Finite Group Databasebeta

← #27287888

Subgroups — #27287888

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×#16810815C211yesmaximal, characteristic
2#20275509C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic, complemented
4C2×#16810374C211yesmaximal, characteristic, complemented
5C2×#16810367C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1957C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×C4×D212C2211yescharacteristic
10#16810815—22no—
11#16810815—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133138—21no—
16C2×#13133137—21no—
17C2×#13133129—21no—
18C2×#13133129—21no—
19#16810385—21no—
20#16810374—22no—
21#16810367—22no—
22C2×#127×C53C2211yescharacteristic
23C2×#102×C53—21no—
24C2×#102×C53—21no—
25C2×C4×C8×C53C2211yescharacteristic
26C53×#384C2211yescharacteristic
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719922—24no—
30C2×#11719922—42no—
31C2×#11719922—42no—
32C22×#88322—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×#11719921—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322D412yes—
47#13133138—24no—
48#13133137—42no—
49#13133129—42no—
50#13133129—42no—
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54C22×C8×C53D411yescharacteristic
55C22×C8×C53D411yescharacteristic
56#127×C53—21no—
57#127×C53—21no—
58#127×C53—21no—
59#127×C53D412yes—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C42×C53C2311yescharacteristic
64#102×C53—42no—
65#102×C53—42no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C24×C53—41no—
69C23×D53—21no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—81no—
85C2×D4×C53—81no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×Dic53—21no—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212—21no—
94C2×D212—21no—
95C2×D212—21no—
96C2×D212SD1612yes—
97C2×C4×D53—41no—
98C2×C4×D53—41no—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C2×C4×D53—41no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C2×C8×C53—21no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53D812yescomplemented
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—81no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53#4111yescharacteristic
118#11719922—48no—
119#11719921—41no—
120#11719921—41no—
121C4⋊Dic53—21no—
122C4⋊Dic53—21no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C23×C53—42no—
133C23×C53—81no—
134C23×C53—81no—
135C23×C53—81no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53—81no—
139C23×C53#12111yescharacteristic
140C22×D53—22no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—41no—
145C22×D53—41no—
146C22×D53—41no—
147D4×C53—42no—
148D4×C53—42no—
149D4×C53—42no—
150D4×C53—42no—
151D4×C53—82no—
152D4×C53—82no—
153D4×C53—81no—
154D4×C53—81no—
155D4×C53—81no—
156D4×C53—81no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53—81no—
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C2×C4×C53#12311yescharacteristic
167C2×C4×C53#12411yescharacteristic
168C2×C4×C53C2×D811yescharacteristic
169C2×C4×C53C2×SD1611yescharacteristic
170C2×C4×C53#13511yescharacteristic
171C2×C4×C53#13511yescharacteristic
172C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179D212—22no—
180D212—22no—
181D212—41no—
182D212—41no—
183C4×D53—81no—
184C4×D53—81no—
185C4×D53—81no—
186C4×D53—81no—
187C8×C53—22no—
188C8×C53—41no—
189#88322—42no—
190#88322—81no—
191#88322—81no—
192#10473—531nomaximal, Sylow
193C22×C53—82no—
194C22×C53—82no—
195C22×C53—82no—
196C22×C53—161no—
197C22×C53—161no—
198C22×C53—161no—
199C22×C53—161no—
200C22×C53#39411yescharacteristic
201C22×C53#45911yescharacteristic
202C22×C53#46311yescharacteristic
203C22×C53#46511yescharacteristic
204C22×C53#46911yescharacteristic
205C22×C53#47111yescharacteristic
206C22×C53#47211yescharacteristic
207C2×D53—42no—
208C2×D53—42no—
209C2×D53—42no—
210C2×D53—81no—
211C2×D53—81no—
212C2×D53—81no—
213C2×D53—81no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—161no—
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—81no—
226C2×#459—531no—
227#1957D5311yescharacteristic, complemented
228#1954—531no—
229#1802—531no—
230C2×#334—531no—
231C2×#331—531no—
232#1296—531no—
233C2×C53—162no—
234C2×C53#166511yescharacteristic
235C2×C53#167711yescharacteristic
236C2×C53#168711yescharacteristic
237C2×C53#170111yescharacteristic
238C2×C53#203411yescharacteristic
239C2×C53#205111yescharacteristic
240C2×C53#205411yescharacteristic
241D53—82no—
242C22×D8—1061no—
243C2×#127C2×D5311yescharacteristic
244C2×C4×D4—531no—
245#459—1062no—
246#459—1062no—
247C2×C4⋊C8—531no—
248C2×C4⋊C8—531no—
249C2×C4⋊C8—531no—
250C2×#102—21no—
251C2×#102—21no—
252C2×#102—1061no—
253C2×#102—1061no—
