Finite Group Databasebeta

← #27286437

Subgroups — #27286437

Order 13568.

628 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
1C2×#16810910C211yesmaximal, characteristic
2#20279633C211yesmaximal, characteristic
3#20275518C211yesmaximal, characteristic, complemented
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#477C211yesmaximal, characteristic, complemented
8C2×#13133193—21no—
9C2×C4×D212C2211yescharacteristic
10#16810910—22no—
11#16810910—22no—
12C2×#13133148—21no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133141—22no—
15C2×#13133137—21no—
16C2×C4×#88322C2211yescharacteristic
17#16810545—22no—
18#16810525—21no—
19#16810521—21no—
20#16810493C2211yescharacteristic
21#16810480—21no—
22Q8⋊#88322—22no—
23#16810350—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×C4×Q8×C53C2211yescharacteristic
26C53×#477—22no—
27C53×#477—22no—
28C2×#107×C53—21no—
29C2×C4⋊C8×C53C2211yescharacteristic
30C2×#103×C53—21no—
31C22×D212D411yescharacteristic
32C22×C4×D53—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133193—42no—
36#13133193—42no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×C4⋊Dic53D412yes—
45C2×#11719919—22no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C2×#11719921—21no—
49#13133148—24no—
50C4⋊#88322—21no—
51C4⋊#88322D412yes—
52#13133141—44no—
53#13133137—24no—
54C4×#88322—21no—
55C4×#88322—21no—
56#13133124—22no—
57C22×Q8×C53—21no—
58C22×Q8×C53D411yescharacteristic
59C22×C8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53D412yes—
64C4×Q8×C53—21no—
65C4×Q8×C53—21no—
66C4×Q8×C53—21no—
67C4×Q8×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C42×C53—21no—
73C2×C42×C53C2311yescharacteristic
74#107×C53—24no—
75C4⋊C8×C53—21no—
76C4⋊C8×C53D412yes—
77#103×C53—24no—
78C23×D53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—22no—
83C2×Q8×C53—21no—
84C2×Q8×C53—21no—
85C2×Q8×C53—42no—
86C2×Q8×C53—42no—
87C2×Q8×C53—41no—
88C2×Q8×C53SD1612yes—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—41no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×Dic53—21no—
96C22×Dic53—22no—
97C2×D212—21no—
98C2×D212—21no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212Q1612yes—
103C2×C4×D53—41no—
104C2×C4×D53—41no—
105C2×C4×D53—41no—
106C2×C4×D53—41no—
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C2×C8×C53—22no—
110C2×C8×C53—41no—
111C2×C8×C53—41no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C42×C53—41no—
121C42×C53—41no—
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53C2×D411yescharacteristic
125C42×C53#4111yescharacteristic
126#11719921—44no—
127#11719921—41no—
128#11719921—41no—
129#11719921—82no—
130C4⋊Dic53—21no—
131C4⋊Dic53—21no—
132C4⋊Dic53—41no—
133C4⋊Dic53—41no—
134C4⋊Dic53—41no—
135C4⋊Dic53—41no—
136C4⋊Dic53—42no—
137#11719919—44no—
138C2×#88322—22no—
139C2×#88322—22no—
140C2×#88322—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143C2×#88322—41no—
144C23×C53#12011yescharacteristic
145C22×D53—22no—
146C22×D53—41no—
147C22×D53—41no—
148C22×D53—41no—
149C22×D53—41no—
150C22×D53—41no—
151C22×D53—41no—
152Q8×C53—22no—
153Q8×C53—22no—
154Q8×C53—44no—
155Q8×C53—41no—
156Q8×C53—41no—
157Q8×C53—82no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—22no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×C4×C53#12011yescharacteristic
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12711yescharacteristic
