Finite Group Databasebeta

← #27276794

Subgroups — #27276794

Order 13568.

568 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277176C211yesmaximal, characteristic
2#20274866C211yesmaximal, 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×#13133146—21no—
13C2×#13133146—21no—
14C2×C4×#88322C2211yescharacteristic
15#16810524C411yescharacteristic
16#16810524C411yescharacteristic
17#16810519C411yescharacteristic
18#16810519C411yescharacteristic
19#16810512—21no—
20#16810512—21no—
21C4×#11719921—21no—
22C42×Dic53C2211yescharacteristic
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—
56#13133146—42no—
57#13133146—42no—
58C4×#88322—21no—
59C4×#88322—21no—
60#13133124—21no—
61#13133124—21no—
62C22×C8×C53—21no—
63C22×C8×C53—21no—
64C2×C4⋊C4×C53C2×C411yescharacteristic
65C2×C4⋊C4×C53C2×C411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72#95×C53—21no—
73C23×D53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C22⋊C411yescharacteristic
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212C4⋊C412yes—
90C2×C4×D53—22no—
91C2×C4×D53—22no—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C8×C53—22no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C42×C53—41no—
107C42×C53—41no—
108C42×C53C2×D411yescharacteristic
109C42×C53C2×Q811yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53#4111yescharacteristic
112#11719921—22no—
113#11719921—22no—
114#11719921—42no—
115#11719921—42no—
116#11719921—42no—
117#11719921—42no—
118C4⋊Dic53—21no—
119C4⋊Dic53C4⋊C412yes—
120#11719919—42no—
121#11719919—42no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C4×Dic53—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—21no—
128C4×Dic53—22no—
129C4×Dic53—41no—
130C4×Dic53D812yes—
131C4×Dic53SD1612yes—
132C2×#88322—22no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C23×C53C2×C22⋊C411yescharacteristic
139C22×D53—22no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C22×D53—41no—
145C22×D53—41no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53C2×C4⋊C411yescharacteristic
155C2×C4×C53#11711yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53C22⋊Q811yescharacteristic
159C2×C4×C53C22⋊Q811yescharacteristic
160C2×C4×C53#13111yescharacteristic
161C2×C4×C53#13111yescharacteristic
162C2×C4×C53#13111yescharacteristic
163C2×C4×C53#13111yescharacteristic
164C2×C4×C53C2×D811yescharacteristic
165C2×C4×C53C2×SD1611yescharacteristic
166C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C53#13711yescharacteristic
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—22no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53#10212yes—
181C2×Dic53#10212yes—
182C2×Dic53#10412yes—
183D212—22no—
184D212—22no—
185D212—41no—
186D212—41no—
187C4×D53—42no—
188C4×D53—42no—
189C4×D53—42no—
190C4×D53—42no—
191C8×C53—42no—
192C8×C53—81no—
193#88322—42no—
194#88322—81no—
195#8567—531nomaximal, Sylow
196C22×C53#38011yescharacteristic
197C22×C53#40811yescharacteristic
198C22×C53#40811yescharacteristic
199C22×C53C2×#10211yescharacteristic
200C22×C53#41711yescharacteristic
201C22×C53C2×#10411yescharacteristic
202C22×C53#42111yescharacteristic
203C2×D53—42no—
204C2×D53—42no—
205C2×D53—42no—
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—81no—
209C2×D53—81no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—81no—
213C4×C53—81no—
214C4×C53#42411yescharacteristic
215C4×C53#42411yescharacteristic
216C4×C53#43311yescharacteristic
217C4×C53#43311yescharacteristic
218C4×C53#45911yescharacteristic
219C4×C53#46111yescharacteristic
220C4×C53#47411yescharacteristic
221C4×C53#47611yescharacteristic
222Dic53—22no—
223Dic53—22no—
224Dic53—22no—
225Dic53—42no—
226Dic53—41no—
227#2293—531no—
228#1842—531no—
229C4×C4⋊C8—531no—
230C2×D4⋊C8—531no—
231C2×D4⋊C8—531no—
232#1279—531no—
233#1279D5311yescharacteristic, complemented
234C2×C53#148011yescharacteristic
235C2×C53#148311yescharacteristic
236C2×C53#149211yescharacteristic
237C2×C53#149511yescharacteristic
238C2×C53#179811yescharacteristic
239C2×C53#180211yescharacteristic
240C2×C53#180611yescharacteristic
241D53—82no—
242C2×C4×D4—531no—
243C2×#117—1061no—
244C2×C4⋊C8—531no—
245C2×C4⋊C8—1061no—
246C2×C4⋊C8—1061no—
247C2×C4⋊C8—1061no—
248C2×C4⋊C8C2×D5311yescharacteristic
249C2×C4×C8—531no—
250C2×C4×C8C2×D5311yescharacteristic
251#390—531no—
252#390—531no—
253#389—531no—
254#389—531no—
255#388—1061no—
256#388—1061no—
257C4×C4⋊C4C2×D5311yescharacteristic
258C4×C22⋊C4—1061no—
259C43—531no—
260#336—1061no—
261D4⋊C8—1062no—
262D4⋊C8—1062no—
