Finite Group Databasebeta

← #27278356

Subgroups — #27278356

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277662C211yesmaximal, characteristic
2#20274842C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic
4C2×#16810350C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1947C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133175—21no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133137—21no—
12C2×C4×#88322C2211yescharacteristic
13#16810525—21no—
14#16810515—22no—
15#16810493C2211yescharacteristic
16#16810480—21no—
17C4×C4⋊Dic53—21no—
18#16810385—21no—
19#16810385—22no—
20#16810350C414yescomplemented
21C2×#107×C53—21no—
22C2×C4⋊C8×C53C2211yescharacteristic
23C53×#384C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C53×#340—21no—
26C22×D212D411yescharacteristic
27C22×C4×D53—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C4×D212—21no—
31C4×D212—21no—
32C4×D212C2×C412yes—
33C4×D212Q812yes—
34#13133175—42no—
35#13133175—42no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D412yes—
40C2×#11719921—22no—
41C2×#11719921—22no—
42C2×#11719921—21no—
43C2×C4×Dic53—22no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46#13133137—42no—
47C4×#88322—21no—
48C4×#88322C2×C412yes—
49#13133124—22no—
50C22×C8×C53—21no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C42×C53—21no—
57C2×C42×C53C2311yescharacteristic
58#107×C53—42no—
59C4⋊C8×C53—21no—
60C4⋊C8×C53C2×C412yes—
61#95×C53—21no—
62C23×D53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—41no—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—22no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212C4⋊C412yes—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C8×C53—22no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53—21no—
89C4⋊C4×C53—42no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53D812yes—
95C42×C53—41no—
96C42×C53—41no—
97C42×C53C22×C411yescharacteristic
98C42×C53C2×Q811yescharacteristic
99C42×C53#4111yescharacteristic
100C42×C53#4111yescharacteristic
101#11719921—22no—
102#11719921—22no—
103#11719921—44no—
104#11719921—82no—
105C4⋊Dic53—21no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110C4⋊Dic53—42no—
111C4⋊Dic53C4⋊C412yes—
112C4×Dic53—44no—
113C2×#88322—22no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C23×C53#12011yescharacteristic
120C22×D53—22no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—41no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53C2×C4⋊C411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53C22⋊Q811yescharacteristic
142C2×C4×C53C22⋊Q811yescharacteristic
143C2×C4×C53#12311yescharacteristic
144C2×C4×C53C2×D811yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53#13511yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53#13711yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C2×Dic53—22no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158D212—22no—
159D212—22no—
160D212—41no—
161D212—41no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—42no—
166C8×C53—42no—
167#88322—42no—
168#88322—81no—
169#8897—531nomaximal, Sylow
170C22×C53#38511yescharacteristic
171C22×C53#44711yescharacteristic
172C22×C53#44911yescharacteristic
173C22×C53#45211yescharacteristic
174C22×C53#45211yescharacteristic
175C22×C53#45411yescharacteristic
176C22×C53#45511yescharacteristic
177C2×D53—42no—
178C2×D53—42no—
179C2×D53—42no—
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C2×D53—81no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—42no—
187C4×C53—81no—
188C4×C53—161no—
189C4×C53#42711yescharacteristic
190C4×C53#42811yescharacteristic
191C4×C53C4×D811yescharacteristic
192C4×C53#44011yescharacteristic
193C4×C53#47411yescharacteristic
194C4×C53#47811yescharacteristic
195C4×C53#48211yescharacteristic
196C4×C53#48511yescharacteristic
197Dic53—42no—
198Dic53—82no—
199#2354—531no—
200#1947D5311yescharacteristic, complemented
201#1834—531no—
202#1802—531no—
203C2×D4⋊C8—531no—
204#1296—531no—
205#1279—531no—
206C2×C53#162511yescharacteristic
207C2×C53#163811yescharacteristic
208C2×C53#164111yescharacteristic
209C2×C53#164611yescharacteristic
210C2×C53#185411yescharacteristic
211C2×C53#186011yescharacteristic
212C2×C53#186211yescharacteristic
213D53—82no—
214C2×C4×D4—531no—
215C2×#117—1061no—
216C2×#107—21no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8C2×D5311yescharacteristic
