Finite Group Databasebeta

← #27538481

Subgroups — #27538481

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
2C2×#16810741C211yesmaximal, characteristic, complemented
3#20276793C212yesmaximal, complemented
4#20276788C212yesmaximal, complemented
5#20276781C214yesmaximal, complemented
6C2×#16810444C211yesmaximal, characteristic, complemented
7#20276464C212yesmaximal, complemented
8C53×#3147C211yesmaximal, characteristic, complemented
9C2×#13133239C2211yescharacteristic
10C2×#13133223—21no—
11#16811064C2214yes—
12#16811063—22no—
13#16811059C2214yes—
14#16811058—22no—
15#16811053—24no—
16C2×#13133157C2212yes—
17#16810741—21no—
18#16810741—21no—
19#16810741C2212yes—
20#16810741C2212yes—
21C2×#13133153C2211yescharacteristic
22C2×#13133145—21no—
23#16810444C2212yescomplemented
24#16810444C2212yescomplemented
25#16810439C2214yescomplemented
26#16810436C2214yescomplemented
27C2×#135×C53—21no—
28C2×#135×C53C2212yes—
29C53×#532—21no—
30C53×#495—21no—
31C53×#495C2212yescomplemented
32C53×#494C2211yescharacteristic, complemented
33C53×#493C2211yescharacteristic, complemented
34C53×#492—21no—
35C2×C4×C8×C53C2211yescharacteristic
36C2×#11719945—21no—
37#13133239—21no—
38#13133239—21no—
39#13133239D412yescomplemented
40#13133239C2312yes—
41C2×Dic212D411yescharacteristic, complemented
42C2×#11719913—21no—
43C2×D424D411yescharacteristic, complemented
44#13133223—22no—
45#13133223—22no—
46#13133223—41no—
47#13133223—41no—
48C2×#11719917—21no—
49C2×#11719912—41no—
50C2×C8×D53—41no—
51#13133163—24no—
52#13133161—24no—
53#13133160—24no—
54#13133158—24no—
55#13133157—22no—
56#13133157C2314yes—
57C2×#11719908—22no—
58#13133153—21no—
59#13133153C2312yes—
60#13133145—22no—
61#13133145—22no—
62C2×#41×C53—22no—
63#135×C53—22no—
64#135×C53—22no—
65#135×C53—22no—
66#135×C53—22no—
67#135×C53—41no—
68#135×C53—41no—
69#135×C53—41no—
70#135×C53—41no—
71C2×Q16×C53—21no—
72C2×Q16×C53C2312yes—
73C2×SD16×C53—22no—
74C2×SD16×C53—22no—
75C2×D8×C53—21no—
76C2×D8×C53C2312yes—
77C22×C8×C53D411yescharacteristic
78C22×C8×C53C2311yescharacteristic
79C4⋊Q8×C53—21no—
80#127×C53—21no—
81#124×C53—22no—
82#121×C53—42no—
83C4×D4×C53—42no—
84C2×C42×C53D411yescharacteristic
85C4×C8×C53D411yescharacteristic
86C4×C8×C53D411yescharacteristic
87C4×C8×C53C2311yescharacteristic
88C4×C8×C53C2311yescharacteristic
89#41×C53—42no—
90#41×C53—42no—
91#41×C53—42no—
92#41×C53—42no—
93C2×Q8×C53—22no—
94C2×D4×C53—22no—
95C2×D4×C53—42no—
96C2×D4×C53—42no—
97C22×C4×C53—21no—
98C22×C4×C53—42no—
99C22×C4×C53C2×D411yescharacteristic
100C2×C22⋊D53—41no—
101#11719945—22no—
102#11719945—22no—
103#11719945—41no—
104#11719945—41no—
105C2×D212—21no—
106C2×C4×D53—41no—
107C2×Dic106—21no—
108Q16×C53—22no—
109Q16×C53—22no—
110Q16×C53—41no—
111Q16×C53—41no—
112SD16×C53—42no—
113SD16×C53—42no—
114SD16×C53—42no—
115SD16×C53—42no—
116D8×C53—22no—
117D8×C53—22no—
118D8×C53—41no—
119D8×C53—41no—
120C2×C8×C53D811yescharacteristic
121C2×C8×C53D811yescharacteristic
122C2×C8×C53D811yescharacteristic
123C2×C8×C53D811yescharacteristic
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53C2×D411yescharacteristic
128C2×C8×C53C2×D411yescharacteristic
129C2×C8×C53#4111yescharacteristic
130C2×C8×C53#4111yescharacteristic
131C2×C8×C53C2411yescharacteristic
132C4⋊C4×C53—42no—
133C22⋊C4×C53—44no—
134C42×C53—21no—
135C42×C53C2×D411yescharacteristic
136C42×C53C2×D411yescharacteristic
137#11719917—22no—
138#11719917—41no—
139C2×#88322—41no—
