Finite Group Databasebeta

← #27285203

Subgroups — #27285203

Order 13568.

838 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810997C211yesmaximal, characteristic, complemented
2C2×#16810910C211yesmaximal, characteristic, complemented
3#20279657C211yesmaximal, characteristic
4C2×#16810366C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1987C211yesmaximal, characteristic, complemented
8C22×#11719924—22no—
9C2×#13133214—21no—
10C2×#13133214C2211yescharacteristic
11C2×#13133192—22no—
12C2×C4×D212C2211yescharacteristic
13#16810997—22no—
14#16810997—22no—
15#16810910—22no—
16#16810910—22no—
17C2×#13133148—21no—
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133137—21no—
20C2×C4×#88322C2211yescharacteristic
21#16810533—22no—
22#16810525—21no—
23#16810366—22no—
24#16810350—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C4×C4⋊C4×C53C2211yescharacteristic
28C53×#340—22no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—24no—
32C22×D212D411yescharacteristic
33C22×D212D412yes—
34C22×C4×D53—21no—
35C2×#11719924—28no—
36C2×#11719924—48no—
37C22×#88322—21no—
38C22×#88322—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214—42no—
44#13133214—42no—
45#13133192—44no—
46#13133192—44no—
47C4×D212—21no—
48C4×D212—21no—
49C4×D212—21no—
50C4×D212—21no—
51C2×C4⋊Dic53D411yescharacteristic
52C2×#11719921—24no—
53C2×#11719921—21no—
54#13133148—24no—
55C4⋊#88322—21no—
56C4⋊#88322D412yescomplemented
57#13133137—24no—
58C4×#88322—21no—
59C4×#88322D412yescomplemented
60C22×Q8×C53—21no—
61C22×C8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53D412yescomplemented
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53D412yes—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73#95×C53—21no—
74C23×D53—21no—
75C23×D53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53—44no—
78C2×Q8×C53—41no—
79C22×C4×C53—22no—
80C22×C4×C53—41no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—22no—
90C2×D212—21no—
91C2×D212—22no—
92C2×D212—44no—
93C2×D212—48no—
94C2×D212—48no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—42no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—42no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—84no—
106C2×D212SD1612yescomplemented
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C2×C4×D53—42no—
110C2×C4×D53—42no—
111C2×C8×C53—41no—
112C2×C8×C53—42no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53SD1614yescomplemented
119C42×C53—41no—
120C42×C53—41no—
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125#11719924—416no—
126#11719921—48no—
127#11719921—41no—
128#11719921—41no—
129C4⋊Dic53—21no—
130C4⋊Dic53—21no—
131C2×#88322—24no—
132C2×#88322—22no—
133C2×#88322—22no—
134C2×#88322—41no—
135C2×#88322—41no—
136C23×C53#12011yescharacteristic
137C22×D53—22no—
138C22×D53—41no—
139C22×D53—42no—
140C22×D53—41no—
141C22×D53—42no—
142C22×D53—42no—
143C22×D53—42no—
144C22×D53—44no—
145C22×D53—42no—
146C22×D53—44no—
147Q8×C53—44no—
148Q8×C53—82no—
149C2×C4×C53—22no—
150C2×C4×C53—24no—
151C2×C4×C53—42no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—82no—
155C2×C4×C53#12011yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12711yescharacteristic
161C2×C4×C53C2×SD1611yescharacteristic
162C2×C4×C53C2×SD1612yes—
163C2×C4×C53#13511yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)12yes—
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C53#13711yescharacteristic
167C2×Dic53—41no—
168C2×Dic53—42no—
169D212—22no—
170D212—22no—
171D212—44no—
172D212—41no—
173D212—41no—
174D212—82no—
175D212—816no—
176D212—81no—
177D212—81no—
178D212—82no—
179D212—81no—
180D212—81no—
181D212—81no—
182D212—81no—
183C4×D53—82no—
184C4×D53—81no—
185C4×D53—81no—
186C8×C53—81no—
187#88322—24no—
188#88322—42no—
189#10054—531nomaximal, Sylow
190C22×C53#39211yescharacteristic
191C22×C53#45011yescharacteristic
192C22×C53#45012yes—
193C22×C53#45211yescharacteristic
194C22×C53#45312yes—
195C2×D53—44no—
196C2×D53—42no—
197C2×D53—82no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—84no—
201C2×D53—84no—
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—82no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—44no—
208C4×C53—82no—
209C4×C53#44811yescharacteristic
210C4×C53#44911yescharacteristic
211C4×C53#46012yes—
212C4×C53#46511yescharacteristic
213C4×C53#46811yescharacteristic
214C4×C53#49211yescharacteristic
215C4×C53#49611yescharacteristic
216Dic53—81no—
217C2×#492—531no—
218C2×#476—531no—
219#2541—531no—
220#1987D5311yescharacteristic, complemented
221C2×#331—531no—
222C2×D4⋊C8—531no—
223#1279—531no—
224C2×C53#162912yes—
225C2×C53#163112yes—
226C2×C53#200011yescharacteristic
227C2×C53#200211yescharacteristic
228C2×C53#201411yescharacteristic
229D53—82no—
230D53—162no—
231C22×SD16—1062no—
232C2×C4⋊Q8C2×D5311yescharacteristic
233C2×#127—531no—
