Finite Group Databasebeta

← #27274977

Subgroups — #27274977

Order 13568.

712 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
1C4×#13133191C211yesmaximal, characteristic
2#20274477C211yesmaximal, characteristic, complemented
3C2×#16810356C212yesmaximal
4#20272594C212yesmaximal, complemented
5C53×#1434C211yesmaximal, characteristic, complemented
6C2×#13133191C2211yescharacteristic
7C2×C4×D212C411yescharacteristic, complemented
8C2×C4×C22⋊D53—21no—
9C2×C22⋊#88322C2212yes—
10C2×#13133128—21no—
11#16810525C411yescharacteristic, complemented
12#16810521C411yescharacteristic, complemented
13#16810514—21no—
14#16810502—21no—
15#16810483C2211yescharacteristic
16C4×C4⋊Dic53C411yescharacteristic, complemented
17C4×#11719921—21no—
18#16810378—21no—
19#16810356C418yescomplemented
20C22×C42×C53C2211yescharacteristic
21C2×#95×C53—21no—
22C2×#95×C53—21no—
23C2×#95×C53C2212yes—
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C22×C4×D53—21no—
27C22×#88322—22no—
28#13133191—22no—
29#13133191C2×C414yes—
30C4×D212—42no—
31C4×C22⋊D53—44no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×C22⋊Dic53—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×C4×Dic53—21no—
42C22⋊#88322—22no—
43C22⋊#88322C2×C414yescomplemented
44#13133128—42no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C23×C4×C53—22no—
49C23×C4×C53D411yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53C2311yescharacteristic
52C2×C42×C53—42no—
53C2×C42×C53C2×C411yescharacteristic
54C2×C42×C53C2×C411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53D411yescharacteristic
57#95×C53—22no—
58#95×C53—42no—
59#95×C53—42no—
60#95×C53C2×C414yescomplemented
61C24×C53C2×D411yescharacteristic
62C23×D53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—24no—
68C22×C4×C53—41no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C22×C412yes—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4112yes—
87C2×C22⋊D53—24no—
88C2×C22⋊D53—42no—
89C2×C22⋊D53—41no—
90C2×C22⋊D53—41no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C2×D212—21no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212C4212yes—
98C2×C4×D53—42no—
99C2×C4×D53—42no—
100C2×C4×D53—42no—
101C42×C53—22no—
102C42×C53—22no—
103C42×C53—41no—
104C42×C53—41no—
105C22⋊Dic53—42no—
106#11719921—24no—
107#11719921—42no—
108#11719921—42no—
109#11719921—42no—
110C4⋊Dic53—21no—
111C4⋊Dic53—41no—
112C4⋊Dic53—41no—
113C4⋊Dic53C4212yes—
114#11719919—42no—
115C4×Dic53—42no—
116C2×#88322—24no—
117C2×#88322—42no—
118C2×#88322—42no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53C2×C22⋊C411yescharacteristic
124C23×C53C4×D412yes—
125C23×C53AGL(1,ℤ/8)12yes—
126C23×C53AGL(1,ℤ/8)11yescharacteristic
127C23×C53#13711yescharacteristic
128C22×D53—22no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—44no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—44no—
156C2×C4×C53—44no—
157C2×C4×C53—44no—
158C2×C4×C53—82no—
159C2×C4×C53—82no—
160C2×C4×C53—82no—
161C2×C4×C53—82no—
162C2×C4×C53#9912yes—
163C2×C4×C53C2×C4211yescharacteristic
164C2×C4×C53#11711yescharacteristic
165C2×C4×C53C4×D412yes—
166C22⋊D53—44no—
167C22⋊D53—44no—
168C2×Dic53—22no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177D212—42no—
178D212—42no—
179C4×D53—84no—
180#88322—44no—
181#8216—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—22no—
190C22×C53C4×C22⋊C411yescharacteristic
191C22×C53C2×#9911yescharacteristic
192C22×C53#41011yescharacteristic
193C22×C53#41711yescharacteristic
194C22×C53#41811yescharacteristic
195C22×C53#42111yescharacteristic
196C22×C53#42111yescharacteristic
197C22×C53#44012yes—
198C22×C53#44212yes—
199C2×D53—42no—
200C2×D53—42no—
201C2×D53—42no—
202C2×D53—81no—
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—81no—
206C4×C53—22no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—22no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53—84no—
214Dic53—42no—
215Dic53—81no—
216Dic53—81no—
217C4×#121—531no—
218#1749—531no—
219C2×#327—532no—
220#1434D5311yescharacteristic, complemented
221#1294—532no—
222C2×C53—22no—
223C2×C53—22no—
224C2×C53#151011yescharacteristic
225C2×C53#151211yescharacteristic
226C2×C53#151311yescharacteristic
227C2×C53#151411yescharacteristic
