Finite Group Databasebeta

← #27538485

Subgroups — #27538485

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287849C212yesmaximal, complemented
2#20285143C211yesmaximal, characteristic, complemented
3#20276796C212yesmaximal, complemented
4#20276791C212yesmaximal, complemented
5#20276784C214yesmaximal, complemented
6C2×#16810444C211yesmaximal, characteristic, complemented
7#20276462C212yesmaximal, complemented
8C2×C53×#443C211yesmaximal, characteristic, complemented
9C2×#13133239—21no—
10C2×#13133239C2212yes—
11#16811173—21no—
12#16811064—22no—
13#16811063C2214yes—
14#16811059—22no—
15#16811058C2214yes—
16#16811053—24no—
17C2×#13133157C2212yes—
18#16811007—21no—
19#16811007C2212yescomplemented
20#16811003C2211yescharacteristic, complemented
21#16811000C2211yescharacteristic, complemented
22#16810996—21no—
23#16810444C2212yescomplemented
24#16810444C2212yescomplemented
25#16810438C2214yescomplemented
26#16810435C2214yescomplemented
27C2×#135×C53C2211yescharacteristic
28C2×#131×C53—21no—
29C53×#443—21no—
30C53×#443—21no—
31C53×#443C2212yes—
32C53×#443C2212yes—
33C2×#108×C53C2211yescharacteristic
34C2×#104×C53—21no—
35C2×C4×C8×C53C2211yescharacteristic
36C2×#11719945—22no—
37#13133239—22no—
38#13133239—22no—
39#13133239—22no—
40#13133239—22no—
41#13133239—41no—
42#13133239—41no—
43#13133239—41no—
44#13133239—41no—
45C2×Dic212—21no—
46C2×Dic212C2312yes—
47C2×#11719913—22no—
48C2×#11719913—22no—
49C2×D424—21no—
50C2×D424C2312yes—
51#13133217—21no—
52#13133214—21no—
53#13133205—22no—
54#13133191—42no—
55C4×D212—42no—
56#13133163—24no—
57#13133161—24no—
58#13133160—24no—
59#13133158—24no—
60#13133157—22no—
61#13133157C2314yes—
62C2×#11719908—22no—
63C2×#41×C53—21no—
64#135×C53—21no—
65#135×C53—21no—
66#135×C53D412yescomplemented
67#135×C53C2312yes—
68C2×Q16×C53D411yescharacteristic, complemented
69C2×SD16×C53—21no—
70C2×D8×C53D411yescharacteristic, complemented
71#131×C53—22no—
72#131×C53—22no—
73#131×C53—41no—
74#131×C53—41no—
75C2×M16×C53—21no—
76C2×M16×C53—41no—
77C22×C8×C53—41no—
78C22×C8×C53D411yescharacteristic
79C22×C8×C53C2311yescharacteristic
80C2×C42×C53D411yescharacteristic
81#108×C53—21no—
82#108×C53C2312yes—
83#104×C53—22no—
84#104×C53—22no—
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89#41×C53—22no—
90#41×C53—22no—
91#41×C53—41no—
92#41×C53—41no—
93C2×Q8×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—41no—
96C22×C4×C53—21no—
97C22×C4×C53—41no—
98C22×C4×C53C2×D411yescharacteristic
99C2×C22⋊D53—42no—
100#11719945—42no—
101#11719945—42no—
102#11719945—42no—
103#11719945—42no—
104C2×D212—22no—
105C2×D212—42no—
106C2×C4×D53—42no—
107C2×Dic106—22no—
108Q16×C53—21no—
109Q16×C53C2×D412yes—
110SD16×C53—22no—
111SD16×C53—22no—
112D8×C53—21no—
113D8×C53C2×D412yes—
114M16×C53—22no—
115M16×C53—41no—
116M16×C53—42no—
117M16×C53—42no—
118C2×C8×C53—41no—
119C2×C8×C53—42no—
120C2×C8×C53—42no—
121C2×C8×C53D811yescharacteristic
122C2×C8×C53D811yescharacteristic
123C2×C8×C53D811yescharacteristic
124C2×C8×C53D811yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53C2×D411yescharacteristic
128C2×C8×C53C2×D411yescharacteristic
129C2×C8×C53C2×D411yescharacteristic
130C2×C8×C53#4111yescharacteristic
131C2×C8×C53#4111yescharacteristic
132C2×C8×C53C2411yescharacteristic
133C42×C53—21no—
134C42×C53C2×D411yescharacteristic
135C42×C53C2×D411yescharacteristic
136#11719921—44no—
137C4⋊Dic53—42no—
138Dic212—22no—
139Dic212—22no—
140Dic212—41no—
