Finite Group Databasebeta

← #27279791

Subgroups — #27279791

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810679C211yesmaximal, characteristic
2#20283068C211yesmaximal, characteristic
3#20274692C211yesmaximal, characteristic
4C2×#16810370C211yesmaximal, characteristic, complemented
5C2×#16810368C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×#13133197—21no—
10C2×C42×D53C2211yescharacteristic
11#16810679—22no—
12#16810679—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133129—21no—
16C2×#13133129—21no—
17C2×#13133127—21no—
18#16810528—21no—
19#16810522—21no—
20#16810522C412yes—
21#16810480—21no—
22#16810480C412yes—
23#16810377—21no—
24#16810370—22no—
25#16810368—22no—
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#384C2211yescharacteristic
29C22×Dic106—21no—
30C22×C4×D53C2×C411yescharacteristic
31C22×C4×D53D411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×#88322—21no—
34C22×#88322—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217—21no—
38#13133217—21no—
39#13133197—42no—
40#13133197—42no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—22no—
43C2×C4⋊Dic53C2×C412yes—
44C2×#11719921—22no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C42×D53—21no—
49C42×D53—21no—
50C42×D53—21no—
51C42×D53—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322—21no—
55C4⋊#88322D412yes—
56#13133129—24no—
57#13133129—42no—
58#13133127—24no—
59#13133124—22no—
60#13133124—22no—
61C22×C8×C53—21no—
62C22×C8×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53C2×C412yes—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53D412yes—
71C23×D53C22⋊C411yescharacteristic
72C22×C4×C53—22no—
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×Dic53—22no—
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C22⋊C411yescharacteristic
80C2×C4×D53—22no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—21no—
87C2×C4×D53—21no—
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×C4×D53—21no—
92C2×C4×D53—21no—
93C2×C4×D53—21no—
94C2×C4×D53—21no—
95C2×C4×D53—41no—
96C2×Dic106—41no—
97C2×Dic106—41no—
98C2×Dic106—41no—
99C2×Dic106—41no—
100C2×Dic106—41no—
101C2×Dic106—41no—
102C2×C8×C53—22no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—42no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53#4111yescharacteristic
116C42×C53#4111yescharacteristic
117#11719921—44no—
118C4⋊Dic53—22no—
119C4⋊Dic53—22no—
120C4⋊Dic53—44no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C4×Dic53—21no—
124C4×Dic53—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—21no—
128C4×Dic53—21no—
129C2×#88322—22no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C23×C53C2×C22⋊C411yescharacteristic
136C22×D53—21no—
137C22×D53—21no—
138C22×D53—21no—
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53#10112yes—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12311yescharacteristic
154C2×C4×C53#13511yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C53#13711yescharacteristic
160C2×C4×C53#13711yescharacteristic
161C2×C4×C53#13711yescharacteristic
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C4×D53—42no—
175C4×D53—42no—
176C4×D53—41no—
177C4×D53—41no—
178C4×D53—41no—
179C4×D53—41no—
180C4×D53—41no—
181C4×D53—41no—
182C4×D53—41no—
183C4×D53—41no—
184Dic106—81no—
185Dic106—81no—
186Dic106—81no—
187Dic106—81no—
188C8×C53—42no—
189C8×C53—81no—
190#88322—42no—
191#88322—81no—
192#9140—531nomaximal, Sylow
193C22×C53#38611yescharacteristic
194C22×C53C2×#10111yescharacteristic
195C22×C53#41311yescharacteristic
196C22×C53#41611yescharacteristic
197C22×C53#41711yescharacteristic
198C22×C53#41811yescharacteristic
199C22×C53#41911yescharacteristic
200C2×D53—22no—
201C2×D53—22no—
202C2×D53—22no—
203C2×D53—41no—
204C2×D53—41no—
205C2×D53—41no—
206C2×D53—41no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—82no—
210C4×C53#44911yescharacteristic
211C4×C53#44911yescharacteristic
212C4×C53#46811yescharacteristic
213C4×C53#46811yescharacteristic
214C4×C53#47011yescharacteristic
215C4×C53#47011yescharacteristic
216C4×C53#48211yescharacteristic
217C4×C53#48211yescharacteristic
218Dic53—41no—
219Dic53—41no—
220Dic53—41no—
221Dic53—82no—
222C2×#432—531no—
223#2857—531no—
224#1797—531no—
225C2×#332—531no—
226C2×#332—531no—
227#1296—531no—
228#1296D5311yescharacteristic, complemented
229C2×C53#155211yescharacteristic
230C2×C53#155411yescharacteristic
231C2×C53#155611yescharacteristic
232C2×C53#155711yescharacteristic
233C2×C53#187511yescharacteristic
234C2×C53#189411yescharacteristic
235C2×C53#190111yescharacteristic
236D53—42no—
237C2×C4⋊Q8—531no—
238C2×C22⋊Q8—1061no—
239C22×C42—531no—
240#432—1062no—
