Finite Group Databasebeta

← #27274648

Subgroups — #27274648

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810695C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic
3#20274494C211yesmaximal, characteristic, complemented
4#20274489C211yesmaximal, characteristic, complemented
5#20272561C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#1459C211yesmaximal, characteristic
8C22×#11719917C411yescharacteristic, complemented
9C23×#88322C2211yescharacteristic
10#16810695—21no—
11#16810695—21no—
12#16810695—21no—
13#16810695—21no—
14C2×C4⋊#88322C411yescharacteristic, complemented
15C2×#13133132C412yes—
16C2×C22⋊#88322C411yescharacteristic, complemented
17C2×C22⋊#88322C2211yescharacteristic
18C2×#13133128C411yescharacteristic, complemented
19C2×#13133127—21no—
20C2×C8×Dic53—21no—
21C4×C22⋊Dic53C2211yescharacteristic
22C4×C22⋊Dic53C2211yescharacteristic
23#16810390C412yes—
24#16810377—21no—
25#16810377—21no—
26C2×C4×D4×C53C2211yescharacteristic
27C2×C22⋊C8×C53C2211yescharacteristic
28C2×C22⋊C8×C53C2211yescharacteristic
29C53×#324C412yescomplemented
30C53×#323C412yescomplemented
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—42no—
34C22×#88322—42no—
35C22×#88322C2×C411yescharacteristic
36C22×#88322C2×C411yescharacteristic
37C22×#88322C2×C411yescharacteristic
38C22×#88322C2×C411yescharacteristic
39C22×#88322D411yescharacteristic
40C22×#88322D411yescharacteristic
41C22×#88322D411yescharacteristic
42C22×#88322D411yescharacteristic
43C2×C22⋊Dic53C2×C412yes—
44C2×C22⋊Dic53C2×C412yes—
45C2×C4×Dic53—21no—
46C2×C4×Dic53—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49#13133132—24no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54#13133128—21no—
55#13133128—21no—
56#13133127—42no—
57C8×Dic53—42no—
58#13133124—21no—
59#13133124—21no—
60C22×D4×C53C2×C411yescharacteristic
61C23×C4×C53C2×C411yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C22×C8×C53—21no—
64C22×C8×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C2×C4⋊C4×C53C2×C411yescharacteristic
70C2×C22⋊C4×C53C2×C411yescharacteristic
71C2×C22⋊C4×C53C2×C411yescharacteristic
72C2×C42×C53C2×C411yescharacteristic
73C22⋊C8×C53—21no—
74C22⋊C8×C53—21no—
75C22⋊C8×C53C2×C412yes—
76C22⋊C8×C53C2×C412yes—
77C24×C53C4211yescharacteristic
78C24×C53C22×C411yescharacteristic
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C4211yescharacteristic
90C22×C4×C53C4211yescharacteristic
91C22×C4×C53C4211yescharacteristic
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C22×C411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C22×Dic53—22no—
99C22×Dic53—22no—
100C2×C8×C53—22no—
101C2×C8×C53—22no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C22⋊C4×C53—21no—
111C22⋊C4×C53—21no—
112C42×C53—21no—
113C42×C53—21no—
114C22⋊Dic53—22no—
115C22⋊Dic53—22no—
116C22⋊Dic53—22no—
117C22⋊Dic53M1614yescomplemented
118C4×Dic53—41no—
119C4×Dic53—41no—
120C4×Dic53—41no—
121C4×Dic53—41no—
122#11719917—42no—
123#11719917—42no—
124C2×#88322—22no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322C22⋊C411yescharacteristic
135C2×#88322C22⋊C411yescharacteristic
136C2×#88322C22⋊C411yescharacteristic
137C2×#88322C22⋊C411yescharacteristic
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53C2×C4211yescharacteristic
146C23×C53C4×D411yescharacteristic
147C23×C53C4×D411yescharacteristic
148C23×C53C2×M1612yes—
149C23×C53#13112yes—
150D4×C53—42no—
151D4×C53—42no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53#11711yescharacteristic
173C2×C4×C53C4×D411yescharacteristic
174C2×C4×C53C4×D411yescharacteristic
175C2×Dic53—24no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C8×C53—42no—
182C8×C53—42no—
183#88322—21no—
184#88322—21no—
185#88322—21no—
186#88322—41no—
187#88322—41no—
188#88322#9912yescomplemented
189#8161—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—42no—
197C22×C53—42no—
198C22×C53—41no—
199C22×C53—41no—
200C22×C53—41no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53C4×C22⋊C411yescharacteristic
204C22×C53C4×M1611yescharacteristic
205C22×C53#40511yescharacteristic
206C22×C53#43512yes—
207C22×C53#43612yes—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53C2×#9911yescharacteristic
215C4×C53#41011yescharacteristic
216Dic53—44no—
217Dic53—82no—
218C2×#435—531no—
219C4×#99—531no—
220#1752—531no—
221C8×C22⋊C4—531no—
222#1459D5311yescharacteristic
223#1282—531no—
224#1282—531no—
225C2×C53—22no—
226C2×C53—22no—
227C2×C53—21no—
228C2×C53—21no—
229C2×C53—81no—
230C2×C53#175311yescharacteristic
231C2×C53C4×#9911yescharacteristic
