Finite Group Databasebeta

← #27276804

Subgroups — #27276804

Order 13568.

548 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
1#20277176C211yesmaximal, characteristic
2#20274831C211yesmaximal, characteristic, complemented
3C4×#13133146C211yesmaximal, characteristic, complemented
4C2×#16810352C211yesmaximal, characteristic
5C2×#16810350C211yesmaximal, characteristic
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133174—21no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133146—21no—
12C2×C4×#88322C2211yescharacteristic
13#16810524C411yescharacteristic
14#16810524C411yescharacteristic
15#16810519C411yescharacteristic
16#16810519C411yescharacteristic
17#16810512—21no—
18#16810512—21no—
19C4×#11719921—21no—
20C42×Dic53C2211yescharacteristic
21#16810377—21no—
22#16810377—21no—
23#16810377—21no—
24#16810352—22no—
25#16810350—22no—
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C4×D53—21no—
31C22×#88322—21no—
32C22×#88322—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212D412yes—
36C4×D212Q812yes—
37#13133174—42no—
38#13133174—42no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×#11719921—21no—
46C2×#11719921—21no—
47C2×C4×Dic53C2×C411yescharacteristic
48C2×C4×Dic53C2×C411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53D411yescharacteristic
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53C4⋊#88322—21no—
54C4⋊#88322—21no—
55#13133146—24no—
56C4×#88322—21no—
57C4×#88322—21no—
58#13133124—21no—
59#13133124—21no—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C4⋊C4×C53C2×C411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70#95×C53—21no—
71C23×D53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53C22⋊C411yescharacteristic
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C22⋊C411yescharacteristic
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212C4⋊C412yes—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C42×C53—41no—
105C42×C53—41no—
106C42×C53C2×D411yescharacteristic
107C42×C53C2×Q811yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110#11719921—22no—
111#11719921—22no—
112#11719921—42no—
113#11719921—42no—
114#11719921—42no—
115#11719921—42no—
116C4⋊Dic53—21no—
117C4⋊Dic53C4⋊C412yes—
118#11719919—42no—
119#11719919—42no—
120C4×Dic53—21no—
121C4×Dic53—21no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C4×Dic53—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—21no—
128C4×Dic53—22no—
129C4×Dic53—41no—
130C2×#88322—22no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C23×C53C2×C22⋊C411yescharacteristic
137C22×D53—22no—
138C22×D53—41no—
139C22×D53—41no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53C2×C4⋊C411yescharacteristic
153C2×C4×C53#11711yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53C22⋊Q811yescharacteristic
157C2×C4×C53C22⋊Q811yescharacteristic
158C2×C4×C53#13111yescharacteristic
159C2×C4×C53#13111yescharacteristic
160C2×C4×C53#13111yescharacteristic
161C2×C4×C53#13111yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53#13511yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—22no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53#10412yes—
181D212—22no—
182D212—22no—
183D212—41no—
184D212—41no—
185C4×D53—42no—
186C4×D53—42no—
187C4×D53—42no—
188C4×D53—42no—
189C8×C53—42no—
190C8×C53—81no—
191#88322—42no—
192#88322—81no—
193#8569—531nomaximal, Sylow
194C22×C53#38011yescharacteristic
195C22×C53#40811yescharacteristic
196C22×C53#40811yescharacteristic
197C22×C53#41611yescharacteristic
198C22×C53#41811yescharacteristic
199C22×C53C2×#10411yescharacteristic
200C22×C53#42111yescharacteristic
201C2×D53—42no—
202C2×D53—42no—
203C2×D53—42no—
204C2×D53—81no—
205C2×D53—81no—
206C2×D53—81no—
207C2×D53—81no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—81no—
211C4×C53—81no—
212C4×C53#42411yescharacteristic
213C4×C53#42411yescharacteristic
214C4×C53#43311yescharacteristic
215C4×C53#43311yescharacteristic
216C4×C53#46311yescharacteristic
217C4×C53#46311yescharacteristic
218C4×C53#47811yescharacteristic
219C4×C53#47811yescharacteristic
220Dic53—22no—
221Dic53—42no—
222Dic53—41no—
223Dic53—41no—
224Dic53—41no—
225#2293—531no—
226#1831—531no—
227C4×C4⋊C8—531no—
228C2×D4⋊C8—531no—
229C2×D4⋊C8—531no—
230#1279—531no—
231#1279D5311yescharacteristic, complemented
232C2×C53#148211yescharacteristic
233C2×C53#148411yescharacteristic
234C2×C53#149411yescharacteristic
235C2×C53#149611yescharacteristic
236C2×C53#179811yescharacteristic
237C2×C53#180411yescharacteristic
238C2×C53#180611yescharacteristic
239D53—82no—
240C2×C4×D4—531no—
241C2×#117—1061no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8—1061no—
244C2×C4⋊C8C2×D5311yescharacteristic
245C2×C4×C8—531no—
246C2×C4×C8C2×D5311yescharacteristic
