Finite Group Databasebeta

← #27278977

Subgroups — #27278977

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284170C211yesmaximal, characteristic
2#20283283C211yesmaximal, characteristic, complemented
3#20273167C211yesmaximal, characteristic, complemented
4#20273115C211yesmaximal, characteristic, complemented
5#20272611C211yesmaximal, characteristic, complemented
6#20272550C211yesmaximal, characteristic, complemented
7C53×#1476C211yesmaximal, characteristic, complemented
8C2×#13133237—21no—
9C2×#13133224—21no—
10#16811173C2211yescharacteristic
11#16810845—21no—
12#16810838—21no—
13#16810833C411yescharacteristic
14#16810828C411yescharacteristic
15#16810823C411yescharacteristic
16#16810818C411yescharacteristic
17#16810650—22no—
18#16810650—21no—
19#16810647—21no—
20#16810647C2211yescharacteristic
21C2×C4⋊#88322—21no—
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24#16810617—21no—
25#16810352—22no—
26#16810352—21no—
27#16810352—21no—
28C4×#41×C53C2211yescharacteristic
29C53×#431C2211yescharacteristic
30C2×C4×C8×C53C2211yescharacteristic
31C53×Q8⋊C8—21no—
32C53×D4⋊C8—21no—
33C2×#11719945—21no—
34#13133237—42no—
35#13133237—42no—
36C2×#11719925—21no—
37C2×#11719923—21no—
38C2×#11719924—21no—
39C2×#11719922—21no—
40#13133224—42no—
41#13133224—42no—
42C2×#11719917—22no—
43C2×#11719917—21no—
44C22×#88322—22no—
45C22×#88322—21no—
46C22×#88322—21no—
47#13133217C2×C411yescharacteristic
48#13133214C2×C411yescharacteristic
49#13133205—21no—
50#13133191—21no—
51#13133191—21no—
52C4×D212—21no—
53C4×D212—21no—
54C4⋊#88322—22no—
55C4⋊#88322—22no—
56C4⋊#88322—21no—
57C4⋊#88322—21no—
58C4⋊#88322—21no—
59C4⋊#88322C2×C411yescharacteristic
60C4⋊#88322C2×C411yescharacteristic
61C4⋊#88322D411yescharacteristic
62C4⋊#88322D411yescharacteristic
63C4⋊#88322D412yes—
64#13133140—21no—
65#13133140—21no—
66#13133137—21no—
67#13133137—21no—
68C2×#41×C53D411yescharacteristic
69C22×C8×C53—21no—
70C4×Q8×C53C2×C411yescharacteristic
71C4×Q8×C53D411yescharacteristic
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53C2×C411yescharacteristic
75C4×D4×C53D411yescharacteristic
76#117×C53—21no—
77#117×C53D411yescharacteristic
78C2×C42×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81C22⋊C8×C53—21no—
82C4×C8×C53—21no—
83C4×C8×C53—21no—
84C4×C8×C53—21no—
85#41×C53—21no—
86#41×C53—21no—
87#41×C53D812yes—
88#41×C53SD1612yes—
89C2×Q8×C53C22⋊C411yescharacteristic
90C2×D4×C53—21no—
91C2×D4×C53C22⋊C411yescharacteristic
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C2×C22⋊D53—41no—
99#11719945—41no—
100#11719945—41no—
101#11719945—41no—
102#11719945—41no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212—21no—
106C2×C4×D53—41no—
107C2×Dic106—21no—
108C2×C8×C53—22no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53C22⋊C411yescharacteristic
115C4⋊C4×C53C22⋊C411yescharacteristic
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C42×C53—21no—
120C42×C53—21no—
121C42×C53—21no—
122C42×C53C22×C411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126#11719925—42no—
127#11719924—42no—
128#11719923—42no—
129#11719922—42no—
130#11719921—41no—
131#11719921—41no—
132C4⋊Dic53—21no—
133C4⋊Dic53—21no—
134#11719917—44no—
135#11719917—41no—
136#11719917—41no—
137C2×#88322—21no—
