Finite Group Databasebeta

← #27287890

Subgroups — #27287890

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810818C211yesmaximal, characteristic
2#20275521C211yesmaximal, characteristic, complemented
3#20274715C211yesmaximal, characteristic, complemented
4C2×#16810374C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1954C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133214C2211yescharacteristic
10#16810818—22no—
11#16810818—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133139—21no—
15C2×#13133139—21no—
16C2×#13133138—21no—
17C2×#13133137—21no—
18#16810493C2211yescharacteristic
19#16810385—21no—
20#16810374—22no—
21#16810366—22no—
22C2×C4×D4×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C22⋊C8×C53—21no—
25C2×C22⋊C8×C53—21no—
26C2×C4×C8×C53C2211yescharacteristic
27C22×D212—21no—
28C22×D212—21no—
29C22×D212—21no—
30C2×#11719922—24no—
31C2×#11719922—42no—
32C2×#11719922—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133214—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214D412yes—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133139—42no—
47#13133139—42no—
48#13133138—42no—
49#13133137—24no—
50C22×D4×C53D411yescharacteristic
51C23×C4×C53—21no—
52C22×C8×C53—21no—
53C22×C8×C53D411yescharacteristic
54C22×C8×C53D411yescharacteristic
55C4×D4×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C2×C4⋊C4×C53D411yescharacteristic
60C2×C22⋊C4×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C22⋊C8×C53—42no—
65C22⋊C8×C53—42no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C24×C53—21no—
69C23×D53—41no—
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53SD1612yes—
76C22×C4×C53—21no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53#4111yescharacteristic
86C22×Dic53—41no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—41no—
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—81no—
102C2×D212—81no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C2×C8×C53D812yescomplemented
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119C42×C53#4111yescharacteristic
120#11719922—48no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C4⋊Dic53—81no—
126C2×#88322—22no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C23×C53—22no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53#12111yescharacteristic
140C22×D53—42no—
141C22×D53—81no—
142C22×D53—81no—
143C22×D53—81no—
144C22×D53—81no—
145C22×D53—81no—
146C22×D53—81no—
147D4×C53—22no—
148D4×C53—22no—
149D4×C53—41no—
150D4×C53—41no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—81no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12311yescharacteristic
164C2×C4×C53#12311yescharacteristic
165C2×C4×C53#12411yescharacteristic
166C2×C4×C53C2×D811yescharacteristic
167C2×C4×C53C2×SD1611yescharacteristic
168C2×C4×C53#13511yescharacteristic
169C2×C4×C53#13511yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
174C2×Dic53—81no—
175C2×Dic53—81no—
176C2×Dic53—81no—
177D212—42no—
178D212—42no—
179D212—42no—
180D212—42no—
181D212—82no—
182D212—82no—
183D212—81no—
184D212—81no—
185D212—81no—
186D212—81no—
187C8×C53—22no—
188C8×C53—41no—
189C8×C53—81no—
190#88322—42no—
191#88322—81no—
192#10473—531nomaximal, Sylow
193C22×C53—42no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53#39411yescharacteristic
201C22×C53#45911yescharacteristic
202C22×C53#46311yescharacteristic
203C22×C53#46511yescharacteristic
204C22×C53#46911yescharacteristic
205C22×C53#47111yescharacteristic
206C22×C53#47211yescharacteristic
207C2×D53—82no—
208C2×D53—82no—
209C2×D53—82no—
210C2×D53—161no—
211C2×D53—161no—
212C2×D53—161no—
213C2×D53—161no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—81no—
217C4×C53#44911yescharacteristic
218C4×C53#45011yescharacteristic
219C4×C53#45911yescharacteristic
220C4×C53#46311yescharacteristic
221C4×C53#46611yescharacteristic
222C4×C53#46911yescharacteristic
223C4×C53#48111yescharacteristic
224C4×C53#48211yescharacteristic
225Dic53—161no—
226C2×#459—531no—
227#1957—531no—
228#1954D5311yescharacteristic, complemented
229#1802—531no—
230C2×#334—531no—
231C2×#331—531no—
232#1296—531no—
233C2×C53—82no—
234C2×C53#166511yescharacteristic
235C2×C53#167711yescharacteristic
236C2×C53#168711yescharacteristic
237C2×C53#170111yescharacteristic
238C2×C53#203411yescharacteristic
239C2×C53#205111yescharacteristic
240C2×C53#205411yescharacteristic
241D53—162no—
242C22×D8—1061no—
243C2×#127—531no—
244C2×C4×D4C2×D5311yescharacteristic
245#459—1062no—
246#459—1062no—
247C2×C4⋊C8—531no—
248C2×C4⋊C8—531no—
249C2×C4⋊C8C2×D5311yescharacteristic
250C2×#102—1061no—
251C2×#102—1061no—