247#390—531no—
248#390—531no—
249#389—531no—
250#389—531no—
251#388—1061no—
252#388—1061no—
253C4×C4⋊C4C2×D5311yescharacteristic
254C4×C22⋊C4—1061no—
255C43—531no—
256#336—1061no—
257#336—1061no—
258#336—1061no—
259D4⋊C8—1062no—
260D4⋊C8—1062no—
261C53#856911yescharacteristic, complemented, Sylow
262C22×D4—531no—
263C23×C4—1061no—
264C22×C8—21no—
265C22×C8—21no—
266C22×C8—1061no—
267C22×C8—1061no—
268C4×D4—532no—
269C4×D4—532no—
270C4×D4—1061no—
271C4×D4—1061no—
272#117—2122no—
273#117—2122no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4C4×D5311yescharacteristic
278C2×C4⋊C4C4×D5311yescharacteristic
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1061no—
281C2×C22⋊C4—1061no—
282C2×C22⋊C4—1061no—
283C2×C22⋊C4—1061no—
284C2×C42—21no—
285C2×C42—531no—
286C2×C42—531no—
287C2×C42—531no—
288C2×C42—531no—
289C2×C42—531no—
290C2×C42—531no—
291C2×C42C22×D5311yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—1064no—
295C4⋊C8—1061no—
296C4⋊C8—1061no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—1061no—
300C4×C8—1061no—
301#95—21no—
302#95—1061no—
303#95—1061no—
304C24—1061no—
305C2×D4—532no—
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—1061no—
309C2×D4—1061no—
310C2×D4—1061no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—531no—
314C22×C4—531no—
315C22×C4—531no—
316C22×C4—531no—
317C22×C4—1061no—
318C22×C4—1062no—
319C22×C4—1062no—
320C22×C4—2121no—
321C22×C4—2121no—
322C22×C4—2121no—
323C22×C4—2121no—
324C22×C4C2×C4×D5311yescharacteristic
325C22×C4D4×D5311yescharacteristic
326C22×C4D4×D5311yescharacteristic
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—1062no—
334C2×C8—2121no—
335C2×C8—2121no—
336C2×C8—2121no—
337C2×C8—2121no—
338C2×C8—2121no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—532no—
344C4⋊C4—1061no—
345C4⋊C4—2122no—
346C4⋊C4—2122no—
347C22⋊C4—1062no—
348C22⋊C4—1062no—
349C22⋊C4—2122no—
350C22⋊C4—2122no—
351C22⋊C4—2122no—
352C22⋊C4—2122no—
353C42—41no—
354C42—41no—
355C42—1062no—
356C42—1061no—
357C42—1061no—
358C42—1061no—
359C42—1061no—
360C42—1061no—
361C42—1061no—
362C42—1061no—
363C42—1061no—
364C42—2121no—
365C42#1171994511yescharacteristic
366C42#1171994511yescharacteristic
367C42D4×D5311yescharacteristic
368C42Q8×D5311yescharacteristic
369C23—1062no—
370C23—2121no—
371C23—2121no—
372C23—2121no—
373C23—2121no—
374C23—2121no—
375C23—2121no—
376C23C22⋊C4×D5311yescharacteristic
377D4—1062no—
378D4—1062no—
379D4—2121no—
380D4—2121no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—532no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—1061no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—1062no—
402C2×C4—2121no—
403C2×C4—2121no—
404C2×C4—2122no—
405C2×C4—2122no—
406C2×C4—2122no—
407C2×C4—2122no—
408C2×C4C4⋊C4×D5311yescharacteristic
409C2×C4#1313317411yescharacteristic
410C2×C4#1313319011yescharacteristic
411C2×C4#1313319011yescharacteristic
412C2×C4#1313319511yescharacteristic
413C2×C4#1313319511yescharacteristic
414C2×C4#1313322311yescharacteristic
415C2×C4#1313322311yescharacteristic
416C2×C4#1313322511yescharacteristic
417C2×C4#1313322511yescharacteristic
418C2×C4#1313323811yescharacteristic
419C2×C4#1313324011yescharacteristic
420C2×C4#1313324111yescharacteristic
421C2×C4#1313324411yescharacteristic
422C8—42no—
423C8—81no—
424C8—4241no—
425C8—2122no—
426C22—4241no—
427C22—4241no—
428C22—4241no—
429C22—4241no—
430C22—2122no—
431C22—2122no—
432C22—2122no—
433C22#1681048211yescharacteristic
434C22#1681058811yescharacteristic
435C22#1681058811yescharacteristic
436C22#1681061811yescharacteristic
437C22#1681062611yescharacteristic
438C22#104×D5311yescharacteristic
439C22#1681063811yescharacteristic
440C4—21no—
441C4—21no—
442C4—81no—
443C4—81no—
444C4—1062no—
445C4—2121no—
446C4—2122no—
447C4—2121no—
448C4—2121no—
449C4#1681064811yescharacteristic
450C4#1681064811yescharacteristic
451C4#1681068211yescharacteristic
452C4#1681068511yescharacteristic
453C4#1681083711yescharacteristic
454C4#1681084111yescharacteristic
455C4#1681092211yescharacteristic
456C4#1681092611yescharacteristic
457C2—4242no—
458C2#2027312211yescharacteristic
459C2#2027313011yescharacteristic
460C2#2027317411yescharacteristic
461C2#2027318211yescharacteristic
462C2#2027469811yescharacteristic
463C2#2027472411yescharacteristic
464C2#2027473411yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#8569
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #20274831C2semidirect
3. C4×#13133146C2semidirect
6. #20272556C2semidirect
7. C53×#1279C2semidirect
231. #1279D53semidirect
261. C53#8569semidirect

Maximal subgroups

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

subgroupindex
#202771762
#202748312
C4×#131331462
C2×#168103522
C2×#168103502
#202725562
C53×#12792
#856953