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—22no—
141C2×#88322—21no—
142C2×#88322—44no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—41no—
146C2×#88322—41no—
147C23×C53—21no—
148C23×C53#12311yescharacteristic
149C22×D53—41no—
150Q8×C53—21no—
151Q8×C53#10212yes—
152D4×C53—22no—
153D4×C53—22no—
154D4×C53—21no—
155D4×C53#10212yes—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—81no—
167C2×C4×C53C2×C22⋊C411yescharacteristic
168C2×C4×C53C4×D411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53C2×D811yescharacteristic
174C2×C4×C53C2×SD1611yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C53#13711yescharacteristic
177C22⋊D53—81no—
178C22⋊D53—81no—
179C2×Dic53—41no—
180D212—42no—
181D212—41no—
182D212—41no—
183D212—41no—
184D212—41no—
185C4×D53—81no—
186C4×D53—81no—
187Dic106—41no—
188Dic106—41no—
189C8×C53—42no—
190C8×C53—81no—
191#88322—41no—
192#88322—41no—
193#88322—41no—
194#88322—42no—
195#88322—41no—
196#8995—531nomaximal, Sylow
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—22no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53#38611yescharacteristic
203C22×C53#48011yescharacteristic
204C22×C53#48111yescharacteristic
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—81no—
208C4×C53—22no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53C2×#10211yescharacteristic
215C4×C53#41711yescharacteristic
216C4×C53#44311yescharacteristic
217C4×C53#44411yescharacteristic
218C4×C53#44711yescharacteristic
219C4×C53#44811yescharacteristic
220Dic53—81no—
221#3035—531no—
222#2906—531no—
223#1492—531no—
224#1480—531no—
225#1476D5311yescharacteristic, complemented
226#1301—531no—
227#1277—531no—
228C2×C53—21no—
229C2×C53—21no—
230C2×C53—42no—
231C2×C53#187411yescharacteristic
232C2×C53#187611yescharacteristic
233C2×C53#188111yescharacteristic
234D53—161no—
235C2×#135—1061no—
236C2×#131—1061no—
237#532—531no—
238C4×#41C2×D5311yescharacteristic
239#464—1061no—
240#463—1061no—
241#462—531no—
242#461—531no—
243#460—531no—
244#459—531no—
245#431C2×D5311yescharacteristic
246#424—1062no—
247#424—1061no—
248#423—531no—
249#423—1061no—
250C2×C4⋊C8—531no—
251C2×C4⋊C8—531no—
252C2×C4⋊C8—1061no—
253#416—1061no—
254C2×C4×C8C2×D5311yescharacteristic
255Q8⋊C8—21no—
256D4⋊C8—21no—
257D4⋊C8—1061no—
258D4⋊C8—1061no—
259D4⋊C8—1062no—
260C53#899511yescharacteristic, complemented, Sylow
261C2×#41—1061no—
262C2×#41D21211yescharacteristic
263#135—2122no—
264#135—2122no—
265C2×Q16—1061no—
266C2×SD16—1061no—
267C2×SD16—1061no—
268C2×D8—1061no—
269#131—2122no—
270#131—2122no—
271C2×M16—1062no—
272C2×M16—1061no—
273C22×C8—21no—
274C22×C8—1062no—
275C22×C8—1061no—
276C22×C8—1061no—
277C4⋊Q8—531no—
278#127—531no—
279#124—1061no—
280#121—1061no—
281#121—1061no—
282C4×Q8C4×D5311yescharacteristic
283C4×Q8C22⋊D5311yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—1061no—
287C4×D4—1061no—
288C4×D4C4×D5311yescharacteristic
289C4×D4C22⋊D5311yescharacteristic
290#117—21no—
291#117D21211yescharacteristic
292C2×C42—21no—
293C2×C42C22×D5311yescharacteristic
294C4⋊C8—21no—
295C4⋊C8—531no—
296C4⋊C8—531no—
297C4⋊C8—531no—
298C4⋊C8—531no—
299C4⋊C8—532no—
300C4⋊C8—1062no—
301C4⋊C8—1062no—
302C4⋊C8—1061no—
303C4⋊C8—1061no—
304C4⋊C8—1061no—
305#103—1061no—
306#103—1061no—
307#102—1061no—
308#102—1061no—
