Finite Group Databasebeta

← #27280499

Subgroups — #27280499

Order 13568.

612 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
1D4×#11719917C211yesmaximal, characteristic, complemented
2#20275556C211yesmaximal, characteristic, complemented
3#20273383C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic, complemented
5#20272590C211yesmaximal, characteristic, complemented
6#20272565C211yesmaximal, characteristic, complemented
7C2×C53×#459C211yesmaximal, characteristic
8C22×#11719917—21no—
9#16810701—21no—
10#16810695—22no—
11D4×#88322—21no—
12#16810676C2211yescharacteristic
13#16810647C2211yescharacteristic
14#16810647C2211yescharacteristic
15C2×#13133134—21no—
16C2×#13133134—21no—
17#16810587—21no—
18C4×#11719917C2211yescharacteristic
19#16810367—22no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C22×D8×C53C412yes—
23C2×#127×C53C2211yescharacteristic
24C2×C4×D4×C53C2211yescharacteristic
25C53×#459—21no—
26C53×#459—21no—
27C53×#459—21no—
28C53×#459—21no—
29C2×C4⋊C8×C53C2211yescharacteristic
30C2×#102×C53C412yes—
31C2×#11719917—22no—
32C2×#11719917—21no—
33C2×#11719917—21no—
34C2×#11719917—42no—
35C2×#11719917—42no—
36C2×#11719917D411yescharacteristic
37C2×#11719917D411yescharacteristic
38C22×#88322—22no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322D412yes—
44#13133134—42no—
45#13133134—42no—
46C22⋊#88322—22no—
47C22⋊#88322—22no—
48C22⋊#88322—21no—
49#13133128—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×D4×C53—21no—
53C22×D4×C53—21no—
54C22×D4×C53C2×C411yescharacteristic
55C22×D4×C53C2×C412yes—
56C23×C4×C53—21no—
57C2×D8×C53—22no—
58C2×D8×C53—22no—
59C2×D8×C53—44no—
60C22×C8×C53C2×C412yes—
61#127×C53—21no—
62#127×C53—21no—
63#127×C53—21no—
64#127×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53D411yescharacteristic
68C4×D4×C53D411yescharacteristic
69C4×D4×C53D411yescharacteristic
70C4×D4×C53D411yescharacteristic
71C2×C4⋊C4×C53C2×C411yescharacteristic
72C2×C22⋊C4×C53—21no—
73C2×C42×C53C2311yescharacteristic
74C4⋊C8×C53—21no—
75C4⋊C8×C53—21no—
76#102×C53—22no—
77#102×C53—22no—
78C24×C53—21no—
79C24×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—21no—
83C2×D4×C53—22no—
84C2×D4×C53—41no—
85C2×D4×C53—42no—
86C2×D4×C53—42no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—82no—
96C2×D4×C53C22⋊C411yescharacteristic
97C2×D4×C53C22⋊C411yescharacteristic
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—41no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53—41no—
107C22×C4×C53C22×C411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110D8×C53—44no—
111D8×C53—44no—
112C2×C8×C53—22no—
113C2×C8×C53—42no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53C22⋊C411yescharacteristic
116C4⋊C4×C53C22⋊C411yescharacteristic
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C22⋊C4×C53—21no—
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125#11719917—21no—
126#11719917—21no—
127#11719917—21no—
128#11719917—44no—
129#11719917—41no—
130#11719917—41no—
131#11719917—41no—
132#11719917—41no—
133#11719917D812yescomplemented
134C2×#88322—21no—
135C2×#88322—21no—
136C2×#88322—22no—
137C2×#88322—44no—
138C2×#88322—41no—
139C2×#88322—41no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—42no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—42no—
146C23×C53—42no—
147C23×C53—42no—
148C23×C53—81no—
149C23×C53—82no—
150C23×C53—82no—
151C23×C53C4×D411yescharacteristic
152D4×C53—21no—
153D4×C53—21no—
154D4×C53—21no—
155D4×C53—42no—
156D4×C53—42no—
157D4×C53—42no—
158D4×C53—82no—
159D4×C53—82no—
160D4×C53—82no—
161D4×C53—81no—
162D4×C53—81no—
163D4×C53—81no—
164D4×C53—81no—
165D4×C53#10012yescomplemented
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53C2×C22⋊C411yescharacteristic
182C2×C4×C53C4×D411yescharacteristic
183C2×C4×C53#12011yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12311yescharacteristic
187C2×C4×C53C2×D811yescharacteristic
188C2×C4×C53#13511yescharacteristic
189C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
191C8×C53—42no—
192#88322—22no—
193#88322—42no—
194#88322—41no—
195#88322—81no—
196#88322—81no—
197#9253—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—22no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—41no—
206C22×C53—82no—
207C22×C53—82no—
208C22×C53—82no—
209C22×C53—82no—
210C22×C53—82no—
211C22×C53—162no—
212C22×C53—162no—
213C22×C53#38611yescharacteristic
214C22×C53C4×D811yescharacteristic
215C22×C53#44211yescharacteristic
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—41no—
220C4×C53—41no—
221C4×C53C2×#10011yescharacteristic
222C4×C53#41311yescharacteristic
223C4×C53#44711yescharacteristic
224C4×C53#44911yescharacteristic
225C4×C53#45911yescharacteristic
226C4×C53#46311yescharacteristic
227C2×#459D5311yescharacteristic
