristic
330D8—4244no—
331M16—2124no—
332M16—2122no—
333C2×C8—22no—
334C2×C8—531no—
335C2×C8—531no—
336C2×C8—531no—
337C2×C8—531no—
338C2×C8—1062no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—1062no—
342C2×C8—2124no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4C2×C22⋊D5312yes—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C42D4×D5311yescharacteristic
352C42#1171994711yescharacteristic
353C42C2×C22⋊D5311yescharacteristic
354C42C23×D5311yescharacteristic
355C23—22no—
356C23—4242no—
357C23—4242no—
358C23—4242no—
359C23—2122no—
360C23#1313318911yescharacteristic
361Q8—42no—
362Q8—82no—
363Q8—82no—
364D4—24no—
365D4—44no—
366D4—42no—
367D4—42no—
368D4—4242no—
369D4—4242no—
370D4—4242no—
371D4—4242no—
372D4—2122no—
373D4—2122no—
374D4—2124no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4#1313318911yescharacteristic
383C2×C4#1313318911yescharacteristic
384C2×C4#1313318911yescharacteristic
385C2×C4#1313323812yes—
386C2×C4#1313324412yes—
387C2×C4C2×D4×D5311yescharacteristic
388C2×C4C22×C22⋊D5311yescharacteristic
389C2×C4C2×#1171994711yescharacteristic
390C8—42no—
391C8—1061no—
392C8—1061no—
393C8—1061no—
394C8—1061no—
395C8—2122no—
396C8—2122no—
397C22—21no—
398C22—21no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—8484no—
403C22—4242no—
404C22—4242no—
405C22—4242no—
406C22#1681084012yes—
407C22C2×#1313318911yescharacteristic
408C4—21no—
409C4—21no—
410C4—42no—
411C4—82no—
412C4#1681084012yes—
413C4#1681089112yes—
414C4C2×#1313323811yescharacteristic
415C4C2×#1313324411yescharacteristic
416C4#1681143112yes—
417C2—22no—
418C2—82no—
419C2—8482no—
420C2C2×#1681084011yescharacteristic
421C2#2028435611yescharacteristic
422C2C2×#1681089111yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3227C2
socleC22×C53C2×#121
prime radical#20285630C2
hypercenter#3227D53
hypocenterC53#19196
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
2. C2×#16810971C2semidirect
3. #20283427C2semidirect
4. #20274250C2semidirect
5. #20274126C2semidirect
6. #20273461C2semidirect
7. #20273396C2semidirect
8. #20273382C2semidirect
9. C53×#3227C2semidirect
21. #16810701C22semidirect
23. D4×#88322C22semidirect
30. #16810375C22semidirect
31. #16810373C22semidirect
32. #16810372C22semidirect
39. C53×#476C22semidirect
206. #3227D53semidirect
234. #476C2×D53semidirect
252. C53#19196semidirect

Maximal subgroups

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

subgroupindex
#202856302
C2×#168109712
#202834272
#202742502
#202741262
#202734612
#202733962
#202733822
C53×#32272
#1919653