140Dic212—21no—
141Dic212C2×D412yes—
142D424—21no—
143D424C2×D412yes—
144#11719913—22no—
145#11719913—22no—
146#11719912—42no—
147#11719912—42no—
148C8×D53—42no—
149C8×D53—42no—
150#11719908—24no—
151C23×C53—42no—
152C23×C53AGL(1,ℤ/8)11yescharacteristic
153C22×D53—41no—
154Q8×C53—42no—
155Q8×C53—42no—
156D4×C53—42no—
157D4×C53—42no—
158D4×C53—84no—
159D4×C53—82no—
160D4×C53—82no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—82no—
166C2×C4×C53—82no—
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53C2×D811yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C53C22×D411yescharacteristic
173C22⋊D53—42no—
174C22⋊D53—42no—
175C2×Dic53—41no—
176D212—22no—
177D212—41no—
178C4×D53—42no—
179C4×D53—42no—
180Dic106—22no—
181Dic106—41no—
182C8×C53#12111yescharacteristic
183C8×C53#12111yescharacteristic
184C8×C53C2×D811yescharacteristic
185C8×C53C2×D811yescharacteristic
186C8×C53C2×D811yescharacteristic
187C8×C53C2×D811yescharacteristic
188C8×C53C22×D411yescharacteristic
189C8×C53C2×#4111yescharacteristic
190#88322—42no—
191#33594—531nomaximal, Sylow
192C22×C53—41no—
193C22×C53—84no—
194C22×C53—82no—
195C22×C53#45911yescharacteristic
196C22×C53#45911yescharacteristic
197C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
198C2×D53—42no—
199C2×D53—82no—
200C4×C53—22no—
201C4×C53—41no—
202C4×C53—82no—
203C4×C53#45911yescharacteristic
204C4×C53#45911yescharacteristic
205C4×C53C2×#12111yescharacteristic
206C4×C53C22×D811yescharacteristic
207Dic53—42no—
208#3414—532no—
209#3147D5311yescharacteristic, complemented
210C2×#443—531no—
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—162no—
220C2×C53C2×#45911yescharacteristic
221D53—82no—
222C2×#135—21no—
223C2×#135—531no—
224C2×#135C2×D5312yes—
225C2×#131—1061no—
226#532—21no—
227#510—534no—
228#510—1062no—
229#509—534no—
230#509—1062no—
231#508—1064no—
232C2×M32—532no—
233#495—21no—
234#495C2×D5312yescomplemented
235#494C2×D5311yescharacteristic, complemented
236#493C2×D5311yescharacteristic, complemented
237#492—21no—
238#443—532no—
239#443—532no—
240#443—1061no—
241#443—1061no—
242C2×#108—531no—
243C2×#104—1061no—
244C2×C4×C8C2×D5311yescharacteristic
245#364—532no—
246#364—532no—
247#362—534no—
248#361—534no—
249C53#3359411yescharacteristic, complemented, Sylow
250C2×#41—22no—
251C2×#41—1061no—
252#135—22no—
253#135—22no—
254#135—22no—
255#135—22no—
256#135—41no—
257#135—41no—
258#135—41no—
259#135—41no—
260#135—532no—
261#135—532no—
262#135—1061no—
263#135—1061no—
264C2×Q16—21no—
265C2×Q16—531no—
266C2×Q16C22×D5312yes—
267C2×SD16—22no—
268C2×SD16—22no—
269C2×SD16—1061no—
270C2×D8—21no—
271C2×D8—531no—
272C2×D8C22×D5312yes—
273#131—1062no—
274#131—1062no—
275#131—2121no—
276#131—2121no—
277C2×M16—1061no—
278C2×M16—2121no—
279C22×C8—2121no—
280C22×C8C22⋊D5311yescharacteristic
281C22×C8C22×D5311yescharacteristic
282C4⋊Q8—21no—
283#127—21no—
284#124—22no—
285#121—42no—
286C4×D4—42no—
287C2×C42C22⋊D5311yescharacteristic
288Q32—1064no—
289SD32—1064no—
290SD32—1064no—
291D16—1064no—
292M32—534no—
293M32—1062no—
294C2×C16—1062no—
295#108—532no—
296#108—1061no—
297#104—1062no—
298#104—1062no—
299C4×C8C22⋊D5311yescharacteristic
300C4×C8C22⋊D5311yescharacteristic
301C4×C8C22×D5311yescharacteristic
302C4×C8C22×D5311yescharacteristic
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307#41—1062no—
308#41—1062no—
309#41—2121no—
310#41—2121no—
311C2×Q8—22no—
312C2×Q8—1061no—
