Finite Group Databasebeta

← #27285202

Subgroups — #27285202

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
2#20279661C211yesmaximal, characteristic
3#20275664C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×#476C211yesmaximal, characteristic, complemented
8C22×#11719924—22no—
9C2×#13133214C2211yescharacteristic
10C2×#13133213—21no—
11C2×#13133191—22no—
12#16810997—22no—
13#16810997—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4×#88322C2211yescharacteristic
16#16810532—22no—
17#16810524—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810385—22no—
20#16810366—22no—
21D4⋊#88322—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#476—22no—
25C53×#476—22no—
26C2×#106×C53—21no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#102×C53—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212D412yes—
32C22×C22⋊D53—24no—
33C2×#11719924—28no—
34C2×#11719924—48no—
35C22×#88322—21no—
36C22×#88322—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133213—42no—
42#13133213—42no—
43#13133191—44no—
44#13133191—44no—
45C2×C4⋊Dic53D412yes—
46C2×#11719921—24no—
47C2×C4×Dic53—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4×#88322—21no—
51C4×#88322D412yescomplemented
52#13133124—21no—
53C22×Q8×C53—21no—
54C22×D4×C53D411yescharacteristic
55C23×C4×C53—21no—
56C22×C8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53D412yescomplemented
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C42×C53C2311yescharacteristic
70#106×C53—24no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53D412yescomplemented
73#102×C53—24no—
74C24×C53—21no—
75C23×D53—22no—
76C2×Q8×C53—22no—
77C2×Q8×C53—44no—
78C2×Q8×C53—41no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53SD1612yescomplemented
85C22×C4×C53—21no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—42no—
90C22×C4×C53—42no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C2×C22⋊D53—48no—
95C2×C22⋊D53—48no—
96C2×C22⋊D53—84no—
97C22×Dic53—22no—
98C2×D212—22no—
99C2×D212—22no—
100C2×D212—44no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—42no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—42no—
109C2×D212—41no—
110C2×D212—41no—
111C2×C8×C53—22no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126#11719924—416no—
127#11719921—48no—
128C4⋊Dic53—22no—
129C4⋊Dic53SD1614yescomplemented
130C4×Dic53—41no—
131C4×Dic53—41no—
132C2×#88322—24no—
133C2×#88322—22no—
134C2×#88322—41no—
135C2×#88322—42no—
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—42no—
139C23×C53—41no—
140C23×C53—42no—
141C23×C53#12011yescharacteristic
142C22×D53—42no—
143C22×D53—42no—
144C22×D53—44no—
145C22×D53—42no—
146C22×D53—44no—
147Q8×C53—44no—
148Q8×C53—82no—
149D4×C53—22no—
150D4×C53—22no—
151D4×C53—41no—
152D4×C53—41no—
153C2×C4×C53—22no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12711yescharacteristic
167C2×C4×C53C2×SD1612yes—
168C2×C4×C53C2×SD1611yescharacteristic
169C2×C4×C53#13511yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)12yes—
172C2×C4×C53#13711yescharacteristic
173C22⋊D53—816no—
174C2×Dic53—24no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177D212—44no—
178D212—82no—
179D212—81no—
180D212—81no—
181D212—82no—
182D212—81no—
183D212—81no—
184D212—81no—
185D212—81no—
186C8×C53—42no—
187#88322—24no—
188#88322—81no—
189#10054—531nomaximal, Sylow
190C22×C53—44no—
191C22×C53—42no—
192C22×C53—82no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53#39211yescharacteristic
196C22×C53#45012yes—
197C22×C53#45011yescharacteristic
198C22×C53#45211yescharacteristic
199C22×C53#45312yes—
200C2×D53—84no—
201C2×D53—84no—
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—82no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—82no—
208C4×C53—81no—
209C4×C53#44811yescharacteristic
210C4×C53#44911yescharacteristic
211C4×C53#46012yes—
212C4×C53#46511yescharacteristic
213C4×C53#46811yescharacteristic
214C4×C53#49211yescharacteristic
215C4×C53#49611yescharacteristic
216Dic53—44no—
217C2×#492—531no—
218C2×#476D5311yescharacteristic, complemented
219#2541—531no—
220#1987—531no—
221C2×#331—531no—
222C2×D4⋊C8—531no—
223#1279—531no—
224C2×C53—82no—
225C2×C53#162912yes—
226C2×C53#163112yes—
227C2×C53#200011yescharacteristic
228C2×C53#200211yescharacteristic
229C2×C53#201411yescharacteristic
230D53—162no—
231C22×SD16—1062no—
232C2×C4⋊Q8C2×D5311yescharacteristic
233C2×#127—531no—
234C2×#127—1061no—
235C2×#121—1062no—
