Finite Group Databasebeta

← #27538680

Subgroups — #27538680

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811058C211yesmaximal, characteristic, complemented
2#20287818C211yesmaximal, characteristic, complemented
3C2×#16810893C211yesmaximal, characteristic, complemented
4#20276847C214yesmaximal, complemented
5#20276833C214yesmaximal, complemented
6#20276509C211yesmaximal, characteristic, complemented
7#20276464C211yesmaximal, characteristic, complemented
8C2×#16810435C211yesmaximal, characteristic, complemented
9C53×#3067C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C22×D424C2211yescharacteristic
12#16811058—22no—
13#16811058—22no—
14C2×#13133160C2212yescomplemented
15C2×#13133158C2212yescomplemented
16#16811040—22no—
17#16811040C2214yescomplemented
18C2×#13133157—22no—
19C2×#13133157C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21#16810893—22no—
22#16810893C2214yes—
23C2×#13133153C2211yescharacteristic
24C2×#13133133—21no—
25#16810452C2214yescomplemented
26#16810439C2212yescomplemented
27#16810436C2212yescomplemented
28#16810435C2214yescomplemented
29C53×#575—21no—
30C2×#135×C53C2211yescharacteristic
31C2×#131×C53C2211yescharacteristic
32C53×#473C2212yescomplemented
33C53×#472C2212yescomplemented
34C53×#471—22no—
35C2×#108×C53C2211yescharacteristic
36C53×#413—21no—
37C22×D212—21no—
38#13133242—24no—
39#13133242—42no—
40C2×#11719924—41no—
41C2×D424—21no—
42C2×D424—21no—
43C2×D424—21no—
44C2×D424—21no—
45C2×D424—21no—
46C2×D424C2312yes—
47C2×#11719922—41no—
48C2×#11719917—21no—
49#13133160—24no—
50#13133158—24no—
51#13133157—21no—
52#13133157—24no—
53#13133157D412yes—
54#13133157D412yes—
55#13133157C2312yes—
56C2×#11719908—21no—
57C2×#11719908—24no—
58C2×#11719908C2312yes—
59#13133153—21no—
60#13133153C2312yes—
61#13133133—24no—
62C2×#41×C53—21no—
63C2×#41×C53D411yescharacteristic
64#137×C53—22no—
65#137×C53—41no—
66AGL(1,ℤ/8)×C53—22no—
67AGL(1,ℤ/8)×C53—41no—
68#135×C53—21no—
69#135×C53—21no—
70#135×C53—21no—
71#135×C53—21no—
72#135×C53—42no—
73C2×Q16×C53C2311yescharacteristic
74C2×SD16×C53—21no—
75C2×D8×C53C2311yescharacteristic
76#131×C53—21no—
77#131×C53—21no—
78#131×C53D412yes—
79#131×C53C2312yes—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53D411yescharacteristic
83C22×C8×C53—21no—
84C22×C8×C53C2311yescharacteristic
85#108×C53—21no—
86#108×C53C2312yes—
87#101×C53—22no—
88#100×C53—22no—
89C23×D53—41no—
90#41×C53—21no—
91#41×C53—21no—
92#41×C53—41no—
93#41×C53—41no—
94#41×C53—41no—
95#41×C53—41no—
96#41×C53D812yes—
97#41×C53C2×D412yes—
98C2×Q8×C53—21no—
99C2×Q8×C53D811yescharacteristic
100C2×D4×C53—21no—
101C2×D4×C53—21no—
102C2×D4×C53—41no—
103C2×D4×C53D811yescharacteristic
104C22×C4×C53—21no—
105C22×C4×C53—41no—
106C22×C4×C53C2×D411yescharacteristic
107C2×D212—22no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—41no—
111C2×D212—41no—
112C2×D212—41no—
113Q16×C53—21no—
114Q16×C53—21no—
115Q16×C53—42no—
116SD16×C53—42no—
117SD16×C53—42no—
118SD16×C53—41no—
119SD16×C53—41no—
120D8×C53—21no—
121D8×C53—21no—
122D8×C53—42no—
123M16×C53—21no—
124M16×C53—22no—
125M16×C53—22no—
126M16×C53—22no—
127M16×C53—41no—
128M16×C53C2×D412yes—
129C2×C8×C53—21no—
130C2×C8×C53—22no—
131C2×C8×C53—22no—
132C2×C8×C53—41no—
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53#4111yescharacteristic
137C2×C8×C53#4111yescharacteristic
138C2×C8×C53C2411yescharacteristic
139#11719924—44no—
