Finite Group Databasebeta

← #27274645

Subgroups — #27274645

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×#16810694C211yesmaximal, characteristic
2C4×#13133131C211yesmaximal, characteristic
3#20274491C211yesmaximal, characteristic, complemented
4#20273031C211yesmaximal, characteristic
5#20272563C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C8×C22⋊C4×C53C211yesmaximal, characteristic, complemented
8C22×#11719912C411yescharacteristic, complemented
9C2×C4×C22⋊D53C2211yescharacteristic
10#16810694—21no—
11#16810694—21no—
12#16810694—21no—
13#16810694—21no—
14C2×#13133146C411yescharacteristic, complemented
15C2×#13133131C412yes—
16C2×C22⋊#88322C2211yescharacteristic
17C2×C22⋊#88322C2211yescharacteristic
18C2×#13133129C411yescharacteristic, complemented
19C2×#13133127—21no—
20C2×#13133127C411yescharacteristic, complemented
21C4×C22⋊Dic53C2211yescharacteristic
22#16810389C412yes—
23#16810377—21no—
24#16810348C412yescomplemented
25#16810346C412yescomplemented
26C23×C8×C53C2211yescharacteristic
27C2×C22⋊C8×C53C2211yescharacteristic
28C2×C4×C8×C53—21no—
29C4×C22⋊C4×C53C2211yescharacteristic
30C53×#336—21no—
31C22×C22⋊D53C2×C411yescharacteristic
32C22×C4×D53C2×C411yescharacteristic
33C2×#11719912—21no—
34C2×#11719912—21no—
35C2×#11719912—42no—
36C22×#88322—21no—
37C22×#88322C2×C411yescharacteristic
38C22×#88322C2×C411yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53—21no—
41C4×C22⋊D53—21no—
42C4×C22⋊D53—21no—
43C2×#11719919C2×C411yescharacteristic
44C2×C22⋊Dic53C2×C412yes—
45C2×C22⋊Dic53C2×C411yescharacteristic
46C2×#11719921C2×C411yescharacteristic
47C2×C4×Dic53—21no—
48C2×C4×Dic53C2×C411yescharacteristic
49#13133146—21no—
50#13133146—21no—
51#13133131—24no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54C22⋊#88322—21no—
55C22⋊#88322C2×C412yes—
56#13133129—21no—
57#13133129—21no—
58#13133127—21no—
59#13133127—21no—
60#13133127—42no—
61#13133124—21no—
62C23×C4×C53C2311yescharacteristic
63C22×C8×C53—21no—
64C22×C8×C53—42no—
65C22×C8×C53C2×C411yescharacteristic
66C22×C8×C53C2×C411yescharacteristic
67C22×C8×C53D411yescharacteristic
68C22×C8×C53D411yescharacteristic
69C22×C8×C53D411yescharacteristic
70C22×C8×C53D411yescharacteristic
71C2×C22⋊C4×C53C2×C412yes—
72C2×C42×C53—21no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53C2×C412yes—
75C4×C8×C53—42no—
76#95×C53—21no—
77C24×C53C22×C411yescharacteristic
78C23×D53C4211yescharacteristic
79C22×C4×C53—22no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C2×C22⋊D53—21no—
89C2×C22⋊D53—21no—
90C2×C22⋊D53—22no—
91C2×C22⋊D53—22no—
92C22×Dic53—22no—
93C22×Dic53C4211yescharacteristic
94C22×Dic53C4211yescharacteristic
95C22×Dic53C4211yescharacteristic
96C2×C4×D53—21no—
97C2×C4×D53—21no—
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C8×C53—22no—
101C2×C8×C53—22no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53C22⋊C411yescharacteristic
110C2×C8×C53C22⋊C411yescharacteristic
111C2×C8×C53C22⋊C411yescharacteristic
112C2×C8×C53C22⋊C411yescharacteristic
113C22⋊C4×C53—22no—
114C22⋊C4×C53M1614yescomplemented
115C42×C53—41no—
116C42×C53—41no—
117C22⋊Dic53—21no—
118C22⋊Dic53—21no—
119C22⋊Dic53—22no—
120C22⋊Dic53—22no—
121#11719921—21no—
122#11719921—21no—
123#11719919—21no—
124#11719919—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—41no—
