Finite Group Databasebeta

← #27285313

Subgroups — #27285313

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810900C211yesmaximal, characteristic, complemented
2#20279662C211yesmaximal, characteristic
3#20275639C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×#486C211yesmaximal, characteristic, complemented
8C2×#13133213—21no—
9C2×#13133188—22no—
10C2×C4×D212C2211yescharacteristic
11#16810900—22no—
12#16810900—22no—
13C2×#13133150—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133141—22no—
17C2×#13133137—21no—
18#16810531—22no—
19#16810526—21no—
20C4×C4⋊Dic53C2211yescharacteristic
21#16810367—22no—
22#16810352—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×#127×C53C2211yescharacteristic
25C53×#486—22no—
26C53×#486—22no—
27C2×#102×C53—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C22⋊D53—24no—
31C22×C4×D53—21no—
32C22×#88322—21no—
33C22×#88322—21no—
34#13133213—42no—
35#13133213—42no—
36#13133188—44no—
37#13133188—44no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4⋊Dic53D412yes—
44C2×C22⋊Dic53—24no—
45C2×#11719921—21no—
46C2×C4×Dic53—21no—
47#13133150—24no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322D412yescomplemented
52#13133141—44no—
53#13133137—24no—
54#13133124—21no—
55C22×Q8×C53—21no—
56C22×D4×C53—21no—
57C22×D4×C53—21no—
58C22×D4×C53D412yes—
59C22×C8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53D412yescomplemented
64#127×C53—21no—
65#127×C53—21no—
66#127×C53—21no—
67#127×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#102×C53—28no—
72C4×C8×C53—21no—
73C4×C8×C53D412yescomplemented
74C24×C53—22no—
75C23×D53—21no—
76C2×Q8×C53—22no—
77C2×Q8×C53—41no—
78C2×Q8×C53—42no—
79C2×Q8×C53—42no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—44no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—42no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—42no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C2×C22⋊D53—48no—
98C2×C22⋊D53—48no—
99C2×C22⋊D53—84no—
100C22×Dic53—22no—
101C22×Dic53—21no—
102C2×D212—21no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212—21no—
106C2×D212—21no—
107C2×D212SD1612yes—
108C2×C4×D53—41no—
109C2×C4×D53—41no—
110C2×C4×D53—42no—
111C2×C4×D53—42no—
112C2×C8×C53—22no—
113C2×C8×C53—42no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C22⋊Dic53—48no—
123#11719921—41no—
124#11719921—41no—
125C4⋊Dic53—22no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128C4⋊Dic53D814yes—
129C4×Dic53—41no—
130C4×Dic53—41no—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—42no—
136C23×C53—42no—
137C23×C53—44no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—44no—
141C23×C53#12011yescharacteristic
142C22×D53—22no—
143C22×D53—41no—
144C22×D53—42no—
145C22×D53—41no—
146C22×D53—42no—
147Q8×C53—44no—
148Q8×C53—81no—
149Q8×C53—81no—
150D4×C53—44no—
151D4×C53—82no—
152D4×C53—81no—
153D4×C53—81no—
154D4×C53—82no—
155D4×C53—81no—
156D4×C53—81no—
157D4×C53—81no—
158D4×C53—81no—
159C2×C4×C53—22no—
160C2×C4×C53—42no—
161C2×C4×C53—82no—
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12711yescharacteristic
168C2×C4×C53C2×D812yes—
169C2×C4×C53C2×SD1611yescharacteristic
170C2×C4×C53#13511yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)12yes—
172C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C53#13711yescharacteristic
174C22⋊D53—816no—
175C2×Dic53—24no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—41no—
179C2×Dic53—42no—
180D212—22no—
181D212—22no—
182D212—41no—
183D212—41no—
184C4×D53—82no—
185C4×D53—81no—
186C4×D53—81no—
187C8×C53—42no—
188#88322—42no—
189#88322—81no—
190#10070—531nomaximal, Sylow
191C22×C53—84no—
192C22×C53—84no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53—82no—
196C22×C53#39211yescharacteristic
197C22×C53#44712yes—
198C22×C53#45011yescharacteristic
199C22×C53#45211yescharacteristic
200C22×C53#45512yes—
201C2×D53—44no—
202C2×D53—42no—
203C2×D53—82no—
204C2×D53—81no—
205C2×D53—81no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—82no—
209C4×C53#44811yescharacteristic
210C4×C53#44911yescharacteristic
211C4×C53#45912yes—
212C4×C53#46511yescharacteristic
213C4×C53#46811yescharacteristic
214C4×C53#49311yescharacteristic
215C4×C53#49611yescharacteristic
216Dic53—44no—
217Dic53—81no—
218C2×#486D5311yescharacteristic, complemented
219C2×#474—531no—
220#2541—531no—
221#1981—531no—
222C2×#331—531no—
223C2×D4⋊C8—531no—
224#1279—531no—
225C2×C53—162no—
226C2×C53#163912yes—
227C2×C53#164112yes—
228C2×C53#200011yescharacteristic
229C2×C53#200411yescharacteristic
230C2×C53#201111yescharacteristic
231D53—82no—
232C2×C4⋊Q8C2×D5311yescharacteristic
233C2×#127—1061no—
234C2×#127C2×D5311yescharacteristic
235C2×#121—1062no—