140#11719922—44no—
141#11719917—22no—
142#11719917—41no—
143C2×#88322—41no—
144D424—22no—
145D424—22no—
146D424—41no—
147D424—41no—
148#11719908—22no—
149#11719908C2×D414yes—
150C23×C53—21no—
151C23×C53—41no—
152C23×C53#13511yescharacteristic
153C22×D53—42no—
154C22×D53—81no—
155C22×D53—81no—
156C22×D53—162no—
157Q8×C53—21no—
158Q8×C53—41no—
159Q8×C53—41no—
160Q8×C53C2×D812yes—
161D4×C53—22no—
162D4×C53—22no—
163D4×C53—21no—
164D4×C53—41no—
165D4×C53—41no—
166D4×C53—81no—
167D4×C53—81no—
168D4×C53C2×D812yes—
169C2×C4×C53—21no—
170C2×C4×C53—22no—
171C2×C4×C53—22no—
172C2×C4×C53—41no—
173C2×C4×C53—81no—
174C2×C4×C53—81no—
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53C2×D811yescharacteristic
178C2×C4×C53C2×D811yescharacteristic
179C2×C4×C53#13511yescharacteristic
180C2×C4×C53C22×D411yescharacteristic
181D212—42no—
182D212—42no—
183D212—81no—
184D212—81no—
185C8×C53—22no—
186C8×C53—42no—
187C8×C53#12111yescharacteristic
188C8×C53#12111yescharacteristic
189C8×C53C22×D411yescharacteristic
190C8×C53C2×#4111yescharacteristic
191#88322—42no—
192#33618—531nomaximal, Sylow
193C22×C53—22no—
194C22×C53—41no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—161no—
198C22×C53#46611yescharacteristic
199C22×C53#46611yescharacteristic
200C22×C53C2×#13511yescharacteristic
201C2×D53—82no—
202C2×D53—162no—
203C2×D53—162no—
204C2×D53—162no—
205C4×C53—22no—
206C4×C53—81no—
207C4×C53#46611yescharacteristic
208C4×C53#46611yescharacteristic
209C4×C53C2×#12111yescharacteristic
210C4×C53C22×D811yescharacteristic
211C2×#509—531no—
212#3407—531no—
213#3067D5311yescharacteristic, complemented
214C2×#472—531no—
215#2223—534no—
216#2221—534no—
217#2161—531no—
218#2148—531no—
219C2×#361—531no—
220C2×C53—21no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—42no—
224C2×C53—161no—
225C2×C53C2×#46611yescharacteristic
226D53—162no—
227#575—21no—
228C2×AGL(1,ℤ/8)—1061no—
229C2×#135C2×D5311yescharacteristic
230C22×D8—531no—
231C2×#131C2×D5311yescharacteristic
232#509—1062no—
233#509—1062no—
234C2×SD32—532no—
235C2×D16—532no—
236#504—534no—
237#504—1062no—
238C2×M32—531no—
239C2×M32—531no—
240C2×M32—1062no—
241#473C2×D5312yescomplemented
242#472—534no—
243#472—1062no—
244#472C2×D5312yescomplemented
245#471—22no—
246C2×#108—531no—
247C2×#108C2×D5311yescharacteristic
248#413—21no—
249C2×#100—1061no—
250#368—534no—
251#362—532no—
252#361—532no—
253#361—534no—
254C53#3361811yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C2×#41C22⋊D5311yescharacteristic
257C22×D4—1061no—
258#137—22no—
259#137—41no—
260AGL(1,ℤ/8)—22no—
261AGL(1,ℤ/8)—41no—
262AGL(1,ℤ/8)—1064no—
263AGL(1,ℤ/8)—2122no—
264#135—21no—
265#135—21no—
266#135—21no—
267#135—21no—
268#135—42no—
269C2×Q16C22×D5311yescharacteristic
270C2×SD16—21no—
271C2×SD16—2121no—
272C2×D8—532no—
273C2×D8—1061no—
274C2×D8—1061no—
275C2×D8—1061no—
276C2×D8—1061no—
277C2×D8—1061no—
278C2×D8—2121no—
279C2×D8C22×D5311yescharacteristic
280#131—21no—
281#131—21no—
282#131C22⋊D5312yes—
283#131C22×D5312yes—
284C2×M16—21no—
285C2×M16—21no—
286C2×M16—1061no—
287C2×M16C22⋊D5311yescharacteristic
288C22×C8—21no—
289C22×C8C22×D5311yescharacteristic
290SD32—1064no—
291D16—1064no—
292M32—532no—
293M32—532no—
294M32—532no—
295M32—1064no—
296M32—1061no—
297C2×C16—532no—
298C2×C16—1064no—
299C2×C16—1061no—
300#108—21no—
301#108—532no—
302#108—1061no—
303#108C22×D5312yes—
304#101—22no—
