Finite Group Databasebeta

← #27538359

Subgroups — #27538359

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811053C212yesmaximal, complemented
2#20285143C211yesmaximal, characteristic, complemented
3#20276777C212yesmaximal, complemented
4#20276770C212yesmaximal, complemented
5#20276765C211yesmaximal, characteristic, complemented
6#20276760C211yesmaximal, characteristic, complemented
7#20276755C211yesmaximal, characteristic, complemented
8#20276750C211yesmaximal, characteristic, complemented
9C2×C4⋊#11719908C211yesmaximal, characteristic, complemented
10#20276420C212yesmaximal, complemented
11C4×#135×C53C211yesmaximal, characteristic, complemented
12C2×#13133239—21no—
13C2×#13133239C2212yes—
14#16811173—21no—
15#16811053—28no—
16C2×#13133163C2212yes—
17C2×#13133161C2212yes—
18C2×#13133160C2212yes—
19C2×#13133158C2212yes—
20C22×#11719908C2212yes—
21#16811007—21no—
22#16811007C2212yes—
23#16811003C2211yescharacteristic
24#16811000C2211yescharacteristic
25#16810996—21no—
26C4⋊#11719908C2211yescharacteristic
27C4⋊#11719908C2211yescharacteristic
28C4⋊#11719908C2212yes—
29#16810426C2212yes—
30#16810426C2212yes—
31#16810423C2212yes—
32#16810423C2212yes—
33C2×#135×C53C2211yescharacteristic
34C4×#41×C53—21no—
35C4×Q16×C53C2211yescharacteristic
36C4×Q16×C53C2211yescharacteristic
37C4×SD16×C53—21no—
38C4×SD16×C53—21no—
39C4×D8×C53C2211yescharacteristic
40C4×D8×C53C2211yescharacteristic
41C53×#427C2211yescharacteristic
42C53×#416—21no—
43C2×C4×C8×C53C2211yescharacteristic
44C2×#11719945—22no—
45#13133239—24no—
46#13133239—24no—
47#13133239—44no—
48C2×Dic212—21no—
49C2×Dic212C2312yes—
50C2×#11719913—22no—
51C2×#11719913—22no—
52C2×D424—21no—
53C2×D424C2312yes—
54#13133217—21no—
55#13133214—21no—
56#13133205—22no—
57#13133191—41no—
58#13133191—41no—
59C4×D212—41no—
60C4×D212—41no—
61#13133163—24no—
62#13133161—24no—
63#13133160—24no—
64#13133158—24no—
65C2×#11719908—24no—
66C2×#11719908C2312yes—
67C2×#11719908C2312yes—
68C2×#41×C53—21no—
69#135×C53—22no—
70#135×C53D414yescomplemented
71C2×Q16×C53C2311yescharacteristic
72C2×SD16×C53—21no—
73C2×D8×C53C2311yescharacteristic
74C22×C8×C53D411yescharacteristic
75C22×C8×C53C2311yescharacteristic
76C4×Q8×C53—21no—
77C4×Q8×C53—21no—
78C4×D4×C53—21no—
79C4×D4×C53—21no—
80C4×D4×C53—41no—
81C4×D4×C53—41no—
82#117×C53—21no—
83#117×C53—41no—
84C2×C42×C53—41no—
85C2×C42×C53D411yescharacteristic
86#107×C53C2311yescharacteristic
87#107×C53C2311yescharacteristic
88#106×C53—21no—
89#103×C53—21no—
90#103×C53—21no—
91#102×C53—21no—
92#102×C53—21no—
93C4×C8×C53D411yescharacteristic
94C4×C8×C53D411yescharacteristic
95C4×C8×C53C2311yescharacteristic
96C4×C8×C53C2311yescharacteristic
97#41×C53—24no—
98#41×C53—42no—
99C2×Q8×C53—21no—
100C2×D4×C53—21no—
101C2×D4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53—41no—
105C22×C4×C53D811yescharacteristic
106C22×C4×C53D811yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C2×C22⋊D53—42no—
109#11719945—44no—
110#11719945—44no—
111C2×D212—22no—
112C2×D212—41no—
113C2×D212—41no—
114C2×C4×D53—42no—
115C2×Dic106—22no—
116Q16×C53—21no—
117Q16×C53C2×D412yes—
118SD16×C53—24no—
119D8×C53—21no—
120D8×C53C2×D412yes—
121C2×C8×C53—22no—
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D412yes—
126C2×C8×C53#4112yes—
127C2×C8×C53C2411yescharacteristic
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—41no—
131C4⋊C4×C53—41no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C42×C53—41no—
136C42×C53—41no—
137C42×C53—41no—
138C42×C53D811yescharacteristic