175C2×C4×C53C2×SD1611yescharacteristic
176C2×C4×C53C2×Q1611yescharacteristic
177C2×C4×C53#13511yescharacteristic
178C2×C4×C53#13511yescharacteristic
179C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C53#13711yescharacteristic
181C2×C4×C53#13711yescharacteristic
182C2×C4×C53#13711yescharacteristic
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—42no—
186C2×Dic53—41no—
187C2×Dic53—42no—
188C2×Dic53—42no—
189D212—22no—
190D212—22no—
191D212—41no—
192D212—41no—
193C4×D53—81no—
194C4×D53—81no—
195C4×D53—81no—
196C4×D53—81no—
197C8×C53—42no—
198#88322—42no—
199#88322—42no—
200#10236—531nomaximal, Sylow
201C22×C53#39211yescharacteristic
202C22×C53#45011yescharacteristic
203C22×C53#45111yescharacteristic
204C22×C53#45211yescharacteristic
205C22×C53#45211yescharacteristic
206C22×C53#45311yescharacteristic
207C22×C53#45411yescharacteristic
208C2×D53—42no—
209C2×D53—42no—
210C2×D53—42no—
211C2×D53—81no—
212C2×D53—81no—
213C2×D53—81no—
214C2×D53—81no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—42no—
218C4×C53—81no—
219C4×C53—82no—
220C4×C53#44811yescharacteristic
221C4×C53#44911yescharacteristic
222C4×C53#45111yescharacteristic
223C4×C53#45211yescharacteristic
224C4×C53#46511yescharacteristic
225C4×C53#46711yescharacteristic
226C4×C53#46811yescharacteristic
227C4×C53#47011yescharacteristic
228Dic53—82no—
229Dic53—81no—
230C2×#477D5311yescharacteristic, complemented
231C2×#476—531no—
232#2538—531no—
233#1956—531no—
234C2×Q8⋊C8—531no—
235C2×D4⋊C8—531no—
236#1296—531no—
237C2×C53#163111yescharacteristic
238C2×C53#163211yescharacteristic
239C2×C53#163711yescharacteristic
240C2×C53#163811yescharacteristic
241C2×C53#200011yescharacteristic
242C2×C53#200111yescharacteristic
243C2×C53#200611yescharacteristic
244D53—82no—
245C2×C4⋊Q8C2×D5311yescharacteristic
246C2×C22⋊Q8—1061no—
247C2×C4×Q8C2×D5311yescharacteristic
248C2×C4×D4—531no—
249#477—22no—
250#477—22no—
251#476—1062no—
252#476—1062no—
253C2×#107—21no—
254C2×#106—1061no—
255C2×C4⋊C8—531no—
256C2×C4⋊C8C2×D5311yescharacteristic
257C2×#103—21no—
258C2×#103—1062no—
259C2×#102—1061no—
260C2×C4×C8—531no—
261#396—1062no—
262#390—1061no—
263#389—1061no—
264#384—531no—
265#380—1061no—
266Q8⋊C8—1062no—
267D4⋊C8—1062no—
268C53#1023611yescharacteristic, complemented, Sylow
269C22×Q8—21no—
270C22×Q8C22⋊D5311yescharacteristic
271C22×D4—531no—
272C23×C4—1061no—
273C22×C8—21no—
274C22×C8—1061no—
275C22×C8—1061no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8C22⋊D5312yes—
280C22⋊Q8—2122no—
281C22⋊Q8—2122no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×D4—1061no—
287C4×D4—1061no—
288C4×D4—1061no—
289C4×D4—1061no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—531no—
294C2×C4⋊C4—532no—
295C2×C4⋊C4—1062no—
296C2×C4⋊C4—1061no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4C22⋊D5311yescharacteristic
299C2×C22⋊C4—1062no—
300C2×C22⋊C4—1062no—
301C2×C22⋊C4—1061no—
302C2×C42—21no—
303C2×C42C22×D5311yescharacteristic
304#107—24no—
305#106—1064no—
306C4⋊C8—21no—
307C4⋊C8—532no—
308C4⋊C8—1061no—
309C4⋊C8C22⋊D5312yes—
310#103—24no—
311#103—2124no—
312#102—1064no—
313C4×C8—1061no—
314C4×C8—1061no—
315#95—1062no—
316C24—1061no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—22no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8#1171992412yes—
327C2×D4—532no—
328C2×D4—1061no—
329C2×D4—1061no—