141Dic212—41no—
142D424—22no—
143D424—22no—
144D424—41no—
145D424—41no—
146#11719913—42no—
147#11719913—42no—
148#11719913—42no—
149#11719913—42no—
150#11719908—24no—
151C23×C53—41no—
152C23×C53AGL(1,ℤ/8)11yescharacteristic
153C22×D53—42no—
154Q8×C53—22no—
155Q8×C53—41no—
156D4×C53—22no—
157D4×C53—42no—
158D4×C53—42no—
159D4×C53—41no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53C2×D811yescharacteristic
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53C22×D411yescharacteristic
172C22⋊D53—82no—
173C22⋊D53—82no—
174C2×Dic53—42no—
175D212—42no—
176D212—42no—
177D212—84no—
178C4×D53—82no—
179C4×D53—82no—
180Dic106—42no—
181Dic106—42no—
182C8×C53—42no—
183C8×C53#12111yescharacteristic
184C8×C53#12111yescharacteristic
185C8×C53C2×D811yescharacteristic
186C8×C53C2×D811yescharacteristic
187C8×C53C2×D811yescharacteristic
188C8×C53C2×D811yescharacteristic
189C8×C53C22×D411yescharacteristic
190C8×C53C2×#4111yescharacteristic
191#33594—531nomaximal, Sylow
192C22×C53—41no—
193C22×C53—42no—
194C22×C53—82no—
195C22×C53#45911yescharacteristic
196C22×C53#45911yescharacteristic
197C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
198C2×D53—84no—
199C2×D53—82no—
200C4×C53—22no—
201C4×C53—41no—
202C4×C53—42no—
203C4×C53#45911yescharacteristic
204C4×C53#45911yescharacteristic
205C4×C53C2×#12111yescharacteristic
206C4×C53C22×D811yescharacteristic
207Dic53—82no—
208#3414—532no—
209#3147—531no—
210C2×#443D5311yescharacteristic, complemented
211#2214—532no—
212#2213—532no—
213#2212—534no—
214C2×#364—531no—
215#2148—532no—
216C2×C53—21no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—82no—
220C2×C53C2×#45911yescharacteristic
221D53—162no—
222C2×#135—532no—
223C2×#135—1061no—
224C2×#135C2×D5311yescharacteristic
225C2×#131—21no—
226#532—1061no—
227#510—534no—
228#510—1062no—
229#509—534no—
230#509—1062no—
231#508—1064no—
232C2×M32—532no—
233#495—532no—
234#495—1061no—
235#494—531no—
236#493—531no—
237#492—1061no—
238#443—21no—
239#443—21no—
240#443C2×D5312yes—
241#443C2×D5312yes—
242C2×#108C2×D5311yescharacteristic
243C2×#104—21no—
244C2×C4×C8C2×D5311yescharacteristic
245#364—532no—
246#364—532no—
247#362—534no—
248#361—534no—
249C53#3359411yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C2×#41—1062no—
252#135—21no—
253#135—21no—
254#135—1062no—
255#135—1062no—
256#135—1062no—
257#135—1062no—
258#135—2121no—
259#135—2121no—
260#135—2121no—
261#135—2121no—
262#135D21212yescomplemented
263#135C22×D5312yes—
264C2×Q16—532no—
265C2×Q16—1061no—
266C2×Q16D21211yescharacteristic, complemented
267C2×SD16—21no—
268C2×SD16—1062no—
269C2×SD16—1062no—
270C2×D8—532no—
271C2×D8—1061no—
272C2×D8D21211yescharacteristic, complemented
273#131—22no—
274#131—22no—
275#131—41no—
276#131—41no—
277C2×M16—21no—
278C2×M16—41no—
279C22×C8—41no—
280C22×C8C22⋊D5311yescharacteristic
281C22×C8C22×D5311yescharacteristic
282C4⋊Q8—1061no—
283#127—1061no—
284#124—1062no—
285#121—2122no—
286C4×D4—2122no—
287C2×C42C22⋊D5311yescharacteristic
288Q32—1064no—
289SD32—1064no—
290SD32—1064no—
291D16—1064no—
292M32—534no—
293M32—1062no—
294C2×C16—1062no—
295#108—21no—
296#108C22×D5312yes—
297#104—22no—
298#104—22no—
299C4×C8C22⋊D5311yescharacteristic
300C4×C8C22⋊D5311yescharacteristic
301C4×C8C22×D5311yescharacteristic
302C4×C8C22×D5311yescharacteristic
303#41—22no—
304#41—22no—
305#41—41no—
306#41—41no—
307#41—2122no—
308#41—2122no—
309#41—2122no—
310#41—2122no—
311C2×Q8—21no—
