Finite Group Databasebeta

← #27280186

Subgroups — #27280186

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284497C211yesmaximal, characteristic
2#20283676C211yesmaximal, characteristic
3#20274736C211yesmaximal, characteristic
4#20273170C211yesmaximal, characteristic, complemented
5#20272611C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1503C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810927—21no—
10#16810920—21no—
11#16810915C411yescharacteristic, complemented
12#16810910C411yescharacteristic, complemented
13#16810905C411yescharacteristic, complemented
14#16810900C411yescharacteristic, complemented
15#16810750—21no—
16#16810700—21no—
17#16810691—22no—
18#16810674—21no—
19#16810659—21no—
20#16810647C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133145—21no—
23#16810617—21no—
24C4×#11719912—21no—
25C2×C4×#88322C2211yescharacteristic
26#16810383—21no—
27Q8⋊#88322—21no—
28Q8⋊#88322—21no—
29C53×#533C2211yescharacteristic
30C53×#431C2211yescharacteristic
31C4×C4⋊C4×C53C2211yescharacteristic
32C53×#328—22no—
33C2×#11719945D411yescharacteristic
34C2×#11719917—21no—
35C2×#11719912—22no—
36C22×#88322—21no—
37C22×#88322—21no—
38C4×Dic106C2×C411yescharacteristic
39C4×Dic106D411yescharacteristic
40C4×D212C2×C411yescharacteristic
41C4×D212D411yescharacteristic
42C4×C22⋊D53—21no—
43C4×C22⋊D53—21no—
44#13133176D411yescharacteristic
45#13133174—21no—
46C42×D53—21no—
47#13133150—21no—
48#13133150—21no—
49#13133148—21no—
50#13133148—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322C2×C411yescharacteristic
55C4⋊#88322C2×C411yescharacteristic
56#13133146—22no—
57#13133145—42no—
58#13133140—21no—
59#13133140—21no—
60#13133137—21no—
61#13133137—21no—
62#13133129—22no—
63#13133127—22no—
64C4×#88322—21no—
65C4×#88322D411yescharacteristic
66C4×#88322D411yescharacteristic
67C4⋊Q8×C53C2×C411yescharacteristic
68C4⋊Q8×C53C2×C411yescharacteristic
69#125×C53—21no—
70C22⋊Q8×C53—21no—
71C22⋊Q8×C53—21no—
72C4×Q8×C53—21no—
73C4×Q8×C53—21no—
74#117×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C42×C53—21no—
77C2×C42×C53C2311yescharacteristic
78C4⋊C8×C53—21no—
79C22⋊C8×C53D412yes—
80C4×C8×C53—21no—
81#95×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—21no—
84C22×C4×C53—41no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C2×C22⋊D53—21no—
89#11719945—21no—
90#11719945—21no—
91#11719945—21no—
92#11719945—21no—
93C2×D212C22⋊C411yescharacteristic
94C2×C4×D53—21no—
95C2×C4×D53—21no—
96C2×C4×D53—21no—
97C2×Dic106C22⋊C411yescharacteristic
98C2×C8×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C42×C53—41no—
109C42×C53—41no—
110C42×C53—41no—
111C42×C53C22×C411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C22⋊Dic53—21no—
116#11719921—21no—
117#11719921—21no—
118C4⋊Dic53C22⋊C411yescharacteristic
119C4⋊Dic53C22⋊C411yescharacteristic
120#11719919—21no—
121#11719919—21no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C4×Dic53—21no—
125#11719917—41no—
126#11719917—41no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—42no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135#11719912—44no—
136C23×C53#12111yescharacteristic
137C22×D53—21no—
138Q8×C53—41no—
139Q8×C53—41no—
140Q8×C53—81no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12311yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53#13711yescharacteristic
157C2×C4×C53#13711yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C22⋊D53—41no—
160C22⋊D53—41no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164D212—21no—
165D212—21no—
166C4×D53—42no—
167C4×D53—41no—
168C4×D53—41no—
169C4×D53—41no—
170C4×D53—41no—
171Dic106—21no—
172Dic106—21no—
173C8×C53—42no—
174#88322—22no—
175#88322—41no—
176#88322—41no—
177#88322—41no—
178#88322—81no—
179#9209—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53#38611yescharacteristic
183C22×C53#46811yescharacteristic
184C22×C53#47011yescharacteristic
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—82no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53#41711yescharacteristic
195C4×C53#41911yescharacteristic
196C4×C53#44311yescharacteristic
197C4×C53#44411yescharacteristic
198C4×C53#46811yescharacteristic
199C4×C53#47011yescharacteristic
200Dic53—41no—
201Dic53—41no—
202Dic53—41no—
203#3076—531no—
204#2965—531no—
205#1806—531no—
206#1503D5311yescharacteristic, complemented
207#1493—531no—
208#1301—531no—
209#1279—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53#187611yescharacteristic