252C2×#102—1061no—
253C2×#102—1061no—
254C2×C22⋊C8—21no—
255C2×C22⋊C8—21no—
256C2×C4×C8C2×D5311yescharacteristic
257#384—531no—
258#340—1061no—
259#334—1062no—
260#331—1062no—
261C53#1047311yescharacteristic, complemented, Sylow
262C22×D4—1061no—
263C22×D4—1061no—
264C22×D4—1061no—
265C22×D4D21211yescharacteristic
266C23×C4—21no—
267C2×D8—1064no—
268C2×D8—2122no—
269C2×D8—2122no—
270C22×C8—21no—
271C22×C8—1061no—
272C22×C8—1061no—
273C22×C8C22⋊D5311yescharacteristic
274C22×C8C22⋊D5311yescharacteristic
275#127—532no—
276#127—1061no—
277#127—1061no—
278#127—1061no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4—1061no—
285C2×C4⋊C4—1061no—
286C2×C4⋊C4D21211yescharacteristic
287C2×C22⋊C4—21no—
288C2×C42C22×D5311yescharacteristic
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—532no—
292C4⋊C8—1061no—
293C4⋊C8—1061no—
294C4⋊C8—1061no—
295#102—1064no—
296#102—2122no—
297#102—2122no—
298#102—2122no—
299C22⋊C8—42no—
300C22⋊C8—42no—
301C4×C8—21no—
302C4×C8—21no—
303C24—21no—
304C24—2121no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—1062no—
311C2×D4—1062no—
312C2×D4—1062no—
313C2×D4—4241no—
314C2×D4—4241no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4#1171991312yes—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—2121no—
335C22×C4C2×D21211yescharacteristic
336C22×C4#1171994511yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338D8—2128no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—1062no—
348C2×C8—2121no—
349C2×C8—2121no—
350C2×C8—2121no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8#1171992212yescomplemented
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—4241no—
357C4⋊C4—2121no—
358C4⋊C4—2121no—
359C4⋊C4—2121no—
360C4⋊C4—2121no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C42D4×D5311yescharacteristic
364C42D4×D5311yescharacteristic
365C42#1171994711yescharacteristic
366C42#1171994711yescharacteristic
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—4241no—
375C23—4241no—
376C23—4241no—
377C23—4241no—
378C23—4241no—
379C23—4241no—
380C23—2122no—
381C23#1313319111yescharacteristic
382D4—22no—
383D4—22no—
384D4—41no—
385D4—41no—
386D4—4242no—
387D4—4242no—
388D4—4241no—
389D4—4241no—
390D4—4241no—
391D4—4241no—
392D4—2122no—
393D4—2122no—
394D4—2122no—
395D4—2122no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—4241no—
406C2×C4—4241no—
407C2×C4—4241no—
408C2×C4#1313318711yescharacteristic
409C2×C4#1313318811yescharacteristic
410C2×C4#1313318911yescharacteristic
411C2×C4#1313319911yescharacteristic
412C2×C4#1313320111yescharacteristic
413C2×C4#1313320311yescharacteristic
414C2×C4C2×#1171992211yescharacteristic
415C2×C4C2×#1171991311yescharacteristic
416C2×C4#1313323711yescharacteristic
417C2×C4#1313323911yescharacteristic
418C2×C4#1313324211yescharacteristic
419C2×C4#1313324311yescharacteristic
420C2×C4#1313324511yescharacteristic
421C2×C4#1313324511yescharacteristic
422C8—22no—
423C8—41no—
424C8—81no—
425C8—4241no—
426C8—2122no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—8481no—
435C22—8481no—
436C22—8481no—
437C22—8481no—
438C22—4242no—
439C22—4242no—
440C22—4242no—
441C22#1681053911yescharacteristic
442C22#1681081811yescharacteristic
443C22#1681083811yescharacteristic
444C22#1681085211yescharacteristic
445C22#1681087611yescharacteristic
446C22#1681088611yescharacteristic
447C22#1681089311yescharacteristic
448C4—21no—
449C4—21no—
450C4—81no—
451C4—8481no—
452C4#1681077011yescharacteristic
453C4#1681077611yescharacteristic
454C4#1681081511yescharacteristic
455C4#1681083511yescharacteristic
456C4#1681085711yescharacteristic
457C4#1681087411yescharacteristic
458C4#1681094211yescharacteristic
459C4#1681094811yescharacteristic
460C2—82no—
461C2—8482no—
462C2#2027405611yescharacteristic
463C2#2027412411yescharacteristic
464C2#2027419411yescharacteristic
465C2#2027427611yescharacteristic
466C2#2027590511yescharacteristic
467C2#2027600611yescharacteristic
468C2#2027602511yescharacteristic
subgroupG/H
centerC23#13133191
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1954C2
socleC23×C53#121
prime radicalC2×#16810818C2
hypercenter#1954D53
hypocenterC53#10473
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
2. #20275521C2semidirect
3. #20274715C2semidirect
4. C2×#16810374C2semidirect
5. C2×#16810366C2semidirect
6. #20272599C2semidirect
7. C53×#1954C2semidirect
111. C2×C8×C53D8semidirect
228. #1954D53semidirect
261. C53#10473semidirect
353. C2×C8#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168108182
#202755212
#202747152
C2×#168103742
C2×#168103662
#202725992
C53×#19542
#1047353