Finite Group Databasebeta

← #27278901

Subgroups — #27278901

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810904C211yesmaximal, characteristic
2#20277662C211yesmaximal, characteristic
3#20275676C211yesmaximal, characteristic, complemented
4#20275638C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×C4×D212C2211yescharacteristic
10C2×#13133175—21no—
11#16810904—22no—
12#16810904C414yescomplemented
13C2×#13133151—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133140—21no—
16C2×#13133140—21no—
17C2×#13133139—21no—
18#16810525—21no—
19#16810515—22no—
20#16810493C2211yescharacteristic
21#16810480—21no—
22C4×C4⋊Dic53—21no—
23#16810386—22no—
24#16810352—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4×C8×C53C2211yescharacteristic
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×Dic106—21no—
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C22×#88322—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217C2×C412yes—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212C2×C412yes—
39C4×D212D412yes—
40#13133175—42no—
41#13133175—42no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D411yescharacteristic
47C2×C4⋊Dic53D412yes—
48C2×#11719921—22no—
49C2×#11719921—22no—
50C2×#11719921—21no—
51C2×C4×Dic53—22no—
52#13133151—24no—
53C4⋊#88322—21no—
54C4⋊#88322—21no—
55#13133140—42no—
56#13133140—42no—
57#13133139—24no—
58#13133124—22no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C42×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53C2×C412yes—
67C4×C8×C53—21no—
68C4×C8×C53—21no—
69#95×C53—21no—
70C23×D53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—21no—
77C22×Dic53—22no—
78C22×Dic53—41no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212C22⋊C412yes—
85C2×C4×D53—22no—
86C2×C4×D53—22no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×Dic106—22no—
92C2×Dic106—41no—
93C2×Dic106—41no—
94C2×Dic106—41no—
95C2×Dic106—41no—
96C2×Dic106—41no—
97C2×C8×C53—22no—
98C2×C8×C53—22no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53D812yes—
107C42×C53—41no—
108C42×C53—41no—
109C42×C53C22×C411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112C42×C53#4111yescharacteristic
113#11719921—22no—
114#11719921—22no—
115#11719921—44no—
116#11719921—82no—
117C4⋊Dic53—21no—
118C4⋊Dic53—22no—
119C4⋊Dic53—22no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C4⋊Dic53—41no—
124C4⋊Dic53—41no—
125C4⋊Dic53—41no—
126C4⋊Dic53—42no—
127C4⋊Dic53C22⋊C412yes—
128C4×Dic53—44no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C23×C53#12011yescharacteristic
133C22×D53—22no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—41no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12311yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53#13511yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53#13711yescharacteristic
161C2×C4×C53#13711yescharacteristic
162C2×Dic53—22no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—42no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171D212—22no—
172D212—22no—
173D212—41no—
174D212—41no—
175C4×D53—42no—
176C4×D53—42no—
177C4×D53—42no—
178C4×D53—42no—
179Dic106—42no—
180Dic106—42no—
181Dic106—81no—
182Dic106—81no—
183C8×C53—42no—
184C8×C53—42no—
185#88322—81no—
186#8984—531nomaximal, Sylow
187C22×C53#38611yescharacteristic
188C22×C53#44711yescharacteristic
189C22×C53#44911yescharacteristic
190C22×C53#45211yescharacteristic
191C22×C53#45211yescharacteristic
192C22×C53#45511yescharacteristic
193C22×C53#45611yescharacteristic
194C2×D53—42no—
195C2×D53—42no—
196C2×D53—42no—
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—42no—
204C4×C53—81no—
205C4×C53#41611yescharacteristic
206C4×C53#41911yescharacteristic
207C4×C53C4×D811yescharacteristic
208C4×C53#44211yescharacteristic
209C4×C53#45911yescharacteristic
210C4×C53#46311yescharacteristic
211C4×C53#48511yescharacteristic
212C4×C53#48511yescharacteristic
213Dic53—42no—
214Dic53—82no—
215Dic53—82no—
216C2×#474—531no—
217#2354—531no—
218#1990—531no—
219#1981—531no—
220C2×D4⋊C8—531no—
221#1296—531no—
222#1279D5311yescharacteristic, complemented
223C2×C53#164111yescharacteristic
224C2×C53#164611yescharacteristic
225C2×C53#164711yescharacteristic
226C2×C53#165211yescharacteristic
227C2×C53#187311yescharacteristic
228C2×C53#187911yescharacteristic
229C2×C53#188211yescharacteristic
230D53—82no—
231C2×C4⋊Q8—531no—
232C2×C4×D4—531no—
233C2×#117—1061no—
234#474—534no—
235#474—1062no—
236C2×#107—1061no—
237C2×C4⋊C8—531no—
238C2×C4⋊C8C2×D5311yescharacteristic
239C2×#103—1061no—
240C2×#103—1061no—
241C2×#102—1061no—
