Finite Group Databasebeta

← #27279372

Subgroups — #27279372

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic
2#20277775C211yesmaximal, characteristic, complemented
3#20275641C211yesmaximal, characteristic
4#20275641C211yesmaximal, characteristic
5#20272554C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×Q8×Dic53—21no—
10#16810971—22no—
11#16810971C414yescomplemented
12C2×#13133139—21no—
13C2×#13133139—21no—
14C2×#13133139—21no—
15C2×#13133139—21no—
16C2×#13133137—22no—
17C2×C4×#88322C2211yescharacteristic
18#16810530—21no—
19#16810522—21no—
20#16810509—22no—
21C4⋊C4×Dic53—21no—
22C4×C4⋊Dic53C2211yescharacteristic
23C4×C4⋊Dic53C2211yescharacteristic
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C53×#332—22no—
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×#88322—21no—
32#13133214—21no—
33#13133214—21no—
34#13133214—21no—
35#13133214C2×C412yes—
36Q8×Dic53—42no—
37Q8×Dic53—42no—
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×C4⋊Dic53D411yescharacteristic
41C2×C4⋊Dic53D411yescharacteristic
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4×Dic53—22no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46#13133139—42no—
47#13133139—42no—
48#13133139—42no—
49#13133139—42no—
50#13133137—28no—
51C4×#88322—21no—
52C4×#88322C2×C412yes—
53#13133124—22no—
54#13133124—22no—
55#13133124—21no—
56#13133124—21no—
57C22×Q8×C53—21no—
58C22×C8×C53—21no—
59C22×C8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53C2×C412yes—
63C4⋊Q8×C53D412yes—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D412yes—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C23×D53—41no—
72C2×Q8×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C22×C4×C53—22no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C2×D212—22no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—42no—
90C2×D212—42no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53C22⋊C414yes—
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C42×C53#4111yescharacteristic
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53—44no—
123C4⋊Dic53D812yes—
124C4⋊Dic53SD1612yes—
125C4×Dic53—44no—
126C4×Dic53—41no—
127C4×Dic53—41no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130C2×#88322—22no—
131C2×#88322—41no—
132C2×#88322—41no—
133C23×C53#12011yescharacteristic
134C22×D53—42no—
135C22×D53—81no—
136C22×D53—81no—
137C22×D53—81no—
138C22×D53—81no—
139C22×D53—81no—
140C22×D53—81no—
141Q8×C53—42no—
142Q8×C53—42no—
143Q8×C53—42no—
144Q8×C53—42no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—24no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53C2×C22⋊C411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12311yescharacteristic
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53C2×SD1611yescharacteristic
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53#13711yescharacteristic
164C2×Dic53—22no—
165C2×Dic53—22no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176D212—44no—
177D212—42no—
178D212—42no—
179D212—42no—
180D212—42no—
181D212—82no—
182D212—81no—
183D212—81no—
184D212—81no—
185D212—81no—
186C8×C53—81no—
187C8×C53—81no—
188#88322—42no—
189#9062—531nomaximal, Sylow
190C22×C53#38611yescharacteristic
191C22×C53#44711yescharacteristic
192C22×C53#44811yescharacteristic
193C22×C53#44911yescharacteristic
194C22×C53#44911yescharacteristic
195C22×C53#45411yescharacteristic
196C22×C53#45411yescharacteristic
197C2×D53—82no—
198C2×D53—82no—
199C2×D53—82no—
200C2×D53—161no—
201C2×D53—161no—
202C2×D53—161no—
203C2×D53—161no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—44no—
207C4×C53C4×D811yescharacteristic
208C4×C53C4×SD1611yescharacteristic
209C4×C53#44011yescharacteristic
210C4×C53#44211yescharacteristic
211C4×C53#45911yescharacteristic
212C4×C53#46011yescharacteristic
213C4×C53#46311yescharacteristic
214C4×C53#46411yescharacteristic
215Dic53—42no—
216Dic53—42no—
217Dic53—81no—
218Dic53—81no—
219C2×#486—531no—
220#2365—531no—
221#1982—531no—
222#1982—531no—
223C2×#332D5311yescharacteristic, complemented
224#1279—531no—
225#1279—531no—
226C2×C53#162511yescharacteristic
227C2×C53#162611yescharacteristic
228C2×C53#162711yescharacteristic
229C2×C53#162811yescharacteristic
230C2×C53#187911yescharacteristic
231C2×C53#188011yescharacteristic
232C2×C53#188811yescharacteristic
233D53—162no—
234C2×C4⋊Q8C2×D5311yescharacteristic
235C2×#127—531no—
236C2×C4×Q8—1061no—
237#486—534no—
238#486—1062no—
239C2×C4⋊C8C2×D5311yescharacteristic
240C2×C4⋊C8C2×D5311yescharacteristic