228C2×C53C4×#9911yescharacteristic
229C2×C53#175911yescharacteristic
230C2×C53#176211yescharacteristic
231D53—82no—
232C2×#121—531no—
233C2×C4×D4—531no—
234C2×C4×D4—1061no—
235C22×C42C2×D5311yescharacteristic
236C2×C22⋊C8—532no—
237C2×#97—1061no—
238#390—531no—
239#389—531no—
240#388—1061no—
241#386—1061no—
242#380—531no—
243C4×C4⋊C4—531no—
244C4×C22⋊C4—1061no—
245C2×#95—21no—
246C2×#95—21no—
247C2×#95C2×D5312yes—
248#336—1061no—
249#327—538no—
250C53#821611yescharacteristic, complemented, Sylow
251C22×D4—531no—
252C22×D4—1061no—
253C23×C4—22no—
254C23×C4—1061no—
255C23×C4C22⋊D5311yescharacteristic
256C23×C4C22⋊D5311yescharacteristic
257C23×C4C22×D5311yescharacteristic
258C22×C8—1062no—
259#121—534no—
260#121—1062no—
261C4×D4—2122no—
262C4×D4—2124no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—1061no—
267C2×C22⋊C4—1061no—
268C2×C22⋊C4—1061no—
269C2×C22⋊C4—1061no—
270C2×C22⋊C4—1061no—
271C2×C22⋊C4—1061no—
272C2×C42—42no—
273C2×C42—1061no—
274C2×C42C4×D5311yescharacteristic
275C2×C42C4×D5311yescharacteristic
276C2×C42D21211yescharacteristic
277C2×C42D21211yescharacteristic
278C22⋊C8—534no—
279C22⋊C8—1062no—
280#97—2122no—
281#95—22no—
282#95—42no—
283#95—42no—
284#95—1061no—
285#95—1061no—
286#95—1061no—
287#95C4×D5314yescomplemented
288C24—1061no—
289C24C2×C22⋊D5311yescharacteristic
290C2×D4—532no—
291C2×D4—1064no—
292C2×D4—1061no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2122no—
296C2×D4—2121no—
297C2×D4—2121no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4—2122no—
318C22×C4—2122no—
319C22×C4—2122no—
320C22×C4#1171992111yescharacteristic
321C22×C4#1171992111yescharacteristic
322C22×C4#1171992111yescharacteristic
323C22×C4#1171992111yescharacteristic
324C22×C4C2×C4×D5311yescharacteristic
325C22×C4C2×C4×D5312yes—
326C22×C4C2×D21211yescharacteristic
327C22×C4#1171994512yes—
328C2×C8—1064no—
329C2×C8—2122no—
330C2×C8—2122no—
331C4⋊C4—532no—
332C4⋊C4—1061no—
333C4⋊C4—2121no—
334C4⋊C4—2121no—
335C4⋊C4—2122no—
336C22⋊C4—1064no—
337C22⋊C4—2122no—
338C22⋊C4—2122no—
339C22⋊C4—2122no—
340C22⋊C4—2122no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—41no—
345C42—2122no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—1062no—
351C23—2121no—
352C23—2121no—
353C23—2121no—
354C23—2121no—
355C23—2121no—
356C23—2121no—
357C23C2×#1171992111yescharacteristic
358C23C4×C22⋊D5312yes—
359C23#1313324212yes—
360C23#1313324311yescharacteristic
361C23#1313324811yescharacteristic
362D4—2122no—
363D4—2122no—
364D4—2124no—
365D4—2124no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—1062no—
394C2×C4—4244no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4#1313313112yes—
404C2×C4C42×D5311yescharacteristic
405C2×C4#1313317411yescharacteristic
406C2×C4C4×D21212yes—
407C8—2124no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—4241no—
417C22—4241no—
418C22—4241no—
419C22—4241no—
420C22—2122no—
421C22—2122no—
422C22—2122no—
423C22C4×#1171992111yescharacteristic
424C22C2×#1313313111yescharacteristic
425C22#1681059611yescharacteristic
426C22#1681062211yescharacteristic
427C22#1681062711yescharacteristic
428C22#1681064111yescharacteristic
429C22#1681064111yescharacteristic
430C22#1681072112yes—
431C22#1681073312yes—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—84no—
440C4—4241no—
441C4—4241no—
442C4—2122no—
443C2—22no—
444C2—22no—
445C2—4242no—
446C2#2027324311yescharacteristic
447C2#2027325111yescharacteristic
448C2#2027325611yescharacteristic
449C2#2027326111yescharacteristic
450C2C4×#1313313111yescharacteristic
451C2#2027452611yescharacteristic
452C2#2027454311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1434C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1434D53
hypocenterC53#8216
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #20274477C2semidirect
4. #20272594C2semidirect
5. C53×#1434C2semidirect
7. C2×C4×D212C4semidirect
11. #16810525C4semidirect
12. #16810521C4semidirect
16. C4×C4⋊Dic53C4semidirect
19. #16810356C4semidirect
43. C22⋊#88322C2×C4semidirect
60. #95×C53C2×C4semidirect
220. #1434D53semidirect
250. C53#8216semidirect
287. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
#202744772
C2×#168103562
#202725942
C53×#14342
#821653