Finite Group Databasebeta

← #27279786

Subgroups — #27279786

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810645C211yesmaximal, characteristic
2#20283068C211yesmaximal, characteristic
3#20274692C211yesmaximal, characteristic
4C2×#16810370C211yesmaximal, characteristic, complemented
5C2×#16810368C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C2×#13133217C2211yescharacteristic
10C2×#13133197—21no—
11C2×C42×D53C2211yescharacteristic
12#16810645—22no—
13#16810645—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133146—21no—
17C2×#13133129—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
33C2×#11719912—24no—
34C2×#11719912—42no—
35C2×#11719912—42no—
36C22×#88322—21no—
37C22×#88322—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217—21no—
41#13133217—21no—
42#13133197—42no—
43#13133197—42no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—22no—
46C2×C4⋊Dic53C2×C412yes—
47C2×#11719921—22no—
48C2×C4×Dic53C2×C411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53D411yescharacteristic
51C42×D53—21no—
52C42×D53—21no—
53C42×D53—21no—
54C42×D53—21no—
55C4⋊#88322—21no—
56C4⋊#88322—21no—
57C4⋊#88322—21no—
58C4⋊#88322D412yes—
59#13133146—24no—
60#13133129—42no—
61#13133124—22no—
62#13133124—22no—
63C22×C8×C53—21no—
64C22×C8×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53C2×C412yes—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53D412yes—
73C23×D53C22⋊C411yescharacteristic
74C22×C4×C53—22no—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—22no—
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C22⋊C411yescharacteristic
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—22no—
95C2×C4×D53—41no—
96C2×C4×D53D812yes—
97C2×C4×D53SD1612yes—
98C2×Dic106—41no—
99C2×Dic106—41no—
100C2×Dic106—41no—
101C2×Dic106—41no—
102C2×Dic106—41no—
103C2×Dic106—41no—
104C2×C8×C53—22no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C4⋊C4×C53—42no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53#4111yescharacteristic
119#11719921—44no—
120C4⋊Dic53—22no—
121C4⋊Dic53—22no—
122C4⋊Dic53—41no—
123C4⋊Dic53—41no—
124C4⋊Dic53—44no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—21no—
128C4×Dic53—21no—
129C4×Dic53—21no—
130C4×Dic53—21no—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C2×#88322—41no—
137#11719912—48no—
138C23×C53C2×C22⋊C411yescharacteristic
139C22×D53—21no—
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53#10212yes—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C2×C4×C53C2×D811yescharacteristic
158C2×C4×C53C2×SD1611yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53#13711yescharacteristic
162C2×C4×C53#13711yescharacteristic
163C2×C4×C53#13711yescharacteristic
164C2×C4×C53#13711yescharacteristic
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53#10112yes—
176C2×Dic53#10212yes—
177C4×D53—22no—
178C4×D53—22no—
179C4×D53—22no—
180C4×D53—22no—
181C4×D53—42no—
182C4×D53—42no—
183C4×D53—41no—
184C4×D53—41no—
185C4×D53—41no—
186C4×D53—41no—
187Dic106—81no—
188Dic106—81no—
189Dic106—81no—
190Dic106—81no—
191C8×C53—42no—
192C8×C53—81no—
193#88322—42no—
194#88322—81no—
195#9139—531nomaximal, Sylow
196C22×C53#38611yescharacteristic
197C22×C53C2×#10111yescharacteristic
198C22×C53#41311yescharacteristic
199C22×C53C2×#10211yescharacteristic
200C22×C53#41711yescharacteristic
201C22×C53#41711yescharacteristic
202C22×C53#41911yescharacteristic
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—22no—
206C2×D53—41no—
207C2×D53—41no—
208C2×D53—41no—
209C2×D53—41no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—82no—
213C4×C53#44711yescharacteristic
214C4×C53#44811yescharacteristic
215C4×C53#46811yescharacteristic
216C4×C53#46811yescharacteristic
217C4×C53#47011yescharacteristic
218C4×C53#47011yescharacteristic
219C4×C53#48011yescharacteristic
220C4×C53#48111yescharacteristic
221Dic53—22no—
222Dic53—22no—
223Dic53—41no—
224Dic53—82no—
225C2×#423—531no—
226#2857—531no—
227#1797—531no—
228C2×#332—531no—
229C2×#332—531no—
230#1296—531no—
231#1296D5311yescharacteristic, complemented
232C2×C53#154611yescharacteristic
233C2×C53#154711yescharacteristic
234C2×C53#155911yescharacteristic
235C2×C53#156111yescharacteristic
236C2×C53#187411yescharacteristic
237C2×C53#189411yescharacteristic
238C2×C53#190111yescharacteristic
239D53—42no—
240C22×M16—1061no—
241C2×C4⋊Q8—531no—
242C2×C22⋊Q8—1061no—
243C22×C42—531no—
244#423—1062no—
245#423—1062no—
246C2×C4⋊C8—531no—
247C2×C4⋊C8—531no—
248C2×C4⋊C8—1061no—
249C2×C4⋊C8C2×D5311yescharacteristic
250C2×C4⋊C8C2×D5311yescharacteristic
251C2×C22⋊C8—1061no—
252#391—1061no—
253#389—532no—