330C2×D4—1061no—
331C2×D4—1061no—
332C2×D4—1061no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—41no—
336C22×C4—1061no—
337C22×C4—1062no—
338C22×C4—2121no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4D4×D5311yescharacteristic
345C22×C4D4×D5311yescharacteristic
346C22×C4C2×C22⋊D5311yescharacteristic
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—1062no—
351C2×C8—1062no—
352C2×C8—2121no—
353C2×C8—2121no—
354C2×C8—2121no—
355C2×C8—2121no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—1061no—
365C4⋊C4—1061no—
366C4⋊C4—2122no—
367C4⋊C4—2124no—
368C4⋊C4—2121no—
369C4⋊C4—2121no—
370C4⋊C4—2121no—
371C4⋊C4—2121no—
372C22⋊C4—4242no—
373C22⋊C4—2124no—
374C22⋊C4—2121no—
375C22⋊C4—2121no—
376C42—41no—
377C42—41no—
378C42D4×D5311yescharacteristic
379C42#1171994711yescharacteristic
380C42C2×C22⋊D5311yescharacteristic
381C42C2×C22⋊D5311yescharacteristic
382C23—1062no—
383C23—2121no—
384C23—2121no—
385C23—2121no—
386C23—2121no—
387C23—2121no—
388C23—2121no—
389C23#1313318511yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—44no—
393Q8—41no—
394Q8—41no—
395Q8—82no—
396D4—1062no—
397D4—1062no—
398D4—2121no—
399D4—2121no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—4241no—
414C2×C4—4241no—
415C2×C4—2122no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4—2122no—
419C2×C4—2121no—
420C2×C4—2122no—
421C2×C4#1313318511yescharacteristic
422C2×C4#1313318611yescharacteristic
423C2×C4#1313318811yescharacteristic
424C2×C4#1313318911yescharacteristic
425C2×C4#1313318911yescharacteristic
426C2×C4#1313321311yescharacteristic
427C2×C4C2×#1171992411yescharacteristic
428C2×C4Q16×D5311yescharacteristic
429C2×C4#1313323711yescharacteristic
430C2×C4#1313324011yescharacteristic
431C2×C4#1313324211yescharacteristic
432C2×C4#1313324811yescharacteristic
433C2×C4#1313324911yescharacteristic
434C2×C4#1313324911yescharacteristic
435C8—42no—
436C8—2122no—
437C8—2122no—
438C22—4241no—
439C22—4241no—
440C22—4241no—
441C22—4241no—
442C22—2122no—
443C22—2122no—
444C22—2122no—
445C22#1681053111yescharacteristic
446C22#1681077311yescharacteristic
447C22#1681077711yescharacteristic
448C22#1681078311yescharacteristic
449C22#1681078411yescharacteristic
450C22#1681078911yescharacteristic
451C22#1681079411yescharacteristic
452C4—21no—
453C4—21no—
454C4—42no—
455C4—81no—
456C4—82no—
457C4—4241no—
458C4—4242no—
459C4#1681076211yescharacteristic
460C4#1681076811yescharacteristic
461C4#1681077711yescharacteristic
462C4#1681078411yescharacteristic
463C4#1681085411yescharacteristic
464C4#1681086111yescharacteristic
465C4#1681087111yescharacteristic
466C4#1681087811yescharacteristic
467C2—4242no—
468C2#2027384011yescharacteristic
469C2#2027384811yescharacteristic
470C2#2027387811yescharacteristic
471C2#2027388611yescharacteristic
472C2#2027571611yescharacteristic
473C2#2027572211yescharacteristic
474C2#2027575511yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#477C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenterC2×#477D53
hypocenterC53#10236
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #20275518C2semidirect
4. C2×Q8⋊#88322C2semidirect
6. #20272599C2semidirect
7. C2×C53×#477C2semidirect
230. C2×#477D53semidirect
268. C53#10236semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
#202796332
#202755182
C2×Q8⋊#883222
C2×#168103502
#202725992
C2×C53×#4772
#1023653