Finite Group Databasebeta

← #27280497

Subgroups — #27280497

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
1C2×#16810818C211yesmaximal, characteristic
2#20275555C211yesmaximal, characteristic, complemented
3#20273384C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic, complemented
5#20272589C211yesmaximal, characteristic, complemented
6#20272566C211yesmaximal, characteristic, complemented
7D4×M16×C53C211yesmaximal, characteristic, complemented
8C22×#11719922C412yes—
9C2×#13133214C2211yescharacteristic
10#16810818—21no—
11#16810818—21no—
12#16810818—21no—
13#16810818—21no—
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137C412yes—
18C2×#13133133—21no—
19C2×#13133133—21no—
20#16810366—22no—
21D4⋊#88322—21no—
22D4⋊#88322—21no—
23C22×M16×C53—21no—
24C2×C4×D4×C53C2211yescharacteristic
25C53×#436—21no—
26C53×#435—22no—
27C8×D4×C53—21no—
28C53×#423C2211yescharacteristic
29C53×#407—21no—
30C4×M16×C53C2211yescharacteristic
31C22×D212—21no—
32C22×D212—21no—
33C22×D212C2×C412yes—
34C2×#11719922—22no—
35C2×#11719922—22no—
36C2×#11719922—44no—
37C2×#11719917—21no—
38C22×#88322C2×C412yes—
39#13133214—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133137—22no—
49#13133137—22no—
50#13133133—42no—
51#13133133—42no—
52C22⋊#88322—21no—
53C4×#88322—21no—
54C22×D4×C53C2×C411yescharacteristic
55C23×C4×C53—21no—
56C2×M16×C53—22no—
57C2×M16×C53—21no—
58C2×M16×C53—42no—
59C2×M16×C53—42no—
60C2×M16×C53D411yescharacteristic
61C2×M16×C53D411yescharacteristic
62C22×C8×C53—22no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53D411yescharacteristic
66C4×D4×C53D411yescharacteristic
67C4×D4×C53D411yescharacteristic
68C4×D4×C53D411yescharacteristic
69C2×C4⋊C4×C53C2×C411yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—21no—
73C4⋊C8×C53D412yes—
74C22⋊C8×C53—22no—
75C22⋊C8×C53—22no—
76#97×C53—21no—
77C4×C8×C53—21no—
78C24×C53—21no—
79C23×D53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C2×D4×C53C22⋊C411yescharacteristic
84C2×D4×C53C22⋊C411yescharacteristic
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53C22×C411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C2×D212—22no—
98C2×D212—22no—
99C2×D212—42no—
100C2×D212—42no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—82no—
110M16×C53—21no—
111M16×C53—21no—
112M16×C53—21no—
113M16×C53—44no—
114M16×C53—41no—
115M16×C53—41no—
116M16×C53D812yescomplemented
117C2×C8×C53—21no—
118C2×C8×C53—21no—
119C2×C8×C53—22no—
120C2×C8×C53—44no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53C22⋊C411yescharacteristic
123C4⋊C4×C53C22⋊C411yescharacteristic
124C22⋊C4×C53—21no—
125C22⋊C4×C53—21no—
126C22⋊C4×C53—21no—
127C22⋊C4×C53—21no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C42×C53#4111yescharacteristic
132#11719922—44no—
133#11719922—44no—
134#11719917—41no—
135#11719917—41no—
136C2×#88322—22no—
137C2×#88322—42no—
138C2×#88322—41no—
139C2×#88322—41no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—42no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—81no—
146C23×C53C4×D411yescharacteristic
147C22×D53—42no—
148C22×D53—42no—
149C22×D53—42no—
150C22×D53—82no—
151C22×D53—82no—
152D4×C53—21no—
153D4×C53—21no—
154D4×C53—21no—
155D4×C53—42no—
156D4×C53#10012yescomplemented
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53C2×C22⋊C411yescharacteristic
173C2×C4×C53C4×D411yescharacteristic
174C2×C4×C53#12011yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12311yescharacteristic
178C2×C4×C53C2×D811yescharacteristic
179C2×C4×C53#13511yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
182D212—42no—
183D212—42no—
184D212—82no—
185D212—82no—
186D212—82no—
187D212—81no—
188D212—81no—
189D212—81no—
190D212—81no—
191C8×C53—22no—
192C8×C53—42no—
193C8×C53—41no—
194#88322—42no—
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#38611yescharacteristic
209C22×C53C4×D811yescharacteristic
210C22×C53#44211yescharacteristic
211C2×D53—82no—
212C2×D53—82no—
213C2×D53—82no—
214C2×D53—162no—
215C2×D53—162no—
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×#459—531no—
228D4×M16D5311yescharacteristic, complemented
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#187911yescharacteristic
239C2×C53#190011yescharacteristic
240C2×C53#190311yescharacteristic
241D53—162no—
242C22×D8—532no—
243C22×M16—21no—
244C2×#127—531no—
245C2×C4×D4C2×D5311yescharacteristic
246#459—1061no—
247#459—1061no—
248#459—1061no—
249#459—1061no—
250#436—21no—
251#435—22no—
252C8×D4—21no—
253#431—531no—
254#423—531no—
255#423C2×D5311yescharacteristic
256C2×C4⋊C8—531no—
257C2×#102—532no—
258C2×#100—1061no—