128C4×Dic53—41no—
129C2×#88322—21no—
130C2×#88322—21no—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136#11719912—42no—
137#11719912—42no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53C2×C4211yescharacteristic
141C23×C53C4×D411yescharacteristic
142C23×C53C4×D411yescharacteristic
143C23×C53C2×M1612yes—
144C23×C53#13112yes—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—41no—
149C22×D53—41no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—24no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53C2×C22⋊C411yescharacteristic
163C2×C4×C53#11711yescharacteristic
164C2×C4×C53C4×D411yescharacteristic
165C2×C4×C53C4×D411yescharacteristic
166C22⋊D53—42no—
167C22⋊D53—42no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C4×D53—42no—
179C4×D53—41no—
180C4×D53—41no—
181C8×C53—21no—
182C8×C53—21no—
183C8×C53—21no—
184C8×C53—42no—
185C8×C53#9912yescomplemented
186#88322—42no—
187#88322—41no—
188#88322—41no—
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×C53C4×C22⋊C411yescharacteristic
199C22×C53C4×M1611yescharacteristic
200C22×C53#40511yescharacteristic
201C22×C53#43512yes—
202C22×C53#43612yes—
203C2×D53—41no—
204C2×D53—41no—
205C2×D53—41no—
206C2×D53—81no—
207C2×D53—81no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—44no—
212C4×C53C2×#9911yescharacteristic
213C4×C53#41011yescharacteristic
214Dic53—41no—
215Dic53—41no—
216Dic53—41no—
217Dic53—82no—
218C2×#435—531no—
219C4×#99—531no—
220#1752—531no—
221C8×C22⋊C4D5311yescharacteristic, complemented
222#1459—531no—
223#1282—531no—
224#1282—531no—
225C2×C53—22no—
226C2×C53—22no—
227C2×C53—21no—
228C2×C53—21no—
229C2×C53#175311yescharacteristic
230C2×C53C4×#9911yescharacteristic
231C2×C53#175711yescharacteristic
232D53—81no—
233C22×M16—531no—
234C23×C8C2×D5311yescharacteristic
235C2×C4×D4—531no—
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⋊C8—531no—
245C2×C22⋊C8C2×D5311yescharacteristic
246C2×#97—531no—
247C2×#97—1061no—
248C2×C4×C8—21no—
249C4×C22⋊C4—531no—
250C4×C22⋊C4C2×D5311yescharacteristic
251#342—532no—
252#336—21no—
253#336—1061no—
254#324—532no—
255#323—532no—
256C53#816111yescharacteristic, complemented, Sylow
257C22×D4—531no—
258C23×C4—531no—
259C23×C4C22×D5311yescharacteristic
260C2×M16—1061no—
261C2×M16—1061no—
262C2×M16—2122no—
263C22×C8—21no—
264C22×C8—42no—
265C22×C8—531no—
266C22×C8—531no—
267C22×C8—1061no—
268C22×C8C4×D5311yescharacteristic
269C22×C8C4×D5311yescharacteristic
270C22×C8D21211yescharacteristic
271C22×C8D21211yescharacteristic
272C22×C8C22⋊D5311yescharacteristic
273C22×C8C22⋊D5311yescharacteristic
274C4×D4—1061no—
275C4×D4—1061no—
276C4×D4—1061no—
277C4×D4—1061no—
278C2×C4⋊C4—531no—
279C2×C22⋊C4—531no—
280C2×C22⋊C4—531no—
281C2×C22⋊C4—532no—
282C2×C22⋊C4C4×D5312yes—
283C2×C42—21no—
284C2×C42—531no—
285C2×C42—1061no—
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288#99—1064no—
289C22⋊C8—21no—
290C22⋊C8—532no—
291C22⋊C8—1061no—
292C22⋊C8—1061no—
293C22⋊C8—1061no—
294C22⋊C8—1061no—
295C22⋊C8—1061no—
296C22⋊C8C4×D5312yes—
297#97—1061no—
298#97—1061no—
299#97—2122no—
300C4×C8—42no—
301#95—21no—
302#95—1061no—
303C24—531no—
304C24C2×C4×D5311yescharacteristic
305C2×D4—1061no—
306C2×D4—1061no—
307C2×D4—1062no—