219C2×#102—1061no—
220C2×C4×C8—531no—
221#390—1061no—
222#388—1062no—
223#384—531no—
224#384C2×D5311yescharacteristic
225#380—1061no—
226C4×C4⋊C4—1061no—
227C4×C4⋊C4C2×D5311yescharacteristic
228#340—21no—
229#340—1061no—
230#340—1062no—
231D4⋊C8—534no—
232C53#889711yescharacteristic, complemented, Sylow
233C22×D4—531no—
234C23×C4—1061no—
235C22×C8—21no—
236C22×C8—1061no—
237C22×C8—1061no—
238C4×D4—532no—
239C4×D4—532no—
240C4×D4—1061no—
241C4×D4—1061no—
242#117—2122no—
243#117—2122no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—532no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4C22⋊D5311yescharacteristic
252C2×C4⋊C4C22⋊D5311yescharacteristic
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1062no—
255C2×C22⋊C4—1062no—
256C2×C42—21no—
257C2×C42—1062no—
258C2×C42C22×D5311yescharacteristic
259#107—42no—
260C4⋊C8—21no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C4⋊C8C4×D5312yes—
264#102—2122no—
265C4×C8—532no—
266C4×C8—1061no—
267#95—21no—
268#95—1062no—
269C24—1061no—
270C2×D4—532no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—1061no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—1062no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4D4×D5311yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C22×C4C2×C22⋊D5311yescharacteristic
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—1062no—
294C2×C8—2121no—
295C2×C8—2121no—
296C2×C8—2121no—
297C2×C8—2121no—
298C2×C8—2121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—532no—
308C4⋊C4—1061no—
309C4⋊C4—2122no—
310C4⋊C4—2121no—
311C4⋊C4—2121no—
312C4⋊C4—2121no—
313C4⋊C4—2121no—
314C4⋊C4#1171992212yes—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—4242no—
318C22⋊C4—2124no—
319C42—41no—
320C42—41no—
321C42—2124no—
322C42C2×C4×D5311yescharacteristic
323C42#1171994511yescharacteristic
324C42Q8×D5311yescharacteristic
325C42#1171994911yescharacteristic
326C23—1062no—
327C23—2121no—
328C23—2121no—
329C23—2121no—
330C23—2121no—
331C23—2121no—
332C23—2121no—
333C23#1313318511yescharacteristic
334D4—1062no—
335D4—1062no—
336D4—2121no—
337D4—2121no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—1062no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4C4⋊C4×D5311yescharacteristic
360C2×C4C4×C22⋊D5311yescharacteristic
361C2×C4#1313317911yescharacteristic
362C2×C4#1313319311yescharacteristic
363C2×C4#1313319511yescharacteristic
364C2×C4#1313320211yescharacteristic
365C2×C4C2×#1171992211yescharacteristic
366C2×C4#1313323611yescharacteristic
367C2×C4#1313323711yescharacteristic
368C2×C4#1313324011yescharacteristic
369C2×C4#1313324111yescharacteristic
370C2×C4#1313324211yescharacteristic
371C2×C4#1313324811yescharacteristic
372C2×C4#1313324911yescharacteristic
373C8—42no—
374C8—4241no—
375C8—2122no—
376C22—4241no—
377C22—4241no—
378C22—4241no—
379C22—4241no—
380C22—2122no—
381C22—2122no—
382C22—2122no—
383C22#1681049411yescharacteristic
384C22#1681075611yescharacteristic
385C22#1681076511yescharacteristic
386C22#1681078311yescharacteristic
387C22#1681078411yescharacteristic
388C22#1681079411yescharacteristic
389C22#1681079911yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—81no—
394C4—161no—
395C4—4242no—
396C4—2122no—
397C4#1681066011yescharacteristic
398C4#1681066411yescharacteristic
399C4C4×#1171992211yescharacteristic
400C4#1681072111yescharacteristic
401C4#1681090011yescharacteristic
402C4#1681092011yescharacteristic
403C4#1681095011yescharacteristic
404C4#1681096611yescharacteristic
405C2—4242no—
406C2#2027380211yescharacteristic
407C2#2027388611yescharacteristic
408C2#2027390611yescharacteristic
409C2#2027394211yescharacteristic
410C2#2027491111yescharacteristic
411C2#2027494511yescharacteristic
412C2#2027496111yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1947C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1947D53
hypocenterC53#8897
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. #20274842C2semidirect
5. #20272599C2semidirect
6. #20272556C2semidirect
7. C53×#1947C2semidirect
20. #16810350C4semidirect
200. #1947D53semidirect
232. C53#8897semidirect

Maximal subgroups

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

subgroupindex
#202776622
#202748422
#202747122
C2×#168103502
#202725992
#202725562
C53×#19472
#889753