254C2×C22⋊C8—1061no—
255C2×C22⋊C8—1061no—
256C2×C4×C8C2×D5311yescharacteristic
257#384C2×D5311yescharacteristic
258#340—1061no—
259#334—1062no—
260#331—1062no—
261C53#1047311yescharacteristic, complemented, Sylow
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4—531no—
266C23×C4—1061no—
267C2×D8—1064no—
268C2×D8—2122no—
269C2×D8—2122no—
270C22×C8—1061no—
271C22×C8—1061no—
272C22×C8—1061no—
273C22×C8C22⋊D5311yescharacteristic
274C22×C8C22⋊D5311yescharacteristic
275#127—21no—
276#127—21no—
277#127—21no—
278#127D21212yes—
279C4×D4—1061no—
280C4×D4—1061no—
281C4×D4—1061no—
282C4×D4—1061no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—531no—
287C2×C22⋊C4—1061no—
288C2×C42C22×D5311yescharacteristic
289C4⋊C8—532no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293C4⋊C8—1061no—
294C4⋊C8—1061no—
295#102—42no—
296#102—42no—
297#102—1064no—
298#102—2122no—
299C22⋊C8—2122no—
300C22⋊C8—2122no—
301C4×C8—21no—
302C4×C8—21no—
303C24—41no—
304C24—1061no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—81no—
320C2×D4—81no—
321C2×D4—532no—
322C2×D4—1061no—
323C2×D4—1061no—
324C2×D4—1061no—
325C2×D4—1061no—
326C2×D4—1061no—
327C22×C4—41no—
328C22×C4—1061no—
329C22×C4—2121no—
330C22×C4—2121no—
331C22×C4—2121no—
332C22×C4—2121no—
333C22×C4—2121no—
334C22×C4—2121no—
335C22×C4D4×D5311yescharacteristic
336C22×C4#1171994711yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338D8—2128no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—1062no—
345C2×C8—2121no—
346C2×C8—2121no—
347C2×C8—2121no—
348C2×C8—2121no—
349C2×C8—2121no—
350C2×C8—2121no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8#1171992212yescomplemented
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—81no—
359C4⋊C4—1061no—
360C4⋊C4—1061no—
361C22⋊C4—2121no—
362C22⋊C4—2121no—
363C42C2×D21211yescharacteristic
364C42#1171994511yescharacteristic
365C42D4×D5311yescharacteristic
366C42#1171994711yescharacteristic
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—1062no—
375C23—2121no—
376C23—2121no—
377C23—2121no—
378C23—2121no—
379C23—2121no—
380C23—2121no—
381C23#1313318811yescharacteristic
382D4—42no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—82no—
387D4—82no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—1062no—
393D4—1062no—
394D4—2121no—
395D4—2121no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—4241no—
402C2×C4—4241no—
403C2×C4—4241no—
404C2×C4—4241no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4#1313318711yescharacteristic
409C2×C4#1313318911yescharacteristic
410C2×C4#1313319111yescharacteristic
411C2×C4#1313319911yescharacteristic
412C2×C4#1313320111yescharacteristic
413C2×C4#1313320311yescharacteristic
414C2×C4C2×#1171992211yescharacteristic
415C2×C4SD16×D5311yescharacteristic
416C2×C4#1313323711yescharacteristic
417C2×C4#1313323811yescharacteristic
418C2×C4#1313324111yescharacteristic
419C2×C4#1313324111yescharacteristic
420C2×C4#1313324211yescharacteristic
421C2×C4#1313324311yescharacteristic
422C8—22no—
423C8—41no—
424C8—4241no—
425C8—4241no—
426C8—2122no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—4241no—
435C22—4241no—
436C22—4241no—
437C22—4241no—
438C22—2122no—
439C22—2122no—
440C22—2122no—
441C22#1681053911yescharacteristic
442C22#1681081511yescharacteristic
443C22#1681083511yescharacteristic
444C22#1681084911yescharacteristic
445C22#1681087311yescharacteristic
446C22#1681088311yescharacteristic
447C22#1681089011yescharacteristic
448C4—21no—
449C4—21no—
450C4—161no—
451C4—4241no—
452C4#1681076911yescharacteristic
453C4#1681077511yescharacteristic
454C4#1681081811yescharacteristic
455C4#1681083811yescharacteristic
456C4#1681085711yescharacteristic
457C4#1681087411yescharacteristic
458C4#1681094011yescharacteristic
459C4#1681094611yescharacteristic
460C2—162no—
461C2—4242no—
462C2#2027405311yescharacteristic
463C2#2027412111yescharacteristic
464C2#2027419111yescharacteristic
465C2#2027427311yescharacteristic
466C2#2027590311yescharacteristic
467C2#2027600811yescharacteristic
468C2#2027602511yescharacteristic
subgroupG/H
centerC23#13133188
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1957C2
socleC23×C53#121
prime radicalC2×#16810815C2
hypercenter#1957D53
hypocenterC53#10473
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. #20275509C2semidirect
3. #20274712C2semidirect
4. C2×#16810374C2semidirect
5. C2×#16810367C2semidirect
6. #20272598C2semidirect
7. C53×#1957C2semidirect
108. C2×C8×C53D8semidirect
227. #1957D53semidirect
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×#168108152
#202755092
#202747122
C2×#168103742
C2×#168103672
#202725982
C53×#19572
#1047353