Finite Group Databasebeta

← #27287889

Subgroups — #27287889

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#20275520C211yesmaximal, characteristic, complemented
3#20275510C211yesmaximal, characteristic, complemented
4C2×#16810373C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1802C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133214C2211yescharacteristic
10#16810818—22no—
11#16810818—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—21no—
15C2×#13133137—21no—
16C2×#13133137—21no—
17C2×C22⋊#88322—21no—
18C2×C22⋊#88322—21no—
19C2×C4×#88322C2211yescharacteristic
20#16810373—22no—
21#16810366—22no—
22C2×C4×D4×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#102×C53—21no—
25C53×#384C2211yescharacteristic
26C53×#340—21no—
27C22×D212—21no—
28C22×D212—21no—
29C22×D212—21no—
30C2×#11719922—24no—
31C2×#11719922—42no—
32C2×#11719922—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35C22×#88322D411yescharacteristic
36C22×#88322D411yescharacteristic
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214D412yes—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322D412yes—
45#13133137—24no—
46#13133137—42no—
47#13133137—42no—
48C22⋊#88322—42no—
49C22⋊#88322—42no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C22×C8×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—21no—
58C4×D4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67#102×C53—42no—
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×C53—41no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
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—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—81no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53#4111yescharacteristic
119#11719922—48no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—22no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322D812yescomplemented
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—81no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12311yescharacteristic
167C2×C4×C53#12311yescharacteristic
168C2×C4×C53#12411yescharacteristic
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53C2×SD1611yescharacteristic
171C2×C4×C53#13511yescharacteristic
172C2×C4×C53#13511yescharacteristic
173C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177D212—42no—
178D212—42no—
179D212—42no—
180D212—42no—
181D212—82no—
182D212—82no—
183D212—81no—
184D212—81no—
185D212—81no—
186D212—81no—
187C8×C53—81no—
188#88322—22no—
189#88322—42no—
190#88322—41no—
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—161no—
218C4×C53#44911yescharacteristic
219C4×C53#45011yescharacteristic
220C4×C53#45911yescharacteristic
221C4×C53#46311yescharacteristic
222C4×C53#46611yescharacteristic
223C4×C53#46911yescharacteristic
224C4×C53#48111yescharacteristic
225C4×C53#48211yescharacteristic
226C2×#459—531no—
227#1957—531no—
228#1954—531no—
229#1802D5311yescharacteristic, complemented
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—21no—
251C2×#102—1061no—
252C2×#102—1061no—
253C2×#102—1061no—
254C2×C22⋊C8—1061no—
255C2×C22⋊C8—1061no—
256C2×C4×C8—531no—
257#384C2×D5311yescharacteristic
258#340—21no—
259#334—1062no—
260#331—1062no—
261C53#1047311yescharacteristic, complemented, Sylow
262C22×D4—1061no—
263C22×D4—1061no—
264C22×D4—1061no—
265C22×D4C22⋊D5311yescharacteristic
266C23×C4—21no—
267C2×D8—1064no—
268C2×D8—2122no—
269C2×D8—2122no—
270C22×C8—21no—
271C22×C8—531no—
272C22×C8—531no—
273C22×C8—1061no—
274C22×C8—1061no—
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—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4C22⋊D5311yescharacteristic
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—42no—
296#102—1064no—
297#102—2122no—
298#102—2122no—
299C22⋊C8—2122no—
300C22⋊C8—2122no—
301C4×C8—1061no—
302C4×C8—1061no—
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#1171992412yes—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4D4×D5311yescharacteristic
336C22×C4#1171994711yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338D8—2128no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—532no—
343C2×C8—1061no—
344C2×C8—1061no—
345C2×C8—1062no—
346C2×C8—1061no—
347C2×C8—1061no—
348C2×C8—1061no—
349C2×C8—2121no—
350C2×C8—2121no—
351C2×C8—2121no—
352C2×C8—2121no—
353C2×C8—2121no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—81no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C42#1171994511yescharacteristic
364C42#1171994511yescharacteristic
365C42D4×D5311yescharacteristic
366C42D4×D5311yescharacteristic
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#1313318911yescharacteristic
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—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4#1313318511yescharacteristic
409C2×C4#1313319011yescharacteristic
410C2×C4#1313319011yescharacteristic
411C2×C4#1313319911yescharacteristic
412C2×C4#1313320011yescharacteristic
413C2×C4#1313320311yescharacteristic
414C2×C4D8×D5311yescharacteristic
415C2×C4C2×#1171992411yescharacteristic
416C2×C4#1313323711yescharacteristic
417C2×C4#1313323811yescharacteristic
418C2×C4#1313324111yescharacteristic
419C2×C4#1313324211yescharacteristic
420C2×C4#1313324311yescharacteristic
421C2×C4#1313324411yescharacteristic
422C8—81no—
423C8—1062no—
424C8—4241no—
425C8—2122no—
426C8—2121no—
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#1681053811yescharacteristic
442C22#1681081611yescharacteristic
443C22#1681084011yescharacteristic
444C22#1681085411yescharacteristic
445C22#1681087411yescharacteristic
446C22#1681088811yescharacteristic
447C22#1681089111yescharacteristic
448C4—21no—
449C4—21no—
450C4—81no—
451C4—161no—
452C4#1681076611yescharacteristic
453C4#1681077311yescharacteristic
454C4#1681081711yescharacteristic
455C4#1681083711yescharacteristic
456C4#1681085811yescharacteristic
457C4#1681087511yescharacteristic
458C4#1681094111yescharacteristic
459C4#1681094711yescharacteristic
460C2—82no—
461C2—8482no—
462C2#2027405811yescharacteristic
463C2#2027412611yescharacteristic
464C2#2027419211yescharacteristic
465C2#2027427811yescharacteristic
466C2#2027590411yescharacteristic
467C2#2027600511yescharacteristic
468C2#2027602411yescharacteristic
subgroupG/H
centerC23#13133189
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1802C2
socleC23×C53#121
prime radicalC2×#16810818C2
hypercenter#1802D53
hypocenterC53#10473
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #20275520C2semidirect
3. #20275510C2semidirect
4. C2×#16810373C2semidirect
5. C2×#16810366C2semidirect
6. #20272598C2semidirect
7. C53×#1802C2semidirect
131. C2×#88322D8semidirect
229. #1802D53semidirect
261. C53#10473semidirect

Maximal subgroups

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

subgroupindex
C2×#168108182
#202755202
#202755102
C2×#168103732
C2×#168103662
#202725982
C53×#18022
#1047353