Finite Group Databasebeta

← #27278354

Subgroups — #27278354

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#20277666C211yesmaximal, characteristic
2#20275471C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic
4C2×#16810352C211yesmaximal, characteristic
5#20272598C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1834C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133175—21no—
10C2×#13133150—21no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133137—21no—
14#16810524—21no—
15#16810513—22no—
16#16810480—21no—
17C4⋊C4×Dic53—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810385—21no—
20#16810385—21no—
21#16810352C414yescomplemented
22C2×C4×C8×C53C2211yescharacteristic
23C53×#384C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C53×#340—22no—
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⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D411yescharacteristic
39C2×#11719921—22no—
40C2×#11719921—22no—
41C2×#11719921—21no—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—21no—
44#13133150—42no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322C2×C412yes—
49#13133137—42no—
50#13133124—22no—
51#13133124—21no—
52C22×C8×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53D412yes—
59C2×C42×C53C2311yescharacteristic
60C4×C8×C53—21no—
61C4×C8×C53C2×C412yes—
62C23×D53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—41no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
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—42no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—41no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C42×C53C22×C411yescharacteristic
97C42×C53C2×Q811yescharacteristic
98C42×C53#4111yescharacteristic
99C42×C53#4111yescharacteristic
100#11719921—22no—
101#11719921—22no—
102#11719921—44no—
103#11719921—82no—
104C4⋊Dic53—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53C4⋊C412yes—
109C4⋊Dic53D812yes—
110C4×Dic53—44no—
111C4×Dic53—41no—
112C4×Dic53—41no—
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—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53C2×C4⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C22⋊Q811yescharacteristic
139C2×C4×C53C22⋊Q811yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53C2×D811yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53#13511yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53#13711yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
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—82no—
187C4×C53—161no—
188C4×C53#42711yescharacteristic
189C4×C53#42811yescharacteristic
190C4×C53C4×D811yescharacteristic
191C4×C53#44011yescharacteristic
192C4×C53#47411yescharacteristic
193C4×C53#47811yescharacteristic
194C4×C53#48211yescharacteristic
195C4×C53#48511yescharacteristic
196Dic53—42no—
197Dic53—42no—
198Dic53—81no—
199#2354—531no—
200#1947—531no—
201#1834D5311yescharacteristic, complemented
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—1061no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8—531no—
219C2×#102—1061no—
220C2×C4×C8C2×D5311yescharacteristic
221#390—1061no—
222#388—1062no—
223#384C2×D5311yescharacteristic
224#384C2×D5311yescharacteristic
225#380—1061no—
226C4×C4⋊C4—531no—
227C4×C4⋊C4—1061no—
228#340—22no—
229#340—1061no—
230#340—1061no—
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—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—531no—
251C2×C4⋊C4—531no—
252C2×C4⋊C4C22⋊D5312yes—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1062no—
255C2×C22⋊C4—1062no—
256C2×C42—1062no—
257C2×C42—1061no—
258C2×C42C22×D5311yescharacteristic
259#107—2122no—
260C4⋊C8—532no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264#102—2122no—
265C4×C8—21no—
266C4×C8C4×D5312yes—
267#95—1062no—
268#95—1061no—
269C24—1061no—
270C2×D4—532no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—1061no—
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—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—532no—
310C4⋊C4—532no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—4242no—
318C22⋊C4—2124no—
319C42—2124no—
320C42—2121no—
321C42—2121no—
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—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
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×C4D8×D5311yescharacteristic
366C2×C4#1313323811yescharacteristic
367C2×C4#1313323811yescharacteristic
368C2×C4#1313324011yescharacteristic
369C2×C4#1313324111yescharacteristic
370C2×C4#1313324411yescharacteristic
371C2×C4#1313324611yescharacteristic
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#1681075511yescharacteristic
385C22#1681076711yescharacteristic
386C22#1681078211yescharacteristic
387C22#1681078411yescharacteristic
388C22#1681079511yescharacteristic
389C22#1681080011yescharacteristic
390C4—21no—
391C4—21no—
392C4—82no—
393C4—161no—
394C4—4241no—
395C4—2122no—
396C4—2122no—
397C4#1681065811yescharacteristic
398C4#1681066211yescharacteristic
399C4#1681070511yescharacteristic
400C4#1681072211yescharacteristic
401C4#1681090211yescharacteristic
402C4#1681092611yescharacteristic
403C4#1681094911yescharacteristic
404C4#1681096711yescharacteristic
405C2—4242no—
406C2#2027380111yescharacteristic
407C2#2027388411yescharacteristic
408C2#2027390511yescharacteristic
409C2#2027394011yescharacteristic
410C2#2027491311yescharacteristic
411C2#2027494711yescharacteristic
412C2#2027495911yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1834C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1834D53
hypocenterC53#8897
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
2. #20275471C2semidirect
5. #20272598C2semidirect
6. #20272555C2semidirect
7. C53×#1834C2semidirect
21. #16810352C4semidirect
201. #1834D53semidirect
232. C53#8897semidirect

Maximal subgroups

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

subgroupindex
#202776662
#202754712
#202747122
C2×#168103522
#202725982
#202725552
C53×#18342
#889753