Finite Group Databasebeta

← #27287481

Subgroups — #27287481

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279884C211yesmaximal, characteristic
2#20275517C211yesmaximal, characteristic, complemented
3#20275478C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic
5C2×#16810371C211yesmaximal, characteristic, complemented
6C2×#16810352C211yesmaximal, characteristic
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×C4×D212C2211yescharacteristic
10C2×#13133148—21no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133142—22no—
13C2×#13133139—21no—
14#16810536—22no—
15#16810525—21no—
16#16810493C2211yescharacteristic
17#16810489—21no—
18#16810480—21no—
19#16810480—21no—
20#16810386—21no—
21#16810385—21no—
22#16810371—22no—
23#16810352—22no—
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C4×C4⋊C4×C53C2211yescharacteristic
27C22×Dic106D412yes—
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C22×#88322—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53D411yescharacteristic
45C2×#11719921—22no—
46C2×#11719921—22no—
47C2×#11719921—21no—
48#13133148—42no—
49C4⋊#88322—21no—
50C4⋊#88322D412yes—
51#13133142—44no—
52#13133139—42no—
53#13133124—22no—
54#13133124—22no—
55C22×C8×C53—21no—
56C22×C8×C53—21no—
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4×C8×C53—21no—
64C4×C8×C53—21no—
65#95×C53—21no—
66C23×D53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—22no—
74C22×Dic53—41no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212SD1612yes—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×Dic106—22no—
88C2×Dic106—22no—
89C2×Dic106—42no—
90C2×Dic106—42no—
91C2×C8×C53—22no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C42×C53—41no—
102C42×C53—41no—
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C42×C53#4111yescharacteristic
106C42×C53#4111yescharacteristic
107#11719921—41no—
108#11719921—41no—
109#11719921—82no—
110#11719921—82no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53—41no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—42no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C2×#88322—22no—
123C2×#88322—41no—
124C2×#88322—41no—
125C23×C53#12011yescharacteristic
126C22×D53—22no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—41no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12311yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53#12411yescharacteristic
147C2×C4×C53C2×SD1611yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×C4×C53#13511yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C53#13711yescharacteristic
153C2×C4×C53#13711yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C2×Dic53—42no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—81no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164D212—22no—
165D212—22no—
166D212—41no—
167D212—41no—
168C4×D53—81no—
169C4×D53—81no—
170C4×D53—81no—
171C4×D53—81no—
172Dic106—44no—
173Dic106—82no—
174C8×C53—42no—
175C8×C53—81no—
176#88322—42no—
177#10412—531nomaximal, Sylow
178C22×C53#39411yescharacteristic
179C22×C53#44811yescharacteristic
180C22×C53#44911yescharacteristic
181C22×C53#45211yescharacteristic
182C22×C53#45211yescharacteristic
183C22×C53#45311yescharacteristic
184C22×C53#45511yescharacteristic
185C2×D53—42no—
186C2×D53—42no—
187C2×D53—42no—
188C2×D53—81no—
189C2×D53—81no—
190C2×D53—81no—
191C2×D53—81no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—81no—
195C4×C53—81no—
196C4×C53#46311yescharacteristic
197C4×C53#46411yescharacteristic
198C4×C53#46511yescharacteristic
199C4×C53#47011yescharacteristic
200C4×C53#48311yescharacteristic
201C4×C53#48511yescharacteristic
202C4×C53#48811yescharacteristic
203C4×C53#48911yescharacteristic
204Dic53—82no—
205Dic53—81no—
206Dic53—161no—
207#2563—531no—
208#1956—531no—
209#1948—531no—
210#1802—531no—
211C2×#333—531no—
212C2×D4⋊C8—531no—
213#1279D5311yescharacteristic, complemented
214C2×C53#162211yescharacteristic
215C2×C53#163411yescharacteristic
216C2×C53#164311yescharacteristic
217C2×C53#164611yescharacteristic
218C2×C53#203511yescharacteristic
219C2×C53#203711yescharacteristic
220C2×C53#204211yescharacteristic
221D53—82no—
222C2×C4⋊Q8—531no—
223C2×C4×D4—531no—
224C2×#106—1061no—
225C2×C4⋊C8—531no—
226C2×C4⋊C8C2×D5311yescharacteristic
227C2×#103—1062no—
228C2×#102—1061no—
229C2×C4×C8C2×D5311yescharacteristic
230#393—1062no—
231#390—1061no—
232#384—531no—
233#382—1061no—
234#380—1061no—