313C2×D4—22no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—1061no—
317C2×D4—2121no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—2121no—
321C22×C4C2×C22⋊D5311yescharacteristic
322Q16—22no—
323Q16—22no—
324Q16—41no—
325Q16—41no—
326Q16—532no—
327Q16—1061no—
328SD16—42no—
329SD16—42no—
330SD16—42no—
331SD16—42no—
332SD16—1062no—
333SD16—1062no—
334D8—22no—
335D8—22no—
336D8—41no—
337D8—41no—
338D8—532no—
339D8—1061no—
340M16—1062no—
341M16—2121no—
342M16—2122no—
343M16—2122no—
344C2×C8—2121no—
345C2×C8—2122no—
346C2×C8—2122no—
347C2×C8#1171992211yescharacteristic
348C2×C8#1171992211yescharacteristic
349C2×C8#1171992211yescharacteristic
350C2×C8#1171992211yescharacteristic
351C2×C8D4×D5311yescharacteristic
352C2×C8D4×D5311yescharacteristic
353C2×C8#1171994711yescharacteristic
354C2×C8#1171994711yescharacteristic
355C2×C8C2×C22⋊D5311yescharacteristic
356C2×C8C2×C22⋊D5311yescharacteristic
357C2×C8C2×C22⋊D5311yescharacteristic
358C2×C8C23×D5311yescharacteristic
359C4⋊C4—42no—
360C22⋊C4—44no—
361C42—21no—
362C42C2×C22⋊D5311yescharacteristic
363C42C2×C22⋊D5311yescharacteristic
364C16—1064no—
365C23—42no—
366C23—2121no—
367C23#1313324311yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—1062no—
371Q8—2121no—
372D4—42no—
373D4—42no—
374D4—84no—
375D4—82no—
376D4—82no—
377D4—1062no—
378D4—2122no—
379D4—2122no—
380D4—2121no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—2121no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4#1313318811yescharacteristic
391C2×C4#1313318811yescharacteristic
392C2×C4C2×#1171992211yescharacteristic
393C2×C4C2×#1171992211yescharacteristic
394C2×C4#1313324311yescharacteristic
395C2×C4C22×C22⋊D5311yescharacteristic
396C8—2122no—
397C8#1313318811yescharacteristic
398C8#1313318811yescharacteristic
399C8C2×#1171992211yescharacteristic
400C8C2×#1171992211yescharacteristic
401C8C2×#1171992211yescharacteristic
402C8C2×#1171992211yescharacteristic
403C8C2×D4×D5311yescharacteristic
404C8C2×#1171994711yescharacteristic
405C22—41no—
406C22—84no—
407C22—82no—
408C22—4242no—
409C22—2122no—
410C22#1681081511yescharacteristic
411C22#1681081511yescharacteristic
412C22C2×#1313324311yescharacteristic
413C4—22no—
414C4—41no—
415C4—82no—
416C4—2122no—
417C4#1681081511yescharacteristic
418C4#1681081511yescharacteristic
419C4C2×#1313318811yescharacteristic
420C4C22×#1171992211yescharacteristic
421C2—21no—
422C2—21no—
423C2—21no—
424C2—162no—
425C2—4242no—
426C2C2×#1681081511yescharacteristic
subgroupG/H
centerC4#16810815
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3147C2
socleC2×C53C2×#459
hypercenter#3147D53
hypocenterC53#33594
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #20287849C2semidirect
2. C2×#16810741C2semidirect
3. #20276793C2semidirect
4. #20276788C2semidirect
5. #20276781C2semidirect
6. C2×#16810444C2semidirect
7. #20276464C2semidirect
8. C53×#3147C2semidirect
23. #16810444C22semidirect
24. #16810444C22semidirect
25. #16810439C22semidirect
26. #16810436C22semidirect
31. C53×#495C22semidirect
32. C53×#494C22semidirect
33. C53×#493C22semidirect
39. #13133239D4semidirect
41. C2×Dic212D4semidirect
43. C2×D424D4semidirect
209. #3147D53semidirect
234. #495C2×D53semidirect
235. #494C2×D53semidirect
236. #493C2×D53semidirect
249. C53#33594semidirect

Maximal subgroups

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

subgroupindex
#202878492
C2×#168107412
#202767932
#202767882
#202767812
C2×#168104442
#202764642
C53×#31472
#3359453