213C2×C53#189411yescharacteristic
214C2×C53#190611yescharacteristic
215D53—81no—
216#533C2×D5311yescharacteristic
217C4×#41—531no—
218#479—1061no—
219#478—1061no—
220#477—531no—
221#476—531no—
222#475—531no—
223#474—531no—
224#446—1061no—
225#436—1061no—
226#435—1062no—
227#431—1061no—
228#431C2×D5311yescharacteristic
229#427—1061no—
230#423—531no—
231C2×C4⋊C8—531no—
232C2×#104—1061no—
233#416—1061no—
234C4×M16—1061no—
235C2×C4×C8—531no—
236C4×C4⋊C4C2×D5311yescharacteristic
237#339—1061no—
238#328—22no—
239Q8⋊C8—1061no—
240Q8⋊C8—1061no—
241C53#920911yescharacteristic, complemented, Sylow
242C2×#41—531no—
243C2×M16—1062no—
244C2×M16—1061no—
245C22×C8—1061no—
246C22×C8—1061no—
247C4⋊Q8C4×D5311yescharacteristic
248C4⋊Q8C4×D5311yescharacteristic
249#125—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—531no—
255C4×Q8—531no—
256C4×D4—531no—
257C4×D4—531no—
258C4×D4—1061no—
259C4×D4—1061no—
260#117—21no—
261#117—531no—
262#117—1061no—
263C2×C4⋊C4—21no—
264C2×C42—21no—
265C2×C42—1061no—
266C2×C42C22×D5311yescharacteristic
267#107—1061no—
268#107—1061no—
269#106—1061no—
270#106—1061no—
271C4⋊C8—21no—
272C4⋊C8—531no—
273C4⋊C8—531no—
274C4⋊C8—1062no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278#104—2122no—
279#103—1061no—
280#103—1061no—
281#102—1061no—
282#102—1061no—
283C22⋊C8—1062no—
284C22⋊C8C22⋊D5312yes—
285#97—1062no—
286C4×C8—21no—
287C4×C8—531no—
288C4×C8—531no—
289C4×C8—1061no—
290#95—21no—
291#41—1061no—
292#41—1061no—
293#41—1061no—
294#41—1061no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—531no—
298C2×D4—531no—
299C2×D4—1061no—
300C22×C4—41no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—1061no—
304C22×C4D4×D5311yescharacteristic
305C22×C4#1171994711yescharacteristic
306C22×C4C2×C22⋊D5311yescharacteristic
307M16—2124no—
308M16—2121no—
309M16—2121no—
310C2×C8—22no—
311C2×C8—1061no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8—1061no—
315C2×C8—2122no—
316C2×C8—2121no—
317C2×C8—2121no—
318C2×C8—2121no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—531no—
328C4⋊C4—531no—
329C4⋊C4—1061no—
330C4⋊C4—1061no—
331C22⋊C4—41no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—1061no—
339C42—1061no—
340C42—1061no—
341C42C2×C4×D5311yescharacteristic
342C42#1171994511yescharacteristic
343C42D4×D5311yescharacteristic
344C42D4×D5311yescharacteristic
345C23—1061no—
346C23#1313318811yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—81no—
350Q8—1061no—
351Q8—1061no—
352D4—1061no—
353D4—1061no—
354D4—2121no—
355D4—2121no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—1061no—
365C2×C4—1061no—
366C2×C4—1061no—
367C2×C4—2122no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4C22⋊C4×D5311yescharacteristic
373C2×C4#1313317711yescharacteristic
374C2×C4C4×C22⋊D5311yescharacteristic
375C2×C4#1313318511yescharacteristic
376C2×C4#1313319011yescharacteristic
377C2×C4#1313319911yescharacteristic
378C2×C4#1313324411yescharacteristic
379C2×C4#1313324611yescharacteristic
380C2×C4#1313324911yescharacteristic
381C2×C4#1313324911yescharacteristic
382C8—42no—
383C8—1062no—
384C8—4241no—
385C8—2121no—
386C8—2121no—
387C8—2121no—
388C22—21no—
389C22—21no—
390C22—2121no—
391C22—2121no—
392C22—2121no—
393C22#1681049911yescharacteristic
394C22#1681086611yescharacteristic
395C22#1681087811yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—2121no—
403C4—2121no—
404C4—2121no—
405C4#1681062311yescharacteristic
406C4#1681063011yescharacteristic
407C4#1681073711yescharacteristic
408C4#1681074211yescharacteristic
409C4#1681087211yescharacteristic
410C4#1681088011yescharacteristic
411C2—21no—
412C2—21no—
413C2—4241no—
414C2#2027503911yescharacteristic
415C2#2027514511yescharacteristic
416C2#2027521311yescharacteristic
subgroupG/H
centerC2×C4#13133190
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1503C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1503D53
hypocenterC53#9209
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
4. #20273170C2semidirect
5. #20272611C2semidirect
6. #20272556C2semidirect
7. C53×#1503C2semidirect
11. #16810915C4semidirect
12. #16810910C4semidirect
13. #16810905C4semidirect
14. #16810900C4semidirect
206. #1503D53semidirect
241. C53#9209semidirect

Maximal subgroups

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

subgroupindex
#202844972
#202836762
#202747362
#202731702
#202726112
#202725562
C53×#15032
#920953