Finite Group Databasebeta

← #27277446

Subgroups — #27277446

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133175C211yesmaximal, characteristic
2#20274692C211yesmaximal, characteristic
3C4×#13133129C211yesmaximal, characteristic
4#20272556C211yesmaximal, characteristic, complemented
5#20272555C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133175C411yescharacteristic
9C2×#13133175C411yescharacteristic
10C2×C42×D53C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133129—21no—
13C2×#13133129—21no—
14C2×C4×#88322C2211yescharacteristic
15C4⋊C4×Dic53C411yescharacteristic
16C4⋊C4×Dic53C411yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18C4×#11719921—21no—
19#16810470—21no—
20#16810468—21no—
21C42×Dic53—21no—
22#16810377—21no—
23#16810377—21no—
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C4×C4⋊C4×C53C2211yescharacteristic
27C22×C4×D53C2×C411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×#88322—21no—
31C22×#88322—21no—
32#13133175—22no—
33#13133175—22no—
34#13133175—22no—
35#13133175C814yescomplemented
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
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—
54#13133129—24no—
55#13133129—42no—
56C4×#88322—21no—
57C4×#88322—21no—
58#13133124—21no—
59#13133124—21no—
60#13133124—21no—
61C22×C8×C53—21no—
62C22×C8×C53—21no—
63C2×C4⋊C4×C53C2×C411yescharacteristic
64C2×C4⋊C4×C53C2×C411yescharacteristic
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4×C8×C53—21no—
70C4×C8×C53—21no—
71#95×C53—21no—
72C23×D53C22⋊C411yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
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—22no—
96C2×C4×D53—41no—
97C2×C4×D53C2×C812yes—
98C2×C4×D53M1612yes—
99C2×C8×C53—22no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53C2×C812yes—
108C4⋊C4×C53M1612yes—
109C42×C53—41no—
110C42×C53—41no—
111C42×C53#4111yescharacteristic
112C42×C53#4111yescharacteristic
113C42×C53#4111yescharacteristic
114C42×C53#4111yescharacteristic
115#11719921—24no—
116#11719921—24no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53C2×C812yes—
120C4⋊Dic53M1612yes—
121C4×Dic53—24no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C4×Dic53—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—21no—
128C4×Dic53—24no—
129C4×Dic53—42no—
130C4×Dic53—41no—
131C4×Dic53—41no—
132C4×Dic53—42no—
133C2×#88322—22no—
134C2×#88322—41no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322—41no—
139C23×C53C2×C22⋊C411yescharacteristic
140C22×D53—21no—
141C22×D53—21no—
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146C22×D53C22⋊C812yes—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53#11711yescharacteristic
156C2×C4×C53#11711yescharacteristic
157C2×C4×C53#12311yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×C4×C53#12311yescharacteristic
161C2×C4×C53C22×C811yescharacteristic
162C2×C4×C53C2×M1611yescharacteristic
163C2×C4×C53C2×M1611yescharacteristic
164C2×C4×C53C2×M1611yescharacteristic
165C2×C4×C53#13511yescharacteristic
166C2×C4×C53#13511yescharacteristic
167C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C53#13711yescharacteristic
169C2×Dic53—21no—
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—22no—
175C2×Dic53—22no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53C22⋊C812yes—
183C2×Dic53#10412yes—
184C4×D53—22no—
185C4×D53—22no—
186C4×D53—22no—
187C4×D53—22no—
188C4×D53—42no—
189C4×D53—42no—
190C4×D53—41no—
191C4×D53—41no—
192C4×D53—41no—
193C4×D53—41no—
194C8×C53—42no—
195C8×C53—81no—
196#88322—42no—
197#88322—81no—
198#8708—531nomaximal, Sylow
199C22×C53#38111yescharacteristic
200C22×C53C2×C22⋊C811yescharacteristic
201C22×C53#40711yescharacteristic
202C22×C53#41611yescharacteristic
203C22×C53#41711yescharacteristic
204C22×C53C2×#10411yescharacteristic
205C22×C53#42111yescharacteristic
206C2×D53—22no—
207C2×D53—22no—
208C2×D53—22no—
209C2×D53—41no—
210C2×D53—41no—
211C2×D53—41no—
212C2×D53—41no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—42no—
216C4×C53—42no—
217C4×C53#43111yescharacteristic
218C4×C53#43111yescharacteristic
219C4×C53#43211yescharacteristic
220C4×C53#43211yescharacteristic
221C4×C53#48211yescharacteristic
222C4×C53#48211yescharacteristic
223C4×C53#48511yescharacteristic
224C4×C53#48511yescharacteristic
225Dic53—22no—
226Dic53—22no—
227Dic53—41no—
228Dic53—42no—
229Dic53—42no—
230C4×#117—531no—
231#1797—531no—
232C4×C22⋊C8—531no—
233#1279—531no—
234#1279—531no—
235#1279—531no—
236#1279D5311yescharacteristic, complemented
237C2×C53#147611yescharacteristic
238C2×C53#147711yescharacteristic
239C2×C53#148811yescharacteristic
240C2×C53#148911yescharacteristic
241C2×C53#181711yescharacteristic
242C2×C53#182511yescharacteristic
243C2×C53#182611yescharacteristic
244D53—42no—
245C2×#117—531no—
246C2×#117—531no—
247C22×C42—531no—
248C2×C4⋊C8—531no—
249C2×C4⋊C8C2×D5311yescharacteristic
250C2×C22⋊C8—1061no—
251C2×C22⋊C8—1061no—
252C2×C4×C8—531no—