242C2×C4×C8C2×D5311yescharacteristic
243#390—1061no—
244#388—1062no—
245#384—531no—
246#380—1061no—
247C4×C4⋊C4—1061no—
248C4×C4⋊C4C2×D5311yescharacteristic
249#340—1062no—
250D4⋊C8—1062no—
251C53#898411yescharacteristic, complemented, Sylow
252C22×Q8—1061no—
253C22×D4—531no—
254C23×C4—1061no—
255C22×C8—21no—
256C22×C8—21no—
257C22×C8—1061no—
258C4⋊Q8—532no—
259C4⋊Q8—1061no—
260C4⋊Q8—1061no—
261C4⋊Q8—1061no—
262C4×D4—532no—
263C4×D4—532no—
264C4×D4—1061no—
265C4×D4—1061no—
266#117—2122no—
267#117—2122no—
268C2×C4⋊C4—531no—
269C2×C4⋊C4—532no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4D21211yescharacteristic
275C2×C4⋊C4D21211yescharacteristic
276C2×C22⋊C4—1061no—
277C2×C22⋊C4—1062no—
278C2×C22⋊C4—1062no—
279C2×C42—21no—
280C2×C42—1062no—
281C2×C42C22×D5311yescharacteristic
282#107—1064no—
283C4⋊C8—21no—
284C4⋊C8—1061no—
285C4⋊C8—1061no—
286C4⋊C8C4×D5312yes—
287#103—2122no—
288#103—2122no—
289#102—1064no—
290C4×C8—21no—
291C4×C8—21no—
292#95—21no—
293#95—1062no—
294C24—1061no—
295C2×Q8—1062no—
296C2×Q8—2121no—
297C2×Q8—2121no—
298C2×Q8—2121no—
299C2×Q8—2121no—
300C2×Q8—2121no—
301C2×D4—532no—
302C2×D4—1061no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C2×D4—1061no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—1061no—
310C22×C4—1062no—
311C22×C4—1062no—
312C22×C4—1062no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4—2121no—
318C22×C4C2×D21211yescharacteristic
319C22×C4D4×D5311yescharacteristic
320C22×C4D4×D5311yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—2121no—
328C2×C8—2121no—
329C2×C8—2121no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—532no—
334C4⋊C4—1061no—
335C4⋊C4—1062no—
336C4⋊C4—1062no—
337C4⋊C4—2122no—
338C4⋊C4—2121no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C4⋊C4D42412yes—
345C22⋊C4—1062no—
346C22⋊C4—1062no—
347C22⋊C4—4242no—
348C22⋊C4—2124no—
349C42—41no—
350C42—41no—
351C42—2124no—
352C42C2×C4×D5311yescharacteristic
353C42#1171994511yescharacteristic
354C42D4×D5311yescharacteristic
355C42#1171994911yescharacteristic
356C23—1062no—
357C23—2121no—
358C23—2121no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23#1313318711yescharacteristic
364Q8—4241no—
365Q8—4241no—
366Q8—2122no—
367Q8—2122no—
368D4—1062no—
369D4—1062no—
370D4—2121no—
371D4—2121no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—1062no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—2122no—
384C2×C4—2122no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4—2121no—
388C2×C4—2121no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4—2122no—
392C2×C4—2122no—
393C2×C4C22⋊C4×D5311yescharacteristic
394C2×C4#1313317911yescharacteristic
395C2×C4C4×D21211yescharacteristic
396C2×C4#1313319011yescharacteristic
397C2×C4#1313319211yescharacteristic
398C2×C4#1313320211yescharacteristic
399C2×C4C2×D42411yescharacteristic
400C2×C4#1313323611yescharacteristic
401C2×C4#1313323811yescharacteristic
402C2×C4#1313323911yescharacteristic
403C2×C4#1313324511yescharacteristic
404C2×C4#1313324511yescharacteristic
405C2×C4#1313324611yescharacteristic
406C2×C4#1313324911yescharacteristic
407C8—42no—
408C8—42no—
409C8—4241no—
410C22—4241no—
411C22—4241no—
412C22—4241no—
413C22—4241no—
414C22—2122no—
415C22—2122no—
416C22—2122no—
417C22#1681050111yescharacteristic
418C22#1681075911yescharacteristic
419C22#1681077011yescharacteristic
420C22#1681078611yescharacteristic
421C22#1681078811yescharacteristic
422C22#1681080211yescharacteristic
423C22#1681080711yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—81no—
428C4—4242no—
429C4—4242no—
430C4—2122no—
431C4#1681061611yescharacteristic
432C4#1681063011yescharacteristic
433C4C4×D42411yescharacteristic
434C4#1681073611yescharacteristic
435C4#1681081911yescharacteristic
436C4#1681083911yescharacteristic
437C4#1681096611yescharacteristic
438C4#1681096711yescharacteristic
439C2—4242no—
440C2#2027391011yescharacteristic
441C2#2027394511yescharacteristic
442C2#2027395211yescharacteristic
443C2#2027398711yescharacteristic
444C2#2027502611yescharacteristic
445C2#2027506211yescharacteristic
446C2#2027508511yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#8984
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #20275676C2semidirect
4. #20275638C2semidirect
6. #20272599C2semidirect
7. C53×#1279C2semidirect
12. #16810904C4semidirect
222. #1279D53semidirect
251. C53#8984semidirect

Maximal subgroups

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

subgroupindex
C2×#168109042
#202776622
#202756762
#202756382
C2×#168103522
#202725992
C53×#12792
#898453