241#432—1062no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8—531no—
244C2×C4⋊C8C2×D5311yescharacteristic
245C2×C4⋊C8C2×D5311yescharacteristic
246C2×C22⋊C8—1061no—
247C2×C22⋊C8—1061no—
248C2×#97—1061no—
249#391—1061no—
250#389—532no—
251#389—1061no—
252#384C2×D5311yescharacteristic
253#380—532no—
254#380—1061no—
255#336—1061no—
256#332—1062no—
257#332—1062no—
258C53#914011yescharacteristic, complemented, Sylow
259C22×Q8—1061no—
260C23×C4—531no—
261C23×C4—531no—
262C23×C4—531no—
263C22×C8—21no—
264C22×C8—21no—
265C22×C8—1061no—
266C22×C8—1061no—
267C4⋊Q8—1061no—
268C4⋊Q8—1061no—
269C4⋊Q8—1061no—
270C4⋊Q8—1061no—
271C22⋊Q8—2122no—
272C22⋊Q8—2122no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—532no—
276C2×C4⋊C4—1062no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4C4×D5312yes—
279C2×C22⋊C4—1062no—
280C2×C42—531no—
281C2×C42—531no—
282C2×C42—531no—
283C2×C42—1061no—
284C2×C42—1061no—
285C2×C42—1061no—
286C2×C42—1061no—
287C2×C42C22×D5311yescharacteristic
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—532no—
292C4⋊C8—1061no—
293C4⋊C8—1061no—
294C4⋊C8—1061no—
295C4⋊C8D21212yes—
296C22⋊C8—1064no—
297C22⋊C8—2122no—
298#97—1064no—
299#95—1062no—
300#95—1062no—
301C24—531no—
302C2×Q8—2121no—
303C2×Q8—2121no—
304C2×Q8—2121no—
305C2×Q8—2121no—
306C2×Q8—2121no—
307C2×Q8—2121no—
308C22×C4—22no—
309C22×C4—531no—
310C22×C4—531no—
311C22×C4—531no—
312C22×C4—1062no—
313C22×C4—1061no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4—1061no—
323C22×C4—1061no—
324C22×C4—1061no—
325C22×C4—1061no—
326C22×C4—1061no—
327C22×C4—1062no—
328C22×C4—2121no—
329C22×C4C2×C4×D5311yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C22×C4D4×D5311yescharacteristic
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—1062no—
339C2×C8—2121no—
340C2×C8—2121no—
341C2×C8—2121no—
342C2×C8—2121no—
343C2×C8—2121no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—1062no—
350C4⋊C4—1062no—
351C4⋊C4—2124no—
352C4⋊C4—2121no—
353C4⋊C4—2121no—
354C22⋊C4—2124no—
355C42—1061no—
356C42—1061no—
357C42—1061no—
358C42—1061no—
359C42—1061no—
360C42—1061no—
361C42C2×D21211yescharacteristic
362C42#1171994511yescharacteristic
363C42D4×D5311yescharacteristic
364C42#1171994911yescharacteristic
365C23—532no—
366C23—1061no—
367C23—1061no—
368C23—1061no—
369C23—1061no—
370C23—1061no—
371C23—1061no—
372C23C22⋊C4×D5311yescharacteristic
373Q8—4241no—
374Q8—4241no—
375Q8—4241no—
376Q8—4241no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—1061no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—1061no—
390C2×C4—1061no—
391C2×C4—2122no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4—2122no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4#1313317911yescharacteristic
405C2×C4C4×D21211yescharacteristic
406C2×C4#1313318711yescharacteristic
407C2×C4#1313319011yescharacteristic
408C2×C4#1313319211yescharacteristic
409C2×C4#1313320211yescharacteristic
410C2×C4#1313323811yescharacteristic
411C2×C4#1313324011yescharacteristic
412C2×C4#1313324111yescharacteristic
413C2×C4#1313324111yescharacteristic
414C2×C4#1313324411yescharacteristic
415C2×C4#1313324611yescharacteristic
416C2×C4#1313324611yescharacteristic
417C2×C4#1313324911yescharacteristic
418C8—42no—
419C8—81no—
420C8—4241no—
421C8—2122no—
422C22—1062no—
423C22—1062no—
424C22—1062no—
425C22—2121no—
426C22—2121no—
427C22—2121no—
428C22—2121no—
429C22#1681050111yescharacteristic
430C22#101×D5311yescharacteristic
431C22#1681060411yescharacteristic
432C22#1681061811yescharacteristic
433C22#1681062111yescharacteristic
434C22#1681062611yescharacteristic
435C22#1681063011yescharacteristic
436C4—21no—
437C4—21no—
438C4—82no—
439C4—4242no—
440C4—2121no—
441C4—2121no—
442C4—2121no—
443C4#1681076911yescharacteristic
444C4#1681077111yescharacteristic
445C4#1681086811yescharacteristic
446C4#1681087011yescharacteristic
447C4#1681088011yescharacteristic
448C4#1681088211yescharacteristic
449C4#1681094911yescharacteristic
450C4#1681095011yescharacteristic
451C2—2122no—
452C2#2027341411yescharacteristic
453C2#2027342411yescharacteristic
454C2#2027343011yescharacteristic
455C2#2027343411yescharacteristic
456C2#2027503611yescharacteristic
457C2#2027514711yescharacteristic
458C2#2027518211yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#9140
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
4. C2×#16810370C2semidirect
5. C2×#16810368C2semidirect
6. #20272598C2semidirect
7. C53×#1296C2semidirect
228. #1296D53semidirect
258. C53#9140semidirect

Maximal subgroups

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

subgroupindex
C2×#168106792
#202830682
#202746922
C2×#168103702
C2×#168103682
#202725982
C53×#12962
#914053