312C2×Q8—1062no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—1062no—
316C2×D4—2122no—
317C2×D4—2122no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—2122no—
321C22×C4C2×C22⋊D5311yescharacteristic
322Q16—21no—
323Q16—1062no—
324Q16—1062no—
325Q16—2121no—
326Q16—2121no—
327Q16C2×D21212yes—
328SD16—22no—
329SD16—22no—
330SD16—2122no—
331SD16—2122no—
332SD16—2122no—
333SD16—2122no—
334D8—21no—
335D8—1062no—
336D8—1062no—
337D8—2121no—
338D8—2121no—
339D8C2×D21212yes—
340M16—22no—
341M16—41no—
342M16—42no—
343M16—42no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8#1171992211yescharacteristic
348C2×C8#1171992211yescharacteristic
349C2×C8#1171992211yescharacteristic
350C2×C8#1171992211yescharacteristic
351C2×C8C2×D21211yescharacteristic
352C2×C8C2×D21211yescharacteristic
353C2×C8#1171994511yescharacteristic
354C2×C8#1171994511yescharacteristic
355C2×C8C2×C22⋊D5311yescharacteristic
356C2×C8C2×C22⋊D5311yescharacteristic
357C2×C8C2×C22⋊D5311yescharacteristic
358C2×C8C23×D5311yescharacteristic
359C4⋊C4—2122no—
360C22⋊C4—2124no—
361C42—21no—
362C42C2×C22⋊D5311yescharacteristic
363C42C2×C22⋊D5311yescharacteristic
364C16—1064no—
365C23—41no—
366C23—2122no—
367C23#1313324211yescharacteristic
368Q8—22no—
369Q8—41no—
370Q8—2122no—
371Q8—2122no—
372D4—22no—
373D4—42no—
374D4—42no—
375D4—41no—
376D4—4244no—
377D4—4242no—
378D4—4242no—
379D4—2122no—
380D4—2122no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—4242no—
388C2×C4—4242no—
389C2×C4—2122no—
390C2×C4#1313319111yescharacteristic
391C2×C4#1313319111yescharacteristic
392C2×C4C2×#1171992211yescharacteristic
393C2×C4C2×#1171992211yescharacteristic
394C2×C4#1313324211yescharacteristic
395C2×C4C22×C22⋊D5311yescharacteristic
396C8—42no—
397C8#1313319111yescharacteristic
398C8#1313319111yescharacteristic
399C8C2×#1171992211yescharacteristic
400C8C2×#1171992211yescharacteristic
401C8C2×#1171992211yescharacteristic
402C8C2×#1171992211yescharacteristic
403C8C22×D21211yescharacteristic
404C8C2×#1171994511yescharacteristic
405C22—41no—
406C22—42no—
407C22—82no—
408C22—4244no—
409C22—4242no—
410C22#1681081811yescharacteristic
411C22#1681081811yescharacteristic
412C22C2×#1313324211yescharacteristic
413C4—22no—
414C4—41no—
415C4—42no—
416C4—4242no—
417C4#1681081811yescharacteristic
418C4#1681081811yescharacteristic
419C4C2×#1313319111yescharacteristic
420C4C22×#1171992211yescharacteristic
421C2—21no—
422C2—21no—
423C2—21no—
424C2—82no—
425C2—8482no—
426C2C2×#1681081811yescharacteristic
subgroupG/H
centerC4#16810818
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#443C2
socleC2×C53C2×#459
hypercenterC2×#443D53
hypocenterC53#33594
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. #20287849C2semidirect
2. #20285143C2semidirect
3. #20276796C2semidirect
4. #20276791C2semidirect
5. #20276784C2semidirect
6. C2×#16810444C2semidirect
7. #20276462C2semidirect
8. C2×C53×#443C2semidirect
19. #16811007C22semidirect
20. #16811003C22semidirect
21. #16811000C22semidirect
23. #16810444C22semidirect
24. #16810444C22semidirect
25. #16810438C22semidirect
26. #16810435C22semidirect
66. #135×C53D4semidirect
68. C2×Q16×C53D4semidirect
70. C2×D8×C53D4semidirect
210. C2×#443D53semidirect
249. C53#33594semidirect
262. #135D212semidirect
266. C2×Q16D212semidirect
272. C2×D8D212semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202851432
#202767962
#202767912
#202767842
C2×#168104442
#202764622
C2×C53×#4432
#3359453