259C2×#100—1061no—
260#407—21no—
261C4×M16C2×D5311yescharacteristic
262#331—1062no—
263D4⋊C8—1061no—
264D4⋊C8—1061no—
265C53#925311yescharacteristic, complemented, Sylow
266C22×D4—532no—
267C22×D4—1061no—
268C22×D4—1061no—
269C22×D4C4×D5311yescharacteristic
270C23×C4—21no—
271C2×D8—1062no—
272C2×D8—1062no—
273C2×D8—2124no—
274C2×M16—22no—
275C2×M16—21no—
276C2×M16—42no—
277C2×M16—42no—
278C2×M16—1061no—
279C2×M16C22⋊D5311yescharacteristic
280C2×M16C22⋊D5311yescharacteristic
281C22×C8—22no—
282C22×C8—532no—
283#127—1061no—
284#127—1061no—
285#127—1061no—
286#127—1061no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4D21211yescharacteristic
290C4×D4D21211yescharacteristic
291C4×D4C22⋊D5311yescharacteristic
292C4×D4C22⋊D5311yescharacteristic
293C2×C4⋊C4C4×D5311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C42C22×D5311yescharacteristic
296C4⋊C8—21no—
297C4⋊C8—1061no—
298C4⋊C8—1061no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301C4⋊C8—1061no—
302C4⋊C8C22⋊D5312yes—
303#102—1062no—
304#102—1062no—
305#100—2122no—
306#100—2122no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—1061no—
310#97—21no—
311C4×C8—21no—
312C4×C8—1061no—
313C24—21no—
314C24—1062no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—1062no—
319C2×D4—1062no—
320C2×D4—4242no—
321C2×D4—2122no—
322C2×D4—2122no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C2×D4—2121no—
327C2×D4—2121no—
328C2×D4—2121no—
329C2×D4—2121no—
330C2×D4—2121no—
331C2×D4#1171992111yescharacteristic
332C2×D4#1171992111yescharacteristic
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×C4C2×C4×D5311yescharacteristic
343C22×C4#1171994511yescharacteristic
344C22×C4C2×C22⋊D5311yescharacteristic
345D8—2124no—
346D8—2124no—
347M16—21no—
348M16—21no—
349M16—21no—
350M16—44no—
351M16—41no—
352M16—41no—
353M16—2121no—
354M16—2121no—
355M16#1171992212yescomplemented
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—22no—
359C2×C8—44no—
360C2×C8—1062no—
361C2×C8—2121no—
362C2×C8—2121no—
363C2×C8—2122no—
364C4⋊C4—21no—
365C4⋊C4#1171992111yescharacteristic
366C4⋊C4#1171992111yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—21no—
370C22⋊C4—21no—
371C42C2×D21211yescharacteristic
372C42#1171994511yescharacteristic
373C42C2×C22⋊D5311yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C23—21no—
376C23—21no—
377C23—42no—
378C23—41no—
379C23—41no—
380C23—81no—
381C23—4242no—
382C23—4242no—
383C23—2122no—
384C23—2122no—
385C23—2122no—
386C23C4×C22⋊D5311yescharacteristic
387D4—21no—
388D4—21no—
389D4—21no—
390D4—42no—
391D4—4242no—
392D4—4242no—
393D4—4242no—
394D4—4241no—
395D4—4241no—
396D4—4241no—
397D4—4241no—
398D4—2122no—
399D4—2122no—
400D4#1313313312yescomplemented
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×#1171992111yescharacteristic
417C2×C4C4×C22⋊D5311yescharacteristic
418C2×C4#1313318611yescharacteristic
419C2×C4#1313319111yescharacteristic
420C2×C4#1313319111yescharacteristic
421C2×C4#1313320011yescharacteristic
422C2×C4C2×#1171992211yescharacteristic
423C2×C4#1313323711yescharacteristic
424C2×C4#1313324211yescharacteristic
425C2×C4#1313324311yescharacteristic
426C8—22no—
427C8—42no—
428C8—41no—
429C8—4241no—
430C8—4241no—
431C8—2122no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—82no—
441C22—82no—
442C22—8482no—
443C22—8482no—
444C22—4242no—
445C22—4242no—
446C22—4242no—
447C22#1681050211yescharacteristic
448C22C4×#1171992211yescharacteristic
449C22#1681073311yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—41no—
454C4—41no—
455C4C2×#1313313311yescharacteristic
456C4#1681060511yescharacteristic
457C4#1681075711yescharacteristic
458C4#1681076811yescharacteristic
459C4#1681081811yescharacteristic
460C4#1681083811yescharacteristic
461C2—21no—
462C2—21no—
463C2—42no—
464C2—81no—
465C2—8482no—
466C2#2027506311yescharacteristic
467C2#2027517811yescharacteristic
468C2#2027519511yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingD4×M16×C53C2
socleC22×C53#386
prime radicalC2×#16810818C2
cosocleC2×C42×C53C23
hypercenterD4×M16D53
hypocenterC53#9253
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
2. #20275555C2semidirect
3. #20273384C2semidirect
4. #20273140C2semidirect
5. #20272589C2semidirect
6. #20272566C2semidirect
7. D4×M16×C53C2semidirect
116. M16×C53D8semidirect
156. D4×C53#100semidirect
228. D4×M16D53semidirect
265. C53#9253semidirect
355. M16#11719922semidirect
400. D4#13133133semidirect

Maximal subgroups

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

subgroupindex
C2×#168108182
#202755552
#202733842
#202731402
#202725892
#202725662
D4×M16×C532
#925353