Finite Group Databasebeta

← #27366806

Subgroups — #27366806

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#20284497C212yesmaximal
2#20283681C211yesmaximal, characteristic
3#20274294C212yesmaximal, complemented
4#20274287C212yesmaximal, complemented
5#20274240C211yesmaximal, characteristic, complemented
6#20274235C211yesmaximal, characteristic, complemented
7#20274206C211yesmaximal, characteristic, complemented
8#20274201C211yesmaximal, characteristic, complemented
9C2×#16810374C211yesmaximal, characteristic, complemented
10#20273372C212yesmaximal, complemented
11C53×#3170C211yesmaximal, characteristic, complemented
12C4×#11719945C2211yescharacteristic
13#16810927—22no—
14#16810920—22no—
15#16810915C2212yes—
16#16810910C2212yes—
17#16810905C2212yes—
18#16810900C2212yes—
19#16810753C2211yescharacteristic
20#16810753C2211yescharacteristic
21#16810702C2211yescharacteristic
22#16810702C2211yescharacteristic
23#16810694—21no—
24#16810694—21no—
25#16810676C2211yescharacteristic
26#16810674—21no—
27#16810659—22no—
28C2×C4⋊#88322C2212yes—
29#16810617—22no—
30C4×#11719912—21no—
31#16810374C2211yescharacteristic
32#16810374C2211yescharacteristic
33#16810374C2212yes—
34#16810372C2212yes—
35#16810372C2212yes—
36#16810369C2212yes—
37#16810369C2212yes—
38C53×#533C2212yes—
39C53×#500C2211yescharacteristic
40C53×#494C2211yescharacteristic
41C53×#487C2212yes—
42C53×#467—21no—
43C53×#467—21no—
44C4×Q16×C53—21no—
45C4×Q16×C53—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×#11719945D411yescharacteristic
48C2×#11719912—21no—
49C2×#11719912—21no—
50C22×#88322—22no—
51C4×Dic106D411yescharacteristic
52C4×Dic106C2311yescharacteristic
53C4×D212D411yescharacteristic
54C4×D212C2311yescharacteristic
55C4×C22⋊D53—21no—
56C4×C22⋊D53—21no—
57#13133176D411yescharacteristic
58#13133174—21no—
59C42×D53—21no—
60#13133150—22no—
61#13133150—22no—
62#13133148—22no—
63#13133148—22no—
64C4⋊#88322—22no—
65C4⋊#88322C2311yescharacteristic
66C4⋊#88322C2311yescharacteristic
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133146—21no—
70#13133146—21no—
71#13133140—22no—
72#13133140—22no—
73#13133137—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76#13133129—21no—
77#13133129—21no—
78#13133127—21no—
79#13133127—21no—
80C4×#88322—21no—
81C2×Q16×C53—21no—
82C2×Q16×C53—21no—
83C22×C8×C53D411yescharacteristic
84C22×C8×C53D411yescharacteristic
85C4⋊Q8×C53C2312yes—
86C4⋊Q8×C53C2312yes—
87#125×C53—22no—
88C22⋊Q8×C53—22no—
89C22⋊Q8×C53—22no—
90C4×Q8×C53—22no—
91C4×Q8×C53—22no—
92#117×C53—22no—
93C2×C42×C53C2311yescharacteristic
94#107×C53—21no—
95#107×C53—21no—
96#103×C53—22no—
97#103×C53—22no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102C2×Q8×C53—22no—
103C2×Q8×C53—22no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C2×C22⋊D53—21no—
108#11719945—24no—
109C2×D212C2×D411yescharacteristic
110C2×C4×D53—21no—
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×Dic106C2×D411yescharacteristic
114Q16×C53—42no—
115Q16×C53—42no—
116C2×C8×C53—22no—
117C2×C8×C53—41no—
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53C2×D411yescharacteristic
120C2×C8×C53C2×D411yescharacteristic
121C2×C8×C53C2×D411yescharacteristic
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—42no—
127C4⋊C4×C53—42no—
128C22⋊C4×C53—42no—
129C42×C53—42no—
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53#4111yescharacteristic
133C42×C53C2411yescharacteristic
134C22⋊Dic53—21no—
135#11719921—21no—
136#11719921—21no—
137C4⋊Dic53C2×D411yescharacteristic
138C4⋊Dic53C2×D411yescharacteristic
139#11719919—21no—
140#11719919—21no—
141C4×Dic53—21no—
142C4×Dic53—21no—
143C4×Dic53—21no—
144C2×#88322—21no—
145C2×#88322—21no—
146C2×#88322—22no—
147C2×#88322—22no—
148C2×#88322—44no—
149#11719912—42no—
150#11719912—42no—
151C23×C53#12111yescharacteristic
152C22×D53—21no—
153Q8×C53—42no—
154Q8×C53—42no—
155Q8×C53—82no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#13714yes—
165C2×C4×C53C22×D411yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C2×#4111yescharacteristic
168C22⋊D53—42no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172D212—22no—
173C4×D53—42no—
174C4×D53—41no—
175C4×D53—41no—
176C4×D53—42no—
177Dic106—22no—
178C8×C53—21no—
179C8×C53—21no—
180C8×C53—21no—
181C8×C53#13512yes—
182#88322—41no—
183#88322—42no—
184#88322—42no—
185#88322—41no—
186#19158—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53#47012yes—
190C22×C53C2×#12111yescharacteristic
191C2×D53—41no—
192C2×D53—41no—
193C2×D53—41no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—82no—
198C4×C53—82no—
199C4×C53#47012yes—
200C4×C53C2×#13511yescharacteristic
201C4×C53C2×#13712yes—