232C2×C53#175711yescharacteristic
233C22×M16—531no—
234C23×C8—531no—
235C2×C4×D4C2×D5311yescharacteristic
236#435—1061no—
237#435—1061no—
238#435—1061no—
239#435—1061no—
240C2×C4⋊C8—531no—
241C2×#99—532no—
242C2×C22⋊C8—531no—
243C2×C22⋊C8—531no—
244C2×C22⋊C8C2×D5311yescharacteristic
245C2×C22⋊C8C2×D5311yescharacteristic
246C2×#97—531no—
247C2×#97—1061no—
248C2×C4×C8—1061no—
249C4×C22⋊C4—531no—
250C4×C22⋊C4—531no—
251#342—532no—
252#336—1061no—
253#336—1061no—
254#324Dic5312yescomplemented
255#323Dic5312yescomplemented
256C53#816111yescharacteristic, complemented, Sylow
257C22×D4C4×D5311yescharacteristic
258C23×C4C4×D5311yescharacteristic
259C23×C4C22×D5311yescharacteristic
260C2×M16—1061no—
261C2×M16—1061no—
262C2×M16—2122no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—531no—
266C22×C8—531no—
267C22×C8—531no—
268C22×C8—531no—
269C22×C8—531no—
270C22×C8—531no—
271C22×C8—531no—
272C22×C8—531no—
273C22×C8—2122no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C2×C4⋊C4C2×Dic5311yescharacteristic
279C2×C22⋊C4—532no—
280C2×C22⋊C4—532no—
281C2×C22⋊C4C4×D5311yescharacteristic
282C2×C22⋊C4C2×Dic5311yescharacteristic
283C2×C42—1061no—
284C2×C42—1061no—
285C2×C42C4×D5311yescharacteristic
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288#99—1064no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—1061no—
292C22⋊C8—1061no—
293C22⋊C8—1061no—
294C22⋊C8—1061no—
295C22⋊C8C2×Dic5312yes—
296C22⋊C8C2×Dic5312yes—
297#97—1061no—
298#97—1061no—
299#97—2122no—
300C4×C8—2122no—
301#95—1061no—
302#95—1061no—
303C24C4×Dic5311yescharacteristic
304C24C2×C4×D5311yescharacteristic
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—22no—
308C2×D4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—1062no—
316C22×C4—1062no—
317C22×C4C4×Dic5311yescharacteristic
318C22×C4C4×Dic5311yescharacteristic
319C22×C4C4×Dic5311yescharacteristic
320C22×C4C2×C4×D5311yescharacteristic
321C22×C4D4×D5311yescharacteristic
322C22×C4D4×D5311yescharacteristic
323C22×C4#1171994711yescharacteristic
324C22×C4#1171994711yescharacteristic
325C22×C4C22×Dic5311yescharacteristic
326M16—2122no—
327M16—2122no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—531no—
335C2×C8—531no—
336C2×C8—531no—
337C2×C8—531no—
338C2×C8—1062no—
339C2×C8—1062no—
340C2×C8—1062no—
341C2×C8—1061no—
342C2×C8—1061no—
343C2×C8—2121no—
344C2×C8—2121no—
345C2×C8—2121no—
346C2×C8—2121no—
347C2×C8—2121no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—534no—
355C22⋊C4—1062no—
356C22⋊C4—1062no—
357C22⋊C4—1062no—
358C42—21no—
359C42—21no—
360C42—2121no—
361C42—2121no—
362C42—2121no—
363C42—2121no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23C2×C4×Dic5311yescharacteristic
372C23#1313317711yescharacteristic
373C23#1313317711yescharacteristic
374C23M16×D5312yes—
375C23#1313322312yes—
376D4—42no—
377D4—42no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—1064no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2122no—
402C2×C4—2122no—
403C2×C4C22⋊C4×D5311yescharacteristic
404C2×C4#1313317311yescharacteristic
405C2×C4D4×Dic5311yescharacteristic
406C2×C4D4×Dic5311yescharacteristic
407C8—42no—
408C8—42no—
409C8—532no—
410C8—1061no—
411C8—1061no—
412C8—1061no—
413C8—2121no—
414C8—2121no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—81no—
427C22—81no—
428C22C22⋊C4×Dic5311yescharacteristic
429C22M16×Dic5311yescharacteristic
430C22#1681057811yescharacteristic
431C22#1681069712yes—
432C22#1681069812yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—4242no—
440C4—2124no—
441C4#99×D5311yescharacteristic
442C4#1681059511yescharacteristic
443C2—22no—
444C2—22no—
445C2—21no—
446C2—21no—
447C2—81no—
448C2#2027450111yescharacteristic
449C2#99×Dic5311yescharacteristic
450C2#2027451811yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1459C2
socleC22×C53C4×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#1459D53
hypocenterC53#8161
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20274494C2semidirect
4. #20274489C2semidirect
5. #20272561C2semidirect
6. #20272561C2semidirect
8. C22×#11719917C4semidirect
14. C2×C4⋊#88322C4semidirect
16. C2×C22⋊#88322C4semidirect
18. C2×#13133128C4semidirect
29. C53×#324C4semidirect
30. C53×#323C4semidirect
117. C22⋊Dic53M16semidirect
188. #88322#99semidirect
254. #324Dic53semidirect
255. #323Dic53semidirect
256. C53#8161semidirect

Maximal subgroups

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

subgroupindex
C2×#168106952
C4×#131331322
#202744942
#202744892
#202725612
C53×#14592
#816153