228D4×M16—531no—
229#1965—531no—
230#1545—531no—
231#1486—531no—
232#1292—531no—
233#1283—531no—
234C2×C53—21no—
235C2×C53—21no—
236C2×C53—42no—
237C2×C53—81no—
238C2×C53—162no—
239C2×C53#187911yescharacteristic
240C2×C53#190011yescharacteristic
241C2×C53#190311yescharacteristic
242C22×D8Dic5312yes—
243C22×M16—1061no—
244C2×#127C2×D5311yescharacteristic
245C2×C4×D4C2×D5311yescharacteristic
246#459—21no—
247#459—21no—
248#459—21no—
249#459—21no—
250#436—1061no—
251#435—1062no—
252C8×D4—1061no—
253#431—531no—
254#423—531no—
255#423—531no—
256C2×C4⋊C8C2×D5311yescharacteristic
257C2×#102Dic5312yes—
258C2×#100—1061no—
259C2×#100—1061no—
260#407—1061no—
261C4×M16—531no—
262#331—1062no—
263D4⋊C8—1061no—
264D4⋊C8—1061no—
265C53#925311yescharacteristic, complemented, Sylow
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C2×Dic5311yescharacteristic
269C22×D4C2×Dic5312yes—
270C23×C4—21no—
271C2×D8—22no—
272C2×D8—22no—
273C2×D8—44no—
274C2×M16—531no—
275C2×M16—531no—
276C2×M16—1062no—
277C2×M16—1061no—
278C2×M16—1061no—
279C2×M16—2122no—
280C2×M16—2122no—
281C22×C8—1062no—
282C22×C8C2×Dic5312yes—
283#127—21no—
284#127—21no—
285#127—21no—
286#127—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C22⋊D5311yescharacteristic
290C4×D4C22⋊D5311yescharacteristic
291C4×D4C22⋊D5311yescharacteristic
292C4×D4C22⋊D5311yescharacteristic
293C2×C4⋊C4C2×Dic5311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C42C22×D5311yescharacteristic
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—532no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301C4⋊C8—1061no—
302C4⋊C8—1061no—
303#102—22no—
304#102—22no—
305#100—2122no—
306#100—2122no—
307C22⋊C8—1062no—
308C22⋊C8—1062no—
309C22⋊C8—1061no—
310#97—1061no—
311C4×C8—1061no—
312C4×C8—1061no—
313C24—21no—
314C24—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—21no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—82no—
331C2×D4C22⋊Dic5311yescharacteristic
332C2×D4C22⋊Dic5311yescharacteristic
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4D4×D5311yescharacteristic
343C22×C4#1171994711yescharacteristic
344C22×C4C22×Dic5311yescharacteristic
345D8—44no—
346D8—44no—
347M16—532no—
348M16—1061no—
349M16—1061no—
350M16—1061no—
351M16—2124no—
352M16—2121no—
353M16—2121no—
354M16—2121no—
355M16—2121no—
356C2×C8—22no—
357C2×C8—42no—
358C2×C8—1061no—
359C2×C8—1061no—
360C2×C8—1062no—
361C2×C8—2124no—
362C2×C8—2121no—
363C2×C8—2121no—
364C4⋊C4—21no—
365C4⋊C4C22⋊Dic5311yescharacteristic
366C4⋊C4C22⋊Dic5311yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—21no—
371C42D4×D5311yescharacteristic
372C42#1171994711yescharacteristic
373C42C2×C22⋊D5311yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C23—21no—
376C23—21no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—81no—
384C23—82no—
385C23—82no—
386C23D4×Dic5311yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—82no—
394D4—82no—
395D4—82no—
396D4—81no—
397D4—81no—
398D4—81no—
399D4—81no—
400D4#1313313412yescomplemented
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4C2×C22⋊Dic5311yescharacteristic
417C2×C4D4×Dic5311yescharacteristic
418C2×C4#1313318511yescharacteristic
419C2×C4#1313318911yescharacteristic
420C2×C4#1313318911yescharacteristic
421C2×C4#1313319811yescharacteristic
422C2×C4D8×D5311yescharacteristic
423C2×C4#1313323611yescharacteristic
424C2×C4#1313324111yescharacteristic
425C2×C4#1313324111yescharacteristic
426C8—42no—
427C8—1062no—
428C8—4241no—
429C8—4241no—
430C8—2122no—
431C8—2121no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—162no—
446C22—162no—
447C22#1681050311yescharacteristic
448C22D8×Dic5311yescharacteristic
449C22#1681073511yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—41no—
454C4—41no—
455C4C2×#1313313411yescharacteristic
456C4#1681060711yescharacteristic
457C4#1681075511yescharacteristic
458C4#1681076511yescharacteristic
459C4#1681081611yescharacteristic
460C4#1681083611yescharacteristic
461C2—21no—
462C2—21no—
463C2—42no—
464C2—81no—
465C2—162no—
466C2#2027506511yescharacteristic
467C2#2027517911yescharacteristic
468C2#2027519711yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#459C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenterC2×#459D53
hypocenterC53#9253
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. D4×#11719917C2semidirect
2. #20275556C2semidirect
3. #20273383C2semidirect
4. #20273140C2semidirect
5. #20272590C2semidirect
6. #20272565C2semidirect
133. #11719917D8semidirect
165. D4×C53#100semidirect
265. C53#9253semidirect
400. D4#13133134semidirect

Maximal subgroups

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

subgroupindex
D4×#117199172
#202755562
#202733832
#202731402
#202725902
#202725652
C2×C53×#4592
#925353