309C22⋊C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313#41—21no—
314#41—21no—
315#41—2121no—
316#41—2121no—
317#41—2121no—
318#41—2121no—
319#41#1171991312yes—
320#41D42412yes—
321C2×Q8—1061no—
322C2×Q8#1171992111yescharacteristic
323C2×D4—21no—
324C2×D4—1061no—
325C2×D4—1061no—
326C2×D4—1061no—
327C2×D4—2121no—
328C2×D4#1171992111yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—2121no—
333C22×C4C2×D21211yescharacteristic
334C22×C4#1171994711yescharacteristic
335C22×C4#1171994711yescharacteristic
336Q16—2122no—
337SD16—2122no—
338SD16—2122no—
339D8—2122no—
340M16—2124no—
341M16—2121no—
342M16—2121no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—1061no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—1061no—
351C2×C8—1062no—
352C2×C8—2124no—
353C2×C8—2121no—
354C2×C8—2121no—
355C2×C8—2121no—
356C2×C8—2121no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—1061no—
360C4⋊C4—1061no—
361C4⋊C4#1171992111yescharacteristic
362C4⋊C4#1171992111yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—2121no—
367C22⋊C4—2121no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42C2×C4×D5311yescharacteristic
372C42D4×D5311yescharacteristic
373C42D4×D5311yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C23—21no—
376C23—2121no—
377C23#1313320111yescharacteristic
378Q8—21no—
379Q8—2121no—
380Q8—2121no—
381Q8#1313313912yes—
382D4—22no—
383D4—22no—
384D4—21no—
385D4—4241no—
386D4—4241no—
387D4—2122no—
388D4—2121no—
389D4—2121no—
390D4—2121no—
391D4—2121no—
392D4#1313313912yes—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—4241no—
405C2×C4—4241no—
406C2×C4—2121no—
407C2×C4C2×#1171992111yescharacteristic
408C2×C4#1313317711yescharacteristic
409C2×C4#1313317711yescharacteristic
410C2×C4#1313318711yescharacteristic
411C2×C4#1313318911yescharacteristic
412C2×C4#1313318911yescharacteristic
413C2×C4C2×D42411yescharacteristic
414C2×C4C2×#1171991311yescharacteristic
415C2×C4#1313324511yescharacteristic
416C2×C4#1313325011yescharacteristic
417C8—42no—
418C8—81no—
419C8—2121no—
420C8—2122no—
421C8—2121no—
422C8—2121no—
423C8—2121no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—4241no—
430C22—4241no—
431C22—4241no—
432C22#1681050011yescharacteristic
433C22#1681093711yescharacteristic
434C22#1681094211yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—4241no—
442C4C2×#1313313911yescharacteristic
443C4#1681062511yescharacteristic
444C4#1681073711yescharacteristic
445C4#1681074211yescharacteristic
446C4#1681075911yescharacteristic
447C4#1681076411yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—8481no—
452C2#2027503011yescharacteristic
453C2#2027504011yescharacteristic
454C2#2027507511yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1476C2
socleC22×C53#386
prime radical#20284170C2
cosocleC2×C42×C53C23
hypercenter#1476D53
hypocenterC53#8995
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #20283283C2semidirect
3. #20273167C2semidirect
4. #20273115C2semidirect
5. #20272611C2semidirect
6. #20272550C2semidirect
7. C53×#1476C2semidirect
225. #1476D53semidirect
260. C53#8995semidirect

Maximal subgroups

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

subgroupindex
#202841702
#202832832
#202731672
#202731152
#202726112
#202725502
C53×#14762
#899553