Finite Group Databasebeta

← #27366812

Subgroups — #27366812

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285301C211yesmaximal, characteristic, complemented
2#20284503C212yesmaximal
3#20274297C212yesmaximal, complemented
4#20274290C212yesmaximal, complemented
5#20274243C211yesmaximal, characteristic, complemented
6#20274238C211yesmaximal, characteristic, complemented
7#20274209C211yesmaximal, characteristic, complemented
8#20274204C211yesmaximal, characteristic, complemented
9C2×#16810374C211yesmaximal, characteristic, complemented
10#20273370C212yesmaximal, complemented
11C53×#2965C211yesmaximal, characteristic
12#16811180C2212yes—
13#16811028C2211yescharacteristic
14#16811003C2211yescharacteristic
15#16810974C2212yes—
16#16810930—22no—
17#16810923—22no—
18#16810918C2212yes—
19#16810913C2212yes—
20#16810908C2212yes—
21#16810903C2212yes—
22#16810864—21no—
23#16810864—21no—
24C4×Dic212—21no—
25C4×Dic212—21no—
26#16810659—22no—
27C2×C4⋊#88322C2212yes—
28#16810619—22no—
29#16810374C2211yescharacteristic
30#16810374C2211yescharacteristic
31#16810374C2212yes—
32#16810371C2212yes—
33#16810371C2212yes—
34#16810370C2212yes—
35#16810370C2212yes—
36C4×#41×C53C2211yescharacteristic
37C53×#446C2211yescharacteristic
38C53×#446C2211yescharacteristic
39C53×#436C2211yescharacteristic
40C53×#436C2211yescharacteristic
41C53×#435—21no—
42C53×#435—21no—
43C53×#431—21no—
44C53×#431C2211yescharacteristic
45C4×M16×C53—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×Dic212—21no—
48C2×Dic212—21no—
49C22×#88322—22no—
50#13133217C2312yes—
51#13133217C2312yes—
52#13133209—22no—
53#13133196—22no—
54#13133196—22no—
55C4×Dic106—22no—
56C4×Dic106—22no—
57#13133176—22no—
58#13133151—21no—
59#13133151—21no—
60#13133150—22no—
61#13133150—22no—
62#13133148—22no—
63#13133148—22no—
64C4⋊#88322—22no—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133142—22no—
68#13133142—22no—
69#13133141—22no—
70#13133141—22no—
71#13133138—22no—
72#13133138—22no—
73C2×#41×C53D411yescharacteristic
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C22×C8×C53D411yescharacteristic
77C22×C8×C53D411yescharacteristic
78C4×Q8×C53D411yescharacteristic
79C4×Q8×C53C2311yescharacteristic
80C4×D4×C53—21no—
81C4×D4×C53—21no—
82C4×D4×C53D411yescharacteristic
83C4×D4×C53C2311yescharacteristic
84#117×C53—21no—
85#117×C53D411yescharacteristic
86C2×C42×C53—21no—
87C2×C42×C53C2311yescharacteristic
88C4⋊C8×C53—21no—
89C4⋊C8×C53—21no—
90C4⋊C8×C53C2311yescharacteristic
91C4⋊C8×C53C2311yescharacteristic
92C22⋊C8×C53—21no—
93C22⋊C8×C53—21no—
94C22⋊C8×C53—21no—
95#97×C53—21no—
96#97×C53—21no—
97C4×C8×C53—21no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102#41×C53—24no—
103C2×Q8×C53C2×D411yescharacteristic
104C2×D4×C53—21no—
105C2×D4×C53C2×D411yescharacteristic
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×Dic106—22no—
113C2×Dic106—22no—
114M16×C53—42no—
115M16×C53—42no—
116C2×C8×C53—22no—
117C2×C8×C53—21no—
118C2×C8×C53—21no—
119C2×C8×C53—41no—
120C2×C8×C53C2×D411yescharacteristic
121C2×C8×C53C2×D411yescharacteristic
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53C2×D411yescharacteristic