253C2×C4×C8C2×D5311yescharacteristic
254C4×C4⋊C4—531no—
255C4×C4⋊C4—531no—
256C4×C4⋊C4—531no—
257C4×C4⋊C4C2×D5311yescharacteristic
258C4×C22⋊C4—1061no—
259C4×C22⋊C4—1061no—
260#376—1061no—
261C43—1061no—
262#336—1061no—
263#336—1061no—
264C53#870811yescharacteristic, complemented, Sylow
265C23×C4—531no—
266C23×C4—531no—
267C23×C4—531no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—1061no—
271C22×C8—1061no—
272#117—534no—
273#117—1062no—
274#117—1062no—
275#117—1062no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—531no—
278C2×C4⋊C4C4×D5311yescharacteristic
279C2×C4⋊C4C4×D5311yescharacteristic
280C2×C22⋊C4—1061no—
281C2×C22⋊C4—1061no—
282C2×C42—21no—
283C2×C42—531no—
284C2×C42—531no—
285C2×C42—531no—
286C2×C42—1061no—
287C2×C42—1061no—
288C2×C42—1061no—
289C2×C42—1061no—
290C2×C42—1061no—
291C2×C42—1061no—
292C2×C42—1061no—
293C2×C42—1061no—
294C2×C42—1061no—
295C2×C42C22×D5311yescharacteristic
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—1061no—
299C4⋊C8—1061no—
300C22⋊C8—1064no—
301C22⋊C8—2122no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—1061no—
305C4×C8—1061no—
306#95—21no—
307#95—1061no—
308#95—1061no—
309#95—1061no—
310C24—531no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—531no—
314C22×C4—531no—
315C22×C4—532no—
316C22×C4—532no—
317C22×C4—531no—
318C22×C4—1062no—
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—1061no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—1061no—
332C22×C4—1061no—
333C22×C4—2121no—
334C22×C4C2×C4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4D4×D5311yescharacteristic
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—1062no—
344C2×C8—2121no—
345C2×C8—2121no—
346C2×C8—2121no—
347C2×C8—2121no—
348C2×C8—2121no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—532no—
352C4⋊C4—532no—
353C4⋊C4—1061no—
354C4⋊C4—1061no—
355C4⋊C4C8×D5312yes—
356C4⋊C4#1171991212yes—
357C22⋊C4—1064no—
358C22⋊C4—1064no—
359C42—41no—
360C42—41no—
361C42—1064no—
362C42—1064no—
363C42—1061no—
364C42—1061no—
365C42—1061no—
366C42—1061no—
367C42—1061no—
368C42—1061no—
369C42—2122no—
370C42—2122no—
371C42—2121no—
372C42—2121no—
373C42#1171994511yescharacteristic
374C42#1171994511yescharacteristic
375C42#1171994711yescharacteristic
376C42#1171994911yescharacteristic
377C23—532no—
378C23—1061no—
379C23—1061no—
380C23—1061no—
381C23—1061no—
382C23—1061no—
383C23—1061no—
384C23C22⋊C4×D5311yescharacteristic
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—532no—
394C2×C4—532no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—1061no—
399C2×C4—1062no—
400C2×C4—1062no—
401C2×C4—1062no—
402C2×C4—1062no—
403C2×C4—1062no—
404C2×C4—1062no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—1061no—
408C2×C4—2122no—
409C2×C4—2122no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2121no—
414C2×C4—2121no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4#1313317411yescharacteristic
419C2×C4#1313317511yescharacteristic
420C2×C4#1313319911yescharacteristic
421C2×C4#1313319911yescharacteristic
422C2×C4#1313320211yescharacteristic
423C2×C4#1313320211yescharacteristic
424C2×C4C2×C8×D5311yescharacteristic
425C2×C4M16×D5311yescharacteristic
426C2×C4M16×D5311yescharacteristic
427C2×C4C2×#1171991211yescharacteristic
428C2×C4#1313323611yescharacteristic
429C2×C4#1313323811yescharacteristic
430C2×C4#1313324411yescharacteristic
431C2×C4#1313324911yescharacteristic
432C8—42no—
433C8—81no—
434C8—4241no—
435C8—2122no—
436C22—1062no—
437C22—1062no—
438C22—1062no—
439C22—2121no—
440C22—2121no—
441C22—2121no—
442C22—2121no—
443C22#1681048511yescharacteristic
444C22C22⋊C8×D5311yescharacteristic
445C22#1681058511yescharacteristic
446C22#1681061611yescharacteristic
447C22#1681062311yescharacteristic
448C22#104×D5311yescharacteristic
449C22#1681063811yescharacteristic
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—1062no—
455C4—1062no—
456C4—2122no—
457C4—2121no—
458C4—2122no—
459C4#1681067411yescharacteristic
460C4#1681067511yescharacteristic
461C4#1681067811yescharacteristic
462C4#1681067911yescharacteristic
463C4#1681094511yescharacteristic
464C4#1681094911yescharacteristic
465C4#1681096211yescharacteristic
466C4#1681096611yescharacteristic
467C2—2122no—
468C2#2027309411yescharacteristic
469C2#2027310111yescharacteristic
470C2#2027314611yescharacteristic
471C2#2027315311yescharacteristic
472C2#2027478211yescharacteristic
473C2#2027480811yescharacteristic
474C2#2027481311yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#8708
LCSG ▷ C2×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
4. #20272556C2semidirect
5. #20272555C2semidirect
6. #20272554C2semidirect
7. C53×#1279C2semidirect
35. #13133175C8semidirect
236. #1279D53semidirect
264. C53#8708semidirect

Maximal subgroups

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

subgroupindex
C4×#131331752
#202746922
C4×#131331292
#202725562
#202725552
#202725542
C53×#12792
#870853