139C42×C53D811yescharacteristic
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142#11719921—42no—
143#11719921—42no—
144C4⋊Dic53—41no—
145C4⋊Dic53—41no—
146Dic212—22no—
147Dic212—22no—
148Dic212—42no—
149D424—22no—
150D424—22no—
151D424—42no—
152#11719913—44no—
153#11719913—44no—
154#11719908—22no—
155#11719908—22no—
156C23×C53—41no—
157C23×C53C2×D811yescharacteristic
158C22×D53—42no—
159Q8×C53—22no—
160Q8×C53—41no—
161D4×C53—22no—
162D4×C53—44no—
163D4×C53—41no—
164C2×C4×C53—22no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—44no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—82no—
171C2×C4×C53—82no—
172C2×C4×C53#12112yes—
173C2×C4×C53C2×D811yescharacteristic
174C2×C4×C53C2×D811yescharacteristic
175C2×C4×C53C2×D811yescharacteristic
176C2×C4×C53C2×D811yescharacteristic
177C2×C4×C53C2×D811yescharacteristic
178C2×C4×C53AGL(1,ℤ/8)12yes—
179C2×C4×C53C22×D411yescharacteristic
180C22⋊D53—84no—
181C2×Dic53—42no—
182D212—42no—
183D212—42no—
184D212—82no—
185D212—82no—
186C4×D53—84no—
187Dic106—42no—
188Dic106—42no—
189C8×C53—21no—
190C8×C53—21no—
191C8×C53#12112yes—
192C8×C53C22×D411yescharacteristic
193C8×C53C2×#4111yescharacteristic
194#33579—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—42no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53#45912yes—
201C22×C53C22×D811yescharacteristic
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—82no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—42no—
209C4×C53—81no—
210C4×C53—81no—
211C4×C53#45912yes—
212C4×C53#50812yescomplemented
213C4×C53C2×#12111yescharacteristic
214C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
215Dic53—82no—
216C2×#508—532no—
217#3147—531no—
218C4×#135D5311yescharacteristic, complemented
219#2211—532no—
220#2210—532no—
221#2209—531no—
222#2208—531no—
223#2207—531no—
224#2206—531no—
225C2×C4⋊C16—531no—
226#2138—532no—
227C2×C53—22no—
228C2×C53—82no—
229C2×C53C2×#45911yescharacteristic
230C2×C53C2×#50811yescharacteristic
231C2×C53#341411yescharacteristic
232D53—162no—
233C2×#135—532no—
234C2×#135—1061no—
235C2×#135C2×D5311yescharacteristic
236#532—1061no—
237C4×#41—21no—
238#508—1068no—
239C2×Q32—532no—
240C2×SD32—532no—
241C2×SD32—532no—
242C2×D16—532no—
243C22×C16—532no—
244#495—532no—
245#495—1061no—
246#494—531no—
247#493—531no—
248#492—1061no—
249C4×Q16C2×D5311yescharacteristic
250C4×Q16C2×D5311yescharacteristic
251C4×SD16—21no—
252C4×SD16—21no—
253C4×D8C2×D5311yescharacteristic
254C4×D8C2×D5311yescharacteristic
255#427C2×D5311yescharacteristic
256#416—21no—
257C2×C4×C8C2×D5311yescharacteristic
258C4⋊C16—531no—
259C4⋊C16—531no—
260C4⋊C16—532no—
261#358—532no—
262#358—532no—
263#357—532no—
264#357—532no—
265C53#3357911yescharacteristic, complemented, Sylow
266C2×#41—21no—
267C2×#41—1062no—
268#135—22no—
269#135—1064no—
270#135—1064no—
271#135—2124no—
272#135D21214yescomplemented
273C2×Q16—532no—
274C2×Q16—1061no—
275C2×Q16C22×D5311yescharacteristic
276C2×SD16—21no—
277C2×SD16—1062no—
278C2×SD16—1062no—
279C2×D8—532no—
280C2×D8—1061no—
281C2×D8C22×D5311yescharacteristic
282C22×C8C22⋊D5311yescharacteristic
283C22×C8C22×D5311yescharacteristic
284C4⋊Q8—1061no—
285#127—1061no—
286#124—1062no—
287#121—2121no—
288#121—2121no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—41no—
294C4×D4—41no—
295C4×D4—2121no—
296C4×D4—2121no—
297#117—21no—
298#117—41no—
299C2×C42—41no—
300C2×C42C22⋊D5311yescharacteristic
301Q32—1064no—
302SD32—1064no—
303SD32—1064no—
304D16—1064no—
305C2×C16—532no—
306C2×C16—532no—