236C2×C4×D4C2×D5311yescharacteristic
237#492—1062no—
238#492—1062no—
239#476—22no—
240#476—22no—
241C2×#106—21no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8C2×D5311yescharacteristic
244C2×#102—21no—
245C2×C4×C8—531no—
246#392—1062no—
247#390—1061no—
248C4×C4⋊C4—531no—
249#340—1062no—
250#331—1062no—
251D4⋊C8—1062no—
252C53#1005411yescharacteristic, complemented, Sylow
253C22×Q8—21no—
254C22×D4—532no—
255C22×D4—1064no—
256C22×D4—1061no—
257C22×D4—1061no—
258C22×D4C22⋊D5311yescharacteristic
259C23×C4—21no—
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⋊Q8C22⋊D5312yescomplemented
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—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—532no—
284C2×C4⋊C4C22⋊D5311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—1064no—
287C2×C42—1061no—
288C2×C42C22×D5311yescharacteristic
289#106—24no—
290C4⋊C8—21no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293C4⋊C8C22⋊D5312yescomplemented
294#102—24no—
295C4×C8—532no—
296C4×C8—1061no—
297#95—1061no—
298C24—21no—
299C24—1062no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—1062no—
309C2×D4—1062no—
310C2×D4—4244no—
311C2×D4—2124no—
312C2×D4—2128no—
313C2×D4—2128no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2122no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2122no—
323C2×D4—2121no—
324C2×D4#1171992412yescomplemented
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—1062no—
332C22×C4D4×D5311yescharacteristic
333C22×C4D4×D5311yescharacteristic
334C22×C4C2×C22⋊D5311yescharacteristic
335SD16—21216no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—1064no—
340C2×C8—1062no—
341C2×C8—2121no—
342C2×C8—2122no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—534no—
350C4⋊C4—1062no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—2128no—
354C42—2121no—
355C42—2121no—
356C42D4×D5311yescharacteristic
357C42#1171994711yescharacteristic
358C42C2×C22⋊D5311yescharacteristic
359C42C2×C22⋊D5311yescharacteristic
360C23—22no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—2124no—
366C23—2122no—
367C23—2122no—
368C23—2122no—
369C23—2124no—
370C23#1313318511yescharacteristic
371Q8—44no—
372Q8—82no—
373D4—22no—
374D4—22no—
375D4—41no—
376D4—41no—
377D4—4242no—
378D4—42416no—
379D4—4241no—
380D4—4241no—
381D4—4241no—
382D4—4242no—
383D4—4241no—
384D4—4241no—
385D4—4241no—
386D4—2124no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—1064no—
396C2×C4—2122no—
397C2×C4—2122no—
398C2×C4#1313318511yescharacteristic
399C2×C4#1313318611yescharacteristic
400C2×C4#1313318811yescharacteristic
401C2×C4#1313318911yescharacteristic
402C2×C4#1313318911yescharacteristic
403C2×C4#1313321311yescharacteristic
404C2×C4SD16×D5312yes—
405C2×C4C2×#1171992411yescharacteristic
406C2×C4#1313323711yescharacteristic
407C2×C4#1313324211yescharacteristic
408C2×C4#1313324412yes—
409C2×C4#1313324811yescharacteristic
410C8—42no—
411C8—1064no—
412C8—4241no—
413C22—42no—
414C22—44no—
415C22—81no—
416C22—81no—
417C22—82no—
418C22—4242no—
419C22—4242no—
420C22—4242no—
421C22—4244no—
422C22—4244no—
423C22#1681053111yescharacteristic
424C22#1681077212yes—
425C22#1681077311yescharacteristic
426C22#1681078311yescharacteristic
427C22#1681078912yes—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—81no—
432C4—2124no—
433C4#1681076211yescharacteristic
434C4#1681076811yescharacteristic
435C4#1681082112yes—
436C4#1681085411yescharacteristic
437C4#1681087111yescharacteristic
438C4#1681099411yescharacteristic
439C4#1681100911yescharacteristic
440C2—82no—
441C2—8482no—
442C2#2027383012yes—
443C2#2027384112yes—
444C2#2027571611yescharacteristic
445C2#2027572711yescharacteristic
446C2#2027580111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#476C2
socleC23×C53#120
prime radical#20279661C2
hypercenterC2×#476D53
hypocenterC53#10054
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#16810997C2semidirect
3. #20275664C2semidirect
4. C2×#16810366C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272554C2semidirect
7. C2×C53×#476C2semidirect
51. C4×#88322D4semidirect
60. C4⋊Q8×C53D4semidirect
72. C4⋊C8×C53D4semidirect
84. C2×D4×C53SD16semidirect
129. C4⋊Dic53SD16semidirect
218. C2×#476D53semidirect
252. C53#10054semidirect
268. C4⋊Q8C22⋊D53semidirect
293. C4⋊C8C22⋊D53semidirect
324. C2×D4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109972
#202796612
#202756642
C2×#168103662
C2×D4⋊#883222
#202725542
C2×C53×#4762
#1005453