305#100—22no—
306#100—1064no—
307C24—2121no—
308#41—21no—
309#41—21no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41#1171992212yes—
315#41C2×C22⋊D5312yes—
316C2×Q8—21no—
317C2×Q8#1171992211yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—1062no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4#1171992211yescharacteristic
328C22×C4—21no—
329C22×C4—41no—
330C22×C4C2×C22⋊D5311yescharacteristic
331Q16—21no—
332Q16—21no—
333Q16—42no—
334SD16—42no—
335SD16—42no—
336SD16—41no—
337SD16—41no—
338SD16—2124no—
339D8—21no—
340D8—21no—
341D8—42no—
342D8—1062no—
343D8—1062no—
344D8—2124no—
345D8—2121no—
346D8—2121no—
347M16—21no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—41no—
352M16—1062no—
353M16—2121no—
354M16C2×C22⋊D5312yes—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—2121no—
360C2×C8D4×D5311yescharacteristic
361C2×C8D4×D5311yescharacteristic
362C2×C8#1171994711yescharacteristic
363C2×C8#1171994711yescharacteristic
364C2×C8C2×C22⋊D5311yescharacteristic
365C2×C8C23×D5311yescharacteristic
366C16—534no—
367C16—1062no—
368C23—21no—
369C23—41no—
370C23—8482no—
371C23—4241no—
372C23—4241no—
373C23—2122no—
374C23#1313323711yescharacteristic
375Q8—21no—
376Q8—41no—
377Q8—41no—
378Q8C2×#1171992212yes—
379D4—22no—
380D4—22no—
381D4—21no—
382D4—41no—
383D4—41no—
384D4—81no—
385D4—81no—
386D4—4241no—
387D4—4241no—
388D4—2122no—
389D4—2122no—
390D4C2×#1171992212yes—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4#1313318911yescharacteristic
398C2×C4#1313318911yescharacteristic
399C2×C4C2×#1171992211yescharacteristic
400C2×C4C2×#1171992211yescharacteristic
401C2×C4#1313323711yescharacteristic
402C2×C4C22×C22⋊D5311yescharacteristic
403C8—22no—
404C8—42no—
405C8—2122no—
406C8#1313318911yescharacteristic
407C8#1313318911yescharacteristic
408C8C2×D4×D5311yescharacteristic
409C8C2×#1171994711yescharacteristic
410C22—22no—
411C22—41no—
412C22—81no—
413C22—81no—
414C22—161no—
415C22—8482no—
416C22—8482no—
417C22—8482no—
418C22—4242no—
419C22#1681085711yescharacteristic
420C22#1681085711yescharacteristic
421C22C2×#1313323711yescharacteristic
422C4—22no—
423C4—81no—
424C4#1681085711yescharacteristic
425C4#1681085711yescharacteristic
426C4C2×#1313318911yescharacteristic
427C4C22×#1171992211yescharacteristic
428C2—21no—
429C2—21no—
430C2—21no—
431C2—42no—
432C2—161no—
433C2—8482no—
434C2C2×#1681085711yescharacteristic
subgroupG/H
centerC2C2×#16810857
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3067C2
socleC2×C53C2×#466
hypercenter#3067D53
hypocenterC53#33618
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C2×#16811058C2semidirect
2. #20287818C2semidirect
3. C2×#16810893C2semidirect
4. #20276847C2semidirect
5. #20276833C2semidirect
6. #20276509C2semidirect
7. #20276464C2semidirect
8. C2×#16810435C2semidirect
9. C53×#3067C2semidirect
14. C2×#13133160C22semidirect
15. C2×#13133158C22semidirect
17. #16811040C22semidirect
25. #16810452C22semidirect
26. #16810439C22semidirect
27. #16810436C22semidirect
28. #16810435C22semidirect
32. C53×#473C22semidirect
33. C53×#472C22semidirect
213. #3067D53semidirect
241. #473C2×D53semidirect
244. #472C2×D53semidirect
254. C53#33618semidirect

Maximal subgroups

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

subgroupindex
C2×#168110582
#202878182
C2×#168108932
#202768472
#202768332
#202765092
#202764642
C2×#168104352
C53×#30672
#3361853