307C2×C16—1064no—
308#107C22×D5311yescharacteristic
309#107C22×D5311yescharacteristic
310#106—21no—
311#103—21no—
312#103—21no—
313#102—21no—
314#102—21no—
315C4×C8C22⋊D5311yescharacteristic
316C4×C8C22⋊D5311yescharacteristic
317C4×C8C22×D5311yescharacteristic
318C4×C8C22×D5311yescharacteristic
319#41—24no—
320#41—42no—
321#41—2124no—
322#41—2124no—
323C2×Q8—21no—
324C2×Q8—1062no—
325C2×D4—21no—
326C2×D4—41no—
327C2×D4—1062no—
328C2×D4—2122no—
329C2×D4—2121no—
330C2×D4—2121no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—2122no—
335C22×C4#1171992211yescharacteristic
336C22×C4#1171992211yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338Q16—21no—
339Q16—1062no—
340Q16—1062no—
341Q16—2122no—
342Q16C2×D21212yes—
343SD16—24no—
344SD16—2124no—
345SD16—2124no—
346D8—21no—
347D8—1062no—
348D8—1062no—
349D8—2122no—
350D8C2×D21212yes—
351C2×C8—22no—
352C2×C8C2×D21212yes—
353C2×C8#1171994512yes—
354C2×C8C2×C22⋊D5311yescharacteristic
355C2×C8C2×C22⋊D5311yescharacteristic
356C2×C8C2×C22⋊D5311yescharacteristic
357C2×C8C23×D5311yescharacteristic
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—2121no—
363C4⋊C4—2121no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—2122no—
368C22⋊C4—2122no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42#1171992211yescharacteristic
373C42#1171992211yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C42C2×C22⋊D5311yescharacteristic
376C16—1062no—
377C16—1062no—
378C23—41no—
379C23—2122no—
380C23C2×#1171992211yescharacteristic
381Q8—22no—
382Q8—41no—
383Q8—2122no—
384Q8—2122no—
385D4—22no—
386D4—44no—
387D4—41no—
388D4—4242no—
389D4—4244no—
390D4—4242no—
391D4—2122no—
392D4—2122no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—4244no—
402C2×C4—2122no—
403C2×C4#1313319112yes—
404C2×C4C2×#1171992211yescharacteristic
405C2×C4C2×#1171992211yescharacteristic
406C2×C4C2×#1171992211yescharacteristic
407C2×C4C2×#1171992211yescharacteristic
408C2×C4C2×#1171992211yescharacteristic
409C2×C4#1313324212yes—
410C2×C4C22×C22⋊D5311yescharacteristic
411C8—21no—
412C8—21no—
413C8#1313319112yes—
414C8C22×D21211yescharacteristic
415C8C2×#1171994511yescharacteristic
416C22—21no—
417C22—21no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—4242no—
422C22—4242no—
423C22—4242no—
424C22#1681081812yes—
425C22C22×#1171992211yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—81no—
431C4—81no—
432C4—4242no—
433C4#1681081812yes—
434C4#1681105312yescomplemented
435C4C2×#1313319111yescharacteristic
436C4C2×#1313324211yescharacteristic
437C2—22no—
438C2—82no—
439C2—8482no—
440C2C2×#1681081811yescharacteristic
441C2C2×#1681105311yescharacteristic
442C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC4×#135×C53C2
socleC22×C53C22×D8
hypercenterC4×#135D53
hypocenterC53#33579
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16811053C2semidirect
2. #20285143C2semidirect
3. #20276777C2semidirect
4. #20276770C2semidirect
5. #20276765C2semidirect
6. #20276760C2semidirect
7. #20276755C2semidirect
8. #20276750C2semidirect
9. C2×C4⋊#11719908C2semidirect
10. #20276420C2semidirect
11. C4×#135×C53C2semidirect
70. #135×C53D4semidirect
212. C4×C53#508semidirect
218. C4×#135D53semidirect
265. C53#33579semidirect
272. #135D212semidirect
434. C4#16811053semidirect

Maximal subgroups

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

subgroupindex
C2×#168110532
#202851432
#202767772
#202767702
#202767652
#202767602
#202767552
#202767502
C2×C4⋊#117199082
#202764202
C4×#135×C532
#3357953