263C53#856711yescharacteristic, complemented, Sylow
264C22×D4—531no—
265C23×C4—1061no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—1061no—
269C22×C8—1061no—
270C4×D4—532no—
271C4×D4—532no—
272C4×D4—1061no—
273C4×D4—1061no—
274#117—2122no—
275#117—2122no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4C4×D5311yescharacteristic
280C2×C4⋊C4C4×D5311yescharacteristic
281C2×C22⋊C4—1061no—
282C2×C22⋊C4—1061no—
283C2×C22⋊C4—1061no—
284C2×C22⋊C4—1061no—
285C2×C22⋊C4—1061no—
286C2×C42—21no—
287C2×C42—531no—
288C2×C42—531no—
289C2×C42—531no—
290C2×C42—531no—
291C2×C42—531no—
292C2×C42—531no—
293C2×C42C22×D5311yescharacteristic
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—1064no—
297C4⋊C8—1061no—
298C4⋊C8—1061no—
299C4⋊C8—2122no—
300C4⋊C8—2122no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—1061no—
304C4×C8—1061no—
305#95—21no—
306#95—1061no—
307#95—1061no—
308C24—1061no—
309C2×D4—532no—
310C2×D4—1061no—
311C2×D4—1061no—
312C2×D4—1061no—
313C2×D4—1061no—
314C2×D4—1061no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—531no—
318C22×C4—531no—
319C22×C4—531no—
320C22×C4—531no—
321C22×C4—1061no—
322C22×C4—1062no—
323C22×C4—1062no—
324C22×C4—2121no—
325C22×C4—2121no—
326C22×C4—2121no—
327C22×C4—2121no—
328C22×C4C2×C4×D5311yescharacteristic
329C22×C4D4×D5311yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—1062no—
338C2×C8—2121no—
339C2×C8—2121no—
340C2×C8—2121no—
341C2×C8—2121no—
342C2×C8—2121no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—532no—
348C4⋊C4—1061no—
349C4⋊C4—2122no—
350C4⋊C4—2122no—
351C22⋊C4—1062no—
352C22⋊C4—1062no—
353C22⋊C4—2122no—
354C22⋊C4—2122no—
355C22⋊C4—2122no—
356C22⋊C4—2122no—
357C42—41no—
358C42—41no—
359C42—532no—
360C42—532no—
361C42—1062no—
362C42—1061no—
363C42—1061no—
364C42—1061no—
365C42—1061no—
366C42—1061no—
367C42—1061no—
368C42—2121no—
369C42#1171994511yescharacteristic
370C42#1171994511yescharacteristic
371C42D4×D5311yescharacteristic
372C42Q8×D5311yescharacteristic
373C23—1062no—
374C23—2121no—
375C23—2121no—
376C23—2121no—
377C23—2121no—
378C23—2121no—
379C23—2121no—
380C23C22⋊C4×D5311yescharacteristic
381D4—1062no—
382D4—1062no—
383D4—2121no—
384D4—2121no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—532no—
394C2×C4—532no—
395C2×C4—532no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—1061no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1062no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2122no—
409C2×C4—2122no—
410C2×C4—2122no—
411C2×C4—2122no—
412C2×C4C4⋊C4×D5311yescharacteristic
413C2×C4#1313317411yescharacteristic
414C2×C4#1313319011yescharacteristic
415C2×C4#1313319011yescharacteristic
416C2×C4#1313319511yescharacteristic
417C2×C4#1313319511yescharacteristic
418C2×C4#1313322311yescharacteristic
419C2×C4#1313322311yescharacteristic
420C2×C4#1313322511yescharacteristic
421C2×C4#1313322511yescharacteristic
422C2×C4D8×D5311yescharacteristic
423C2×C4SD16×D5311yescharacteristic
424C2×C4#1313324411yescharacteristic
425C2×C4#1313324911yescharacteristic
426C8—42no—
427C8—81no—
428C8—4241no—
429C8—2122no—
430C22—4241no—
431C22—4241no—
432C22—4241no—
433C22—4241no—
434C22—2122no—
435C22—2122no—
436C22—2122no—
437C22#1681048211yescharacteristic
438C22#1681058811yescharacteristic
439C22#1681058811yescharacteristic
440C22#102×D5311yescharacteristic
441C22#1681062311yescharacteristic
442C22#104×D5311yescharacteristic
443C22#1681063811yescharacteristic
444C4—21no—
445C4—21no—
446C4—81no—
447C4—81no—
448C4—1062no—
449C4—1062no—
450C4—1062no—
451C4—2122no—
452C4—2121no—
453C4#1681064811yescharacteristic
454C4#1681064811yescharacteristic
455C4#1681068211yescharacteristic
456C4#1681068511yescharacteristic
457C4#1681081711yescharacteristic
458C4#1681082711yescharacteristic
459C4#1681090211yescharacteristic
460C4#1681091211yescharacteristic
461C2—4242no—
462C2#2027311211yescharacteristic
463C2#2027312711yescharacteristic
464C2#2027316411yescharacteristic
465C2#2027317911yescharacteristic
466C2#2027469811yescharacteristic
467C2#2027471411yescharacteristic
468C2#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#8567
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. #20274866C2semidirect
3. C4×#13133146C2semidirect
6. #20272556C2semidirect
7. C53×#1279C2semidirect
233. #1279D53semidirect
263. C53#8567semidirect

Maximal subgroups

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

subgroupindex
#202771762
#202748662
C4×#131331462
C2×#168103522
C2×#168103502
#202725562
C53×#12792
#856753