234C2×#127—1061no—
235C2×#121—1062no—
236C2×C4×D4—531no—
237#492—1062no—
238#492—1062no—
239#476—1062no—
240#476—1062no—
241C2×#106—1061no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8C2×D5311yescharacteristic
244C2×#102—1061no—
245C2×C4×C8—531no—
246#392—1062no—
247#390—1061no—
248C4×C4⋊C4C2×D5311yescharacteristic
249#340—22no—
250#331—1062no—
251D4⋊C8—1062no—
252C53#1005411yescharacteristic, complemented, Sylow
253C22×Q8—21no—
254C22×D4—531no—
255C22×D4—532no—
256C22×D4—1064no—
257C22×D4—1061no—
258C22×D4—1061no—
259C23×C4—1061no—
260C2×SD16—1068no—
261C2×SD16—2128no—
262C22×C8—21no—
263C22×C8—1061no—
264C22×C8—1061no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8D21212yescomplemented
269#127—1061no—
270#127—1061no—
271#127—1061no—
272#127—1061no—
273#127—2122no—
274#127—2122no—
275#121—2124no—
276#121—2124no—
277C4×D4—1061no—
278C4×D4—1061no—
279C4×D4—1061no—
280C4×D4—1061no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—531no—
284C2×C4⋊C4C22⋊D5312yes—
285C2×C22⋊C4—1064no—
286C2×C22⋊C4—1061no—
287C2×C42—21no—
288C2×C42C22×D5311yescharacteristic
289#106—1064no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—532no—
293C4⋊C8—1061no—
294#102—1064no—
295C4×C8—532no—
296C4×C8—1061no—
297#95—21no—
298C24—1061no—
299C24—1062no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×D4—532no—
304C2×D4—1061no—
305C2×D4—1062no—
306C2×D4—1061no—
307C2×D4—1062no—
308C2×D4—1061no—
309C2×D4—1061no—
310C2×D4—1061no—
311C2×D4—4244no—
312C2×D4—2128no—
313C2×D4—2124no—
314C2×D4—2128no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2122no—
321C2×D4—2121no—
322C2×D4—2122no—
323C2×D4—2121no—
324C2×D4—2121no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—1061no—
328C22×C4—2121no—
329C22×C4—2121no—
330C22×C4—2122no—
331C22×C4—2122no—
332C22×C4D4×D5311yescharacteristic
333C22×C4D4×D5311yescharacteristic
334C22×C4C2×C22⋊D5311yescharacteristic
335SD16—21216no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—1064no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—2121no—
342C2×C8—2121no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C4⋊C4#1171992414yescomplemented
351C22⋊C4—2128no—
352C22⋊C4—2121no—
353C22⋊C4—2121no—
354C42—41no—
355C42—41no—
356C42C2×D21211yescharacteristic
357C42#1171994511yescharacteristic
358C42D4×D5311yescharacteristic
359C42D4×D5311yescharacteristic
360C23—1062no—
361C23—2122no—
362C23—2124no—
363C23—2122no—
364C23—2122no—
365C23—2124no—
366C23—2122no—
367C23—2121no—
368C23—2121no—
369C23—2122no—
370C23#1313318511yescharacteristic
371Q8—44no—
372Q8—82no—
373D4—1062no—
374D4—1062no—
375D4—4242no—
376D4—42416no—
377D4—4241no—
378D4—4241no—
379D4—4241no—
380D4—4242no—
381D4—4241no—
382D4—4241no—
383D4—4241no—
384D4—2124no—
385D4—2121no—
386D4—2121no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—4241no—
394C2×C4—4242no—
395C2×C4—4241no—
396C2×C4—2122no—
397C2×C4—2121no—
398C2×C4#1313318711yescharacteristic
399C2×C4#1313318711yescharacteristic
400C2×C4#1313319011yescharacteristic
401C2×C4#1313319011yescharacteristic
402C2×C4#1313319111yescharacteristic
403C2×C4#1313321311yescharacteristic
404C2×C4SD16×D5311yescharacteristic
405C2×C4C2×#1171992412yes—
406C2×C4#1313324011yescharacteristic
407C2×C4#1313324212yes—
408C2×C4#1313324411yescharacteristic
409C2×C4#1313324911yescharacteristic
410C8—81no—
411C8—1064no—
412C8—2122no—
413C22—4242no—
414C22—4242no—
415C22—4242no—
416C22—4244no—
417C22—4244no—
418C22—4241no—
419C22—4241no—
420C22—4242no—
421C22—2122no—
422C22—2124no—
423C22#1681053211yescharacteristic
424C22#1681077211yescharacteristic
425C22#1681077312yes—
426C22#1681078411yescharacteristic
427C22#1681078912yes—
428C4—21no—
429C4—21no—
430C4—44no—
431C4—82no—
432C4—4241no—
433C4#1681076311yescharacteristic
434C4#1681077111yescharacteristic
435C4#1681082312yes—
436C4#1681085511yescharacteristic
437C4#1681087211yescharacteristic
438C4#1681099711yescharacteristic
439C4#1681101311yescharacteristic
440C2—8482no—
441C2—4242no—
442C2#2027383012yes—
443C2#2027384012yes—
444C2#2027571711yescharacteristic
445C2#2027573111yescharacteristic
446C2#2027580211yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1987C2
socleC23×C53#120
prime radical#20279657C2
hypercenter#1987D53
hypocenterC53#10054
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
1. C2×#16810997C2semidirect
2. C2×#16810910C2semidirect
4. C2×#16810366C2semidirect
5. C2×#16810350C2semidirect
6. #20272556C2semidirect
7. C53×#1987C2semidirect
56. C4⋊#88322D4semidirect
59. C4×#88322D4semidirect
65. C4⋊Q8×C53D4semidirect
106. C2×D212SD16semidirect
118. C4⋊C4×C53SD16semidirect
220. #1987D53semidirect
252. C53#10054semidirect
268. C4⋊Q8D212semidirect
350. C4⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109972
C2×#168109102
#202796572
C2×#168103662
C2×#168103502
#202725562
C53×#19872
#1005453