235#380—1061no—
236C4×C4⋊C4C2×D5311yescharacteristic
237#340—1061no—
238#340—1061no—
239#333—1062no—
240D4⋊C8—1062no—
241C53#1041211yescharacteristic, complemented, Sylow
242C22×Q8—532no—
243C22×D4—531no—
244C23×C4—1061no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—1061no—
248C4⋊Q8—1061no—
249C4⋊Q8—1061no—
250C4⋊Q8—1061no—
251C4⋊Q8—1061no—
252C4×D4—1061no—
253C4×D4—1061no—
254C4×D4—1061no—
255C4×D4—1061no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4—1061no—
261C2×C4⋊C4—1061no—
262C2×C4⋊C4D21211yescharacteristic
263C2×C4⋊C4D21211yescharacteristic
264C2×C22⋊C4—1062no—
265C2×C22⋊C4—1062no—
266C2×C22⋊C4—1061no—
267C2×C42—21no—
268C2×C42C22×D5311yescharacteristic
269#106—2122no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—532no—
273C4⋊C8—1061no—
274#103—2124no—
275#102—2122no—
276C4×C8—21no—
277C4×C8—21no—
278#95—21no—
279#95—1062no—
280#95—1062no—
281C24—1061no—
282C2×Q8—1062no—
283C2×Q8—1062no—
284C2×Q8—2122no—
285C2×Q8—2122no—
286C2×D4—532no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—1061no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—1061no—
295C22×C4—1062no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4—2121no—
299C22×C4—2121no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4C2×D21211yescharacteristic
304C22×C4D4×D5311yescharacteristic
305C22×C4D4×D5311yescharacteristic
306C2×C8—22no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—1062no—
313C2×C8—2121no—
314C2×C8—2121no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—1061no—
320C4⋊C4—1061no—
321C4⋊C4—2122no—
322C4⋊C4—2121no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C4⋊C4—2121no—
328C4⋊C4—2121no—
329C4⋊C4—2121no—
330C22⋊C4—4242no—
331C22⋊C4—4242no—
332C22⋊C4—2121no—
333C22⋊C4—2121no—
334C42—41no—
335C42—41no—
336C42#1171994511yescharacteristic
337C42#1171994511yescharacteristic
338C42D4×D5311yescharacteristic
339C42#1171994911yescharacteristic
340C23—1062no—
341C23—2121no—
342C23—2121no—
343C23—2121no—
344C23—2121no—
345C23—2121no—
346C23—2121no—
347C23#1313318711yescharacteristic
348Q8—4242no—
349Q8—2124no—
350D4—1062no—
351D4—1062no—
352D4—2121no—
353D4—2121no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—4241no—
363C2×C4—4241no—
364C2×C4—4241no—
365C2×C4—4241no—
366C2×C4—4241no—
367C2×C4—4241no—
368C2×C4—4241no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2122no—
372C2×C4—2121no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4#1313319011yescharacteristic
376C2×C4#1313319011yescharacteristic
377C2×C4#1313319211yescharacteristic
378C2×C4#1313320211yescharacteristic
379C2×C4#1313320211yescharacteristic
380C2×C4#1313320511yescharacteristic
381C2×C4SD16×D5311yescharacteristic
382C2×C4#1313323611yescharacteristic
383C2×C4#1313323911yescharacteristic
384C2×C4#1313323911yescharacteristic
385C2×C4#1313324511yescharacteristic
386C2×C4#1313324611yescharacteristic
387C2×C4#1313324911yescharacteristic
388C2×C4#1313325011yescharacteristic
389C8—42no—
390C8—81no—
391C8—2122no—
392C22—4241no—
393C22—4241no—
394C22—4241no—
395C22—4241no—
396C22—2122no—
397C22—2122no—
398C22—2122no—
399C22#1681054111yescharacteristic
400C22#1681076311yescharacteristic
401C22#1681077011yescharacteristic
402C22#1681078611yescharacteristic
403C22#1681078711yescharacteristic
404C22#1681079211yescharacteristic
405C22#1681080211yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—8481no—
411C4—4242no—
412C4—4241no—
413C4#1681083911yescharacteristic
414C4#1681084611yescharacteristic
415C4#1681085111yescharacteristic
416C4#1681088011yescharacteristic
417C4#1681095511yescharacteristic
418C4#1681096611yescharacteristic
419C4#1681097811yescharacteristic
420C4#1681098311yescharacteristic
421C2—4242no—
422C2#2027379011yescharacteristic
423C2#2027386211yescharacteristic
424C2#2027392511yescharacteristic
425C2#2027394511yescharacteristic
426C2#2027591411yescharacteristic
427C2#2027592811yescharacteristic
428C2#2027595711yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1279D53
hypocenterC53#10412
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #20275517C2semidirect
3. #20275478C2semidirect
5. C2×#16810371C2semidirect
7. C53×#1279C2semidirect
213. #1279D53semidirect
241. C53#10412semidirect

Maximal subgroups

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

subgroupindex
#202798842
#202755172
#202754782
#202747162
C2×#168103712
C2×#168103522
C53×#12792
#1041253