127C4⋊C4×C53C2×D411yescharacteristic
128C22⋊C4×C53—21no—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C42×C53—21no—
132C42×C53—21no—
133C42×C53—21no—
134C42×C53C2×D411yescharacteristic
135C42×C53C2×D411yescharacteristic
136C42×C53#4111yescharacteristic
137C42×C53C2411yescharacteristic
138C22⋊Dic53—42no—
139C4⋊Dic53—22no—
140C4⋊Dic53—22no—
141C4⋊Dic53—22no—
142C4⋊Dic53—22no—
143#11719919—42no—
144#11719919—42no—
145C4×Dic53—42no—
146C2×#88322—22no—
147C2×#88322—22no—
148C2×#88322—44no—
149Dic212—42no—
150Dic212—42no—
151C23×C53—21no—
152C23×C53#12111yescharacteristic
153Q8×C53—22no—
154D4×C53—22no—
155D4×C53—42no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—42no—
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#13714yes—
170C2×C4×C53C22×D411yescharacteristic
171C2×C4×C53C22×D411yescharacteristic
172C2×C4×C53C2×#4111yescharacteristic
173C2×Dic53—42no—
174C2×Dic53—42no—
175Dic106—42no—
176Dic106—42no—
177Dic106—82no—
178C8×C53—21no—
179C8×C53—21no—
180C8×C53—21no—
181C8×C53—41no—
182C8×C53—41no—
183C8×C53#13512yes—
184#88322—42no—
185#88322—42no—
186#19158—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53#47012yes—
193C22×C53C2×#12111yescharacteristic
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53#47012yes—
201C4×C53C2×#13511yescharacteristic
202C4×C53C2×#13712yes—
203C4×C53#57511yescharacteristic
204Dic53—82no—
205Dic53—82no—
206#3170—531no—
207#3076—532no—
208#2965D5311yescharacteristic
209#1704—532no—
210#1703—532no—
211#1696—531no—
212#1695—531no—
213#1690—531no—
214#1689—531no—
215C2×#334—531no—
216#1542—532no—
217C2×C53—22no—
218C2×C53—81no—
219C2×C53#303511yescharacteristic
220C2×C53C2×#47011yescharacteristic
221C2×C53#306711yescharacteristic
222#533—532no—
223C4×#41C2×D5311yescharacteristic
224#500—531no—
225#494—531no—
226#487—532no—
227#479—1062no—
228#478—1062no—
229#477—532no—
230#476—532no—
231#475—532no—
232#474—532no—
233#467—1061no—
234#467—1061no—
235#446C2×D5311yescharacteristic
236#446C2×D5311yescharacteristic
237C4×Q16—1061no—
238C4×Q16—1061no—
239#436C2×D5311yescharacteristic
240#436C2×D5311yescharacteristic
241#435—21no—
242#435—21no—
243#431—21no—
244#431C2×D5311yescharacteristic
245#427—1062no—
246C2×C4⋊C8—532no—
247#416—1062no—
248C4×M16—21no—
249C2×C4×C8C2×D5311yescharacteristic
250#334—531no—
251#334—531no—
252#334—532no—
253#333—532no—
254#333—532no—
255#332—532no—
256#332—532no—
257C53#1915811yescharacteristic, complemented, Sylow
258C2×#41D21211yescharacteristic
259C2×Q16—1061no—
260C2×Q16—1061no—
261C2×M16—21no—
262C2×M16—21no—
263C22×C8—1062no—
264C22×C8C22⋊D5311yescharacteristic
265C22×C8C22⋊D5311yescharacteristic
266C4⋊Q8—532no—
267C4⋊Q8—532no—
268#125—1062no—
269C22⋊Q8—1062no—
270C22⋊Q8—1062no—
271C4×Q8—1062no—
272C4×Q8—1062no—
273C4×Q8D21211yescharacteristic
274C4×Q8C22×D5311yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4D21211yescharacteristic
278C4×D4C22×D5311yescharacteristic
279#117—21no—
280#117—1062no—
281#117D21211yescharacteristic
282C2×C42—21no—
283C2×C42C22×D5311yescharacteristic
284#107—1061no—
285#107—1061no—
286#107—1062no—
287#107—1062no—
288#106—1062no—