308C2×D4—1062no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—531no—
313C22×C4—531no—
314C22×C4—531no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4—1062no—
318C22×C4—2121no—
319C22×C4—2121no—
320C22×C4C2×C4×D5311yescharacteristic
321C22×C4C2×C4×D5311yescharacteristic
322C22×C4C2×D21211yescharacteristic
323C22×C4#1171994511yescharacteristic
324C22×C4#1171994511yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326M16—2122no—
327M16—2122no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—1061no—
338C2×C8—1061no—
339C2×C8—1062no—
340C2×C8—2121no—
341C2×C8—2121no—
342C2×C8—2121no—
343C2×C8—2121no—
344C2×C8#1171992111yescharacteristic
345C2×C8#1171992111yescharacteristic
346C2×C8#1171992111yescharacteristic
347C2×C8#1171992111yescharacteristic
348C4⋊C4—1061no—
349C4⋊C4—1061no—
350C22⋊C4—22no—
351C22⋊C4—1061no—
352C22⋊C4—1061no—
353C22⋊C4—1061no—
354C22⋊C4—1061no—
355C22⋊C4—1062no—
356C22⋊C4—1062no—
357C22⋊C4#1171991214yescomplemented
358C42—41no—
359C42—41no—
360C42—1061no—
361C42—1061no—
362C42—2121no—
363C42—2121no—
364C23—41no—
365C23—41no—
366C23—1061no—
367C23—1061no—
368C23—1061no—
369C23—2121no—
370C23—2121no—
371C23C42×D5311yescharacteristic
372C23C4×C22⋊D5311yescharacteristic
373C23C4×D21211yescharacteristic
374C23C2×#1171991212yes—
375C23#1313322312yes—
376D4—2122no—
377D4—2122no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—2122no—
396C2×C4—2122no—
397C2×C4—2122no—
398C2×C4—2122no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4C2×#1171992111yescharacteristic
404C2×C4#1313317411yescharacteristic
405C2×C4C4×C22⋊D5311yescharacteristic
406C2×C4C4×D21211yescharacteristic
407C8—21no—
408C8—21no—
409C8—21no—
410C8—42no—
411C8—2122no—
412C8—2121no—
413C8—2121no—
414C8#1313313112yescomplemented
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22—4241no—
424C22—4241no—
425C22—2121no—
426C22—2121no—
427C22—2121no—
428C22C4×#1171992111yescharacteristic
429C22C4×#1171991211yescharacteristic
430C22#1681057711yescharacteristic
431C22#1681069412yes—
432C22#1681070212yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—4242no—
438C4—2121no—
439C4—2121no—
440C4—2121no—
441C4C2×#1313313111yescharacteristic
442C4#1681059611yescharacteristic
443C2—22no—
444C2—22no—
445C2—21no—
446C2—21no—
447C2—4241no—
448C2#2027449811yescharacteristic
449C2C4×#1313313111yescharacteristic
450C2#2027451611yescharacteristic
subgroupG/H
centerC2×C4C2×#11719921
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC8×C22⋊C4×C53C2
socleC22×C53C4×C22⋊C4
prime radicalC22×C22⋊D53C2×C4
cosocleC23×C4×C53C23
hypercenterC8×C22⋊C4D53
hypocenterC53#8161
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20274491C2semidirect
5. #20272563C2semidirect
6. #20272561C2semidirect
7. C8×C22⋊C4×C53C2semidirect
8. C22×#11719912C4semidirect
14. C2×#13133146C4semidirect
18. C2×#13133129C4semidirect
20. C2×#13133127C4semidirect
24. #16810348C4semidirect
25. #16810346C4semidirect
114. C22⋊C4×C53M16semidirect
185. C8×C53#99semidirect
221. C8×C22⋊C4D53semidirect
256. C53#8161semidirect
357. C22⋊C4#11719912semidirect
414. C8#13133131semidirect

Maximal subgroups

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

subgroupindex
C2×#168106942
C4×#131331312
#202744912
#202730312
#202725632
#202725612
C8×C22⋊C4×C532
#816153