202C4×C53#57511yescharacteristic
203Dic53—41no—
204Dic53—41no—
205Dic53—41no—
206#3170D5311yescharacteristic, complemented
207#3076—532no—
208#2965—531no—
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#303511yescharacteristic
219C2×C53C2×#47011yescharacteristic
220C2×C53#306711yescharacteristic
221D53—81no—
222#533C2×D5312yes—
223C4×#41—531no—
224#500C2×D5311yescharacteristic
225#494C2×D5311yescharacteristic
226#487C2×D5312yes—
227#479—1062no—
228#478—1062no—
229#477—532no—
230#476—532no—
231#475—532no—
232#474—532no—
233#467—21no—
234#467—21no—
235#446—531no—
236#446—531no—
237C4×Q16—21no—
238C4×Q16—21no—
239#436—531no—
240#436—531no—
241#435—1061no—
242#435—1061no—
243#431—531no—
244#431—1061no—
245#427—1062no—
246C2×C4⋊C8—532no—
247#416—1062no—
248C4×M16—1061no—
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×#41—531no—
259C2×Q16—21no—
260C2×Q16—21no—
261C2×M16—1061no—
262C2×M16—1061no—
263C22×C8—1062no—
264C22×C8C22⋊D5311yescharacteristic
265C22×C8C22⋊D5311yescharacteristic
266C4⋊Q8C22×D5312yes—
267C4⋊Q8C22×D5312yes—
268#125—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—531no—
274C4×Q8—531no—
275C4×D4—531no—
276C4×D4—531no—
277C4×D4—1061no—
278C4×D4—1061no—
279#117—22no—
280#117—531no—
281#117—1061no—
282C2×C42—1061no—
283C2×C42C22×D5311yescharacteristic
284#107—21no—
285#107—21no—
286#107—1062no—
287#107—1062no—
288#106—1062no—
289#106—1062no—
290C4⋊C8—531no—
291C4⋊C8—531no—
292C4⋊C8—532no—
293C4⋊C8—532no—
294C4⋊C8—1061no—
295C4⋊C8—1061no—
296C4⋊C8—1062no—
297#103—22no—
298#103—22no—
299#103—1062no—
300#103—1062no—
301#102—1062no—
302#102—1062no—
303C22⋊C8—1061no—
304C22⋊C8—1061no—
305C22⋊C8—1061no—
306#97—1061no—
307#97—1061no—
308C4×C8—1061no—
309C4×C8C22⋊D5311yescharacteristic
310C4×C8C22⋊D5311yescharacteristic
311C4×C8C22×D5311yescharacteristic
312C4×C8C22×D5311yescharacteristic
313#41—1064no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—531no—
317C2×D4—531no—
318C2×D4—1061no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4D4×D5311yescharacteristic
323C22×C4#1171994711yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325Q16—42no—
326Q16—42no—
327M16—2122no—
328M16—2122no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—1062no—
332C2×C8—1062no—
333C2×C8—1061no—
334C2×C8—1061no—
335C2×C8—2124no—
336C2×C8C2×C22⋊D5311yescharacteristic
337C2×C8C2×C22⋊D5311yescharacteristic
338C2×C8C2×C22⋊D5311yescharacteristic
339C2×C8C2×C22⋊D5311yescharacteristic
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—531no—
347C4⋊C4—531no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C22⋊C4—42no—
351C22⋊C4—1061no—
352C22⋊C4—1061no—
353C22⋊C4—1061no—
354C42—42no—
355C42—1061no—
356C42—1061no—
357C42—1061no—
358C42D4×D5311yescharacteristic
359C42#1171994711yescharacteristic
360C42C2×C22⋊D5311yescharacteristic
361C42C23×D5311yescharacteristic
362C23—1061no—
363C23#1313318811yescharacteristic
364Q8—42no—
365Q8—42no—
366Q8—82no—
367Q8—1062no—
368D4—1062no—
369D4—2122no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—1061no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4—2121no—
381C2×C4—2121no—
382C2×C4#1313318811yescharacteristic
383C2×C4#1313318811yescharacteristic
384C2×C4#1313318811yescharacteristic
385C2×C4#1313324914yes—
386C2×C4C2×D4×D5311yescharacteristic
387C2×C4C22×C22⋊D5311yescharacteristic
388C2×C4C2×#1171994711yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—2121no—
393C8—2122no—
394C8—2122no—
395C8—2121no—
396C8#1313323712yes—
397C22—21no—
398C22—21no—
399C22—2121no—
400C22—2121no—
401C22—2121no—
402C22#1681087812yes—
403C22C2×#1313318811yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—82no—
408C4—82no—
409C4—2121no—
410C4—2121no—
411C4—2121no—
412C4#1681087812yes—
413C4C2×#1313323711yescharacteristic
414C4C2×#1313324912yes—
415C4#1681142111yescharacteristic
416C2—22no—
417C2—4241no—
418C2#2028416411yescharacteristic
419C2C2×#1681087811yescharacteristic
420C2#2028443111yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3170C2
socleC22×C53C2×#121
prime radicalC2×#11719945D4
hypercenter#3170D53
hypocenterC53#19158
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
3. #20274294C2semidirect
4. #20274287C2semidirect
5. #20274240C2semidirect
6. #20274235C2semidirect
7. #20274206C2semidirect
8. #20274201C2semidirect
9. C2×#16810374C2semidirect
10. #20273372C2semidirect
11. C53×#3170C2semidirect
206. #3170D53semidirect
257. C53#19158semidirect

Maximal subgroups

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

subgroupindex
#202844972
#202836812
#202742942
#202742872
#202742402
#202742352
#202742062
#202742012
C2×#168103742
#202733722
C53×#31702
#1915853