289#106—1062no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—532no—
293C4⋊C8—532no—
294C4⋊C8—1062no—
295C4⋊C8C22×D5311yescharacteristic
296C4⋊C8C22×D5311yescharacteristic
297#103—1062no—
298#103—1062no—
299#103—1062no—
300#103—1062no—
301#102—1062no—
302#102—1062no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306#97—21no—
307#97—21no—
308C4×C8—21no—
309C4×C8C22⋊D5311yescharacteristic
310C4×C8C22⋊D5311yescharacteristic
311C4×C8C22×D5311yescharacteristic
312C4×C8C22×D5311yescharacteristic
313#41—24no—
314C2×Q8—1062no—
315C2×Q8—1062no—
316C2×Q8C2×D21211yescharacteristic
317C2×D4—21no—
318C2×D4C2×D21211yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4C2×D21211yescharacteristic
323C22×C4#1171994511yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325Q16—2122no—
326Q16—2122no—
327M16—42no—
328M16—42no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—1062no—
334C2×C8—1062no—
335C2×C8—2124no—
336C2×C8C2×C22⋊D5311yescharacteristic
337C2×C8C2×C22⋊D5311yescharacteristic
338C2×C8C2×C22⋊D5311yescharacteristic
339C2×C8C2×C22⋊D5311yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—1062no—
343C4⋊C4—1062no—
344C4⋊C4—1062no—
345C4⋊C4—1062no—
346C4⋊C4—2122no—
347C4⋊C4—2122no—
348C4⋊C4C2×D21211yescharacteristic
349C4⋊C4C2×D21211yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—2122no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—2122no—
358C42C2×D21211yescharacteristic
359C42#1171994511yescharacteristic
360C42C2×C22⋊D5311yescharacteristic
361C42C23×D5311yescharacteristic
362C23—21no—
363C23#1313319111yescharacteristic
364Q8—22no—
365Q8—4242no—
366Q8—2122no—
367Q8—2122no—
368D4—22no—
369D4—42no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—2122no—
381C2×C4—2122no—
382C2×C4#1313319111yescharacteristic
383C2×C4#1313319111yescharacteristic
384C2×C4#1313319111yescharacteristic
385C2×C4#1313325014yes—
386C2×C4C22×C22⋊D5311yescharacteristic
387C2×C4C22×D21211yescharacteristic
388C2×C4C2×#1171994511yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—2122no—
395C8—2122no—
396C8#1313323712yes—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#1681088112yes—
403C22C2×#1313319111yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—4242no—
411C4—4242no—
412C4#1681088112yes—
413C4C2×#1313323711yescharacteristic
414C4C2×#1313325012yes—
415C4#1681142111yescharacteristic
416C2—22no—
417C2—81no—
418C2#2028417011yescharacteristic
419C2C2×#1681088111yescharacteristic
420C2#2028443511yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2965C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#2965D53
hypocenterC53#19158
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #20285301C2semidirect
3. #20274297C2semidirect
4. #20274290C2semidirect
5. #20274243C2semidirect
6. #20274238C2semidirect
7. #20274209C2semidirect
8. #20274204C2semidirect
9. C2×#16810374C2semidirect
10. #20273370C2semidirect
257. C53#19158semidirect

Maximal subgroups

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

subgroupindex
#202853012
#202845032
#202742972
#202742902
#202742432
#202742382
#202742092
#202742042
C2×#168103742
#202733702
C53×#29652
#1915853