254#389—1061no—
255#384C2×D5311yescharacteristic
256#380—532no—
257#380—1061no—
258#336—1061no—
259#332—1062no—
260#332—1062no—
261C53#913911yescharacteristic, complemented, Sylow
262C22×Q8—1061no—
263C23×C4—531no—
264C23×C4—531no—
265C23×C4—531no—
266C2×M16—1064no—
267C2×M16—2122no—
268C2×M16—2122no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—1061no—
272C22×C8—1061no—
273C4⋊Q8—1061no—
274C4⋊Q8—1061no—
275C4⋊Q8—1061no—
276C4⋊Q8—1061no—
277C22⋊Q8—2122no—
278C22⋊Q8—2122no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—532no—
282C2×C4⋊C4—1062no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4C4×D5312yes—
285C2×C22⋊C4—1062no—
286C2×C42—531no—
287C2×C42—531no—
288C2×C42—531no—
289C2×C42—1061no—
290C2×C42—1061no—
291C2×C42—1061no—
292C2×C42—1061no—
293C2×C42C22×D5311yescharacteristic
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—532no—
298C4⋊C8—1064no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301C4⋊C8—1061no—
302C4⋊C8D21212yes—
303C22⋊C8—2122no—
304#95—1062no—
305#95—1062no—
306C24—531no—
307C2×Q8—2121no—
308C2×Q8—2121no—
309C2×Q8—2121no—
310C2×Q8—2121no—
311C2×Q8—2121no—
312C2×Q8—2121no—
313C22×C4—22no—
314C22×C4—531no—
315C22×C4—531no—
316C22×C4—532no—
317C22×C4—532no—
318C22×C4—531no—
319C22×C4—1062no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4—1061no—
323C22×C4—1061no—
324C22×C4—1061no—
325C22×C4—1061no—
326C22×C4—1061no—
327C22×C4—1061no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—1061no—
332C22×C4—1062no—
333C22×C4—2121no—
334C22×C4C2×C4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4D4×D5311yescharacteristic
337M16—2128no—
338C2×C8—22no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—1062no—
345C2×C8—2121no—
346C2×C8—2121no—
347C2×C8—2121no—
348C2×C8—2121no—
349C2×C8—2121no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—1062no—
356C4⋊C4—1062no—
357C4⋊C4—2124no—
358C4⋊C4—2121no—
359C4⋊C4—2121no—
360C22⋊C4—2124no—
361C42—1061no—
362C42—1061no—
363C42—1061no—
364C42—1061no—
365C42—1061no—
366C42—1061no—
367C42C2×D21211yescharacteristic
368C42#1171994511yescharacteristic
369C42D4×D5311yescharacteristic
370C42#1171994911yescharacteristic
371C23—532no—
372C23—1061no—
373C23—1061no—
374C23—1061no—
375C23—1061no—
376C23—1061no—
377C23—1061no—
378C23C22⋊C4×D5311yescharacteristic
379Q8—4241no—
380Q8—4241no—
381Q8—4241no—
382Q8—4241no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—532no—
389C2×C4—532no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1062no—
395C2×C4—1062no—
396C2×C4—1062no—
397C2×C4—1062no—
398C2×C4—1061no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—2122no—
402C2×C4—2122no—
403C2×C4—2122no—
404C2×C4—2122no—
405C2×C4—2122no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4#1313317911yescharacteristic
411C2×C4C4×D21211yescharacteristic
412C2×C4#1313318711yescharacteristic
413C2×C4#1313319011yescharacteristic
414C2×C4#1313319211yescharacteristic
415C2×C4#1313320211yescharacteristic
416C2×C4D8×D5311yescharacteristic
417C2×C4SD16×D5311yescharacteristic
418C2×C4#1313324111yescharacteristic
419C2×C4#1313324411yescharacteristic
420C2×C4#1313324611yescharacteristic
421C2×C4#1313324611yescharacteristic
422C2×C4#1313324911yescharacteristic
423C2×C4#1313324911yescharacteristic
424C8—42no—
425C8—81no—
426C8—4241no—
427C8—2122no—
428C22—1062no—
429C22—1062no—
430C22—1062no—
431C22—2121no—
432C22—2121no—
433C22—2121no—
434C22—2121no—
435C22#1681050111yescharacteristic
436C22#101×D5311yescharacteristic
437C22#1681060411yescharacteristic
438C22#102×D5311yescharacteristic
439C22#1681062111yescharacteristic
440C22#1681062311yescharacteristic
441C22#1681063011yescharacteristic
442C4—21no—
443C4—21no—
444C4—82no—
445C4—1062no—
446C4—1062no—
447C4—4242no—
448C4—2121no—
449C4#1681075811yescharacteristic
450C4#1681076311yescharacteristic
451C4#1681086811yescharacteristic
452C4#1681087011yescharacteristic
453C4#1681088011yescharacteristic
454C4#1681088211yescharacteristic
455C4#1681093811yescharacteristic
456C4#1681094311yescharacteristic
457C2—2122no—
458C2#2027338511yescharacteristic
459C2#2027338911yescharacteristic
460C2#2027344211yescharacteristic
461C2#2027345111yescharacteristic
462C2#2027503111yescharacteristic
463C2#2027514711yescharacteristic
464C2#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#9139
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
231. #1296D53semidirect
261. C53#9139semidirect

Maximal subgroups

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

subgroupindex
C2×#168106452
#202830682
#202746922
C2×#168103702
C2×#168103682
#202725982
C53×#12962
#913953