241C2×#102—1062no—
242C2×#102—1061no—
243C2×#102—1061no—
244C2×#102—1061no—
245C2×#102—1061no—
246C2×C4×C8—531no—
247#391—1061no—
248#389—1061no—
249#387—1062no—
250C4×C4⋊C4—531no—
251C4×C4⋊C4—531no—
252C4×C4⋊C4—1061no—
253#332—22no—
254C53#906211yescharacteristic, complemented, Sylow
255C22×Q8—21no—
256C22×D4—1061no—
257C22×D4—1061no—
258C22×D4—1061no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—1061no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8C4×D5312yes—
265C4⋊Q8C22⋊D5312yes—
266#127—532no—
267#127—1061no—
268#127—1061no—
269#127—1061no—
270C4×Q8—2122no—
271C4×Q8—2122no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—531no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—1062no—
278C2×C4⋊C4D21212yes—
279C2×C42—1062no—
280C2×C42—1061no—
281C2×C42—1061no—
282C2×C42C22×D5311yescharacteristic
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287#102—1068no—
288#102—2122no—
289#102—2122no—
290#102—2122no—
291#102—2122no—
292C4×C8—532no—
293C4×C8—1061no—
294#95—1062no—
295#95—1062no—
296#95—1061no—
297#95—1061no—
298C24—2121no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—1062no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—2122no—
309C2×D4—2122no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C22×C4—22no—
322C22×C4—1061no—
323C22×C4—1061no—
324C22×C4—1061no—
325C22×C4—1061no—
326C22×C4C2×D21211yescharacteristic
327C22×C4D4×D5311yescharacteristic
328C22×C4D4×D5311yescharacteristic
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—1062no—
336C2×C8—2121no—
337C2×C8—2121no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—532no—
342C4⋊C4—532no—
343C4⋊C4—1061no—
344C4⋊C4—1061no—
345C4⋊C4—1061no—
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C4⋊C4—1061no—
349C4⋊C4—2124no—
350C4⋊C4#1171992114yes—
351C42—2124no—
352C42—2121no—
353C42—2121no—
354C42—2121no—
355C42—2121no—
356C42C2×C4×D5311yescharacteristic
357C42#1171994711yescharacteristic
358C42#1171994711yescharacteristic
359C42C2×C22⋊D5311yescharacteristic
360C23—4241no—
361C23—4241no—
362C23—4241no—
363C23—4241no—
364C23—4241no—
365C23—4241no—
366C23—2122no—
367C23#1313318711yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372D4—4242no—
373D4—4241no—
374D4—4241no—
375D4—4241no—
376D4—4241no—
377D4—2124no—
378D4—2122no—
379D4—2122no—
380D4—2122no—
381D4—2122no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—1062no—
388C2×C4—1062no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4C2×#1171992111yescharacteristic
400C2×C4#1313317711yescharacteristic
401C2×C4#1313317711yescharacteristic
402C2×C4#1313318911yescharacteristic
403C2×C4#1313318911yescharacteristic
404C2×C4#1313320111yescharacteristic
405C2×C4D8×D5311yescharacteristic
406C2×C4SD16×D5311yescharacteristic
407C2×C4#1313323811yescharacteristic
408C2×C4#1313324011yescharacteristic
409C2×C4#1313324111yescharacteristic
410C2×C4#1313324411yescharacteristic
411C2×C4#1313324411yescharacteristic
412C2×C4#1313324911yescharacteristic
413C8—81no—
414C8—81no—
415C8—2122no—
416C22—8481no—
417C22—8481no—
418C22—8481no—
419C22—8481no—
420C22—4242no—
421C22—4242no—
422C22—4242no—
423C22#1681050011yescharacteristic
424C22#1681075811yescharacteristic
425C22#1681076311yescharacteristic
426C22#1681076911yescharacteristic
427C22#1681077111yescharacteristic
428C22#1681079811yescharacteristic
429C22#1681079811yescharacteristic
430C4—21no—
431C4—21no—
432C4—44no—
433C4—4241no—
434C4—4241no—
435C4—2122no—
436C4—2122no—
437C4#1681070311yescharacteristic
438C4#1681070811yescharacteristic
439C4#1681072611yescharacteristic
440C4#1681073211yescharacteristic
441C4#1681081611yescharacteristic
442C4#1681082111yescharacteristic
443C4#1681084011yescharacteristic
444C4#1681084711yescharacteristic
445C2—8482no—
446C2#2027380511yescharacteristic
447C2#2027381211yescharacteristic
448C2#2027381911yescharacteristic
449C2#2027382611yescharacteristic
450C2#2027506111yescharacteristic
451C2#2027506811yescharacteristic
452C2#2027511611yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC23×C53#120
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenterC2×#332D53
hypocenterC53#9062
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #20277775C2semidirect
5. #20272554C2semidirect
6. #20272554C2semidirect
7. C2×C53×#332C2semidirect
11. #16810971C4semidirect
223. C2×#332D53semidirect
254. C53#9062semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
#202777752
#202756412
#202725542
C2×C53×#3322
#906253