236C2×C4×D4—531no—
237#486—22no—
238#486—22no—
239#474—1062no—
240#474—1062no—
241C2×#107—1061no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8—531no—
244C2×#103—1062no—
245C2×#102—22no—
246C2×#102—1061no—
247C2×C4×C8C2×D5311yescharacteristic
248#392—1062no—
249#390—1061no—
250C4×C4⋊C4—531no—
251#331—1062no—
252D4⋊C8—1062no—
253C53#1007011yescharacteristic, complemented, Sylow
254C22×Q8—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—531no—
258C22×D4—1064no—
259C22×D4C22⋊D5312yes—
260C23×C4—1061no—
261C22×C8—21no—
262C22×C8—1061no—
263C22×C8—1061no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8C22⋊D5312yescomplemented
268#127—21no—
269#127—21no—
270#127—21no—
271#127—21no—
272#127—2122no—
273#127—2122no—
274#121—2124no—
275#121—2124no—
276C4×D4—1061no—
277C4×D4—1061no—
278C4×D4—1061no—
279C4×D4—1061no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—532no—
283C2×C4⋊C4—531no—
284C2×C22⋊C4—1064no—
285C2×C22⋊C4—1061no—
286C2×C42—1061no—
287C2×C42C22×D5311yescharacteristic
288#107—1064no—
289C4⋊C8—532no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293#103—2124no—
294#102—28no—
295#102—1064no—
296C4×C8—21no—
297C4×C8C22⋊D5312yescomplemented
298#95—1061no—
299C24—22no—
300C24—1061no—
301C2×Q8—22no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—44no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—42no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—532no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—1061no—
322C2×D4—1061no—
323C2×D4—1061no—
324C2×D4—4244no—
325C2×D4—2128no—
326C2×D4—2128no—
327C22×C4—41no—
328C22×C4—1062no—
329C22×C4—1061no—
330C22×C4—2121no—
331C22×C4—2121no—
332C22×C4—2122no—
333C22×C4—2122no—
334C22×C4D4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C2×C8—22no—
338C2×C8—42no—
339C2×C8—1062no—
340C2×C8—2121no—
341C2×C8—2122no—
342C2×C8—2121no—
343C2×C8—2121no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—534no—
349C4⋊C4—1062no—
350C4⋊C4—1061no—
351C4⋊C4—1061no—
352C22⋊C4—2128no—
353C22⋊C4—2121no—
354C22⋊C4—2121no—
355C42—2121no—
356C42—2121no—
357C42D4×D5311yescharacteristic
358C42#1171994711yescharacteristic
359C42C2×C22⋊D5311yescharacteristic
360C42C2×C22⋊D5311yescharacteristic
361C23—44no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—44no—
366C23—1062no—
367C23—2122no—
368C23—2121no—
369C23—2121no—
370C23—2122no—
371C23#1313318511yescharacteristic
372Q8—44no—
373Q8—81no—
374Q8—81no—
375D4—44no—
376D4—82no—
377D4—81no—
378D4—81no—
379D4—81no—
380D4—82no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—1062no—
385D4—1062no—
386D4—42416no—
387D4—2121no—
388D4—2121no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—1064no—
393C2×C4—4241no—
394C2×C4—4242no—
395C2×C4—4241no—
396C2×C4—2122no—
397C2×C4—2121no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4#1313318511yescharacteristic
401C2×C4#1313318611yescharacteristic
402C2×C4#1313318811yescharacteristic
403C2×C4#1313318911yescharacteristic
404C2×C4#1313318911yescharacteristic
405C2×C4#1313321311yescharacteristic
406C2×C4D8×D5312yes—
407C2×C4SD16×D5311yescharacteristic
408C2×C4#1313323811yescharacteristic
409C2×C4#1313324112yes—
410C2×C4#1313324411yescharacteristic
411C2×C4#1313324611yescharacteristic
412C8—42no—
413C8—4241no—
414C8—2122no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—84no—
419C22—84no—
420C22—4241no—
421C22—4241no—
422C22—4242no—
423C22—2122no—
424C22—2124no—
425C22#1681053111yescharacteristic
426C22#1681075512yes—
427C22#1681077211yescharacteristic
428C22#1681078211yescharacteristic
429C22#1681080012yes—
430C4—21no—
431C4—21no—
432C4—82no—
433C4—4241no—
434C4—2124no—
435C4#1681076011yescharacteristic
436C4#1681076711yescharacteristic
437C4#1681081612yes—
438C4#1681084911yescharacteristic
439C4#1681086611yescharacteristic
440C4#1681099811yescharacteristic
441C4#1681101111yescharacteristic
442C2—162no—
443C2—4242no—
444C2#2027389112yes—
445C2#2027390512yes—
446C2#2027571511yescharacteristic
447C2#2027574411yescharacteristic
448C2#2027578411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#486C2
socleC23×C53#120
prime radical#20279662C2
hypercenterC2×#486D53
hypocenterC53#10070
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. C2×#16810900C2semidirect
3. #20275639C2semidirect
4. C2×#16810367C2semidirect
5. C2×#16810352C2semidirect
6. #20272555C2semidirect
7. C2×C53×#486C2semidirect
51. C4⋊#88322D4semidirect
63. C4⋊Q8×C53D4semidirect
73. C4×C8×C53D4semidirect
218. C2×#486D53semidirect
253. C53#10070semidirect
267. C4⋊Q8C22⋊D53semidirect
297. C4×C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109002
#202796622
#202756392
C2×#168103672
C2×#168103522
#202725552
C2×C53×#4862
#1007053