Finite Group Databasebeta

← #27280331

Subgroups — #27280331

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810701C211yesmaximal, characteristic, complemented
2#20275522C211yesmaximal, characteristic, complemented
3#20274715C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5#20272599C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#459C211yesmaximal, characteristic
8C22×#11719917—21no—
9#16810701—22no—
10#16810701—22no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133138—21no—
14C2×#13133138—21no—
15C2×#13133138—21no—
16C2×C22⋊#88322—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810493C2211yescharacteristic
19#16810385—21no—
20#16810367—22no—
21C22×D8×C53—21no—
22C2×#127×C53C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#459—22no—
25C53×#459C414yescomplemented
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×#102×C53—21no—
28C2×#11719917—24no—
29C2×#11719917—42no—
30C2×#11719917—42no—
31C22×#88322—21no—
32C22×#88322—21no—
33C22×#88322D411yescharacteristic
34C22×#88322D411yescharacteristic
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322D412yes—
42#13133138—42no—
43#13133138—42no—
44#13133138—42no—
45C22⋊#88322—24no—
46C4×#88322—21no—
47C4×#88322—21no—
48C22×D4×C53—21no—
49C22×D4×C53—21no—
50C22×D4×C53—21no—
51C22×D4×C53D411yescharacteristic
52C23×C4×C53—21no—
53C2×D8×C53—24no—
54C2×D8×C53—42no—
55C2×D8×C53—42no—
56C22×C8×C53—21no—
57#127×C53—21no—
58#127×C53—21no—
59#127×C53—21no—
60#127×C53C2×C412yes—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53C2×C412yes—
64C4×D4×C53D412yes—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53C2×C412yes—
70#102×C53—24no—
71C24×C53—21no—
72C24×C53—41no—
73C2×D4×C53—22no—
74C2×D4×C53—22no—
75C2×D4×C53—22no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—81no—
93C2×D4×C53—81no—
94C2×D4×C53C22⋊C412yes—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×Dic53—41no—
106D8×C53—48no—
107C2×C8×C53—22no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53C22⋊C412yes—
112C22⋊C4×C53—22no—
113C22⋊C4×C53—22no—
114C42×C53C22×C411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C4⋊Dic53—81no—
123#11719917—48no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—22no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322D812yescomplemented
136C23×C53—22no—
137C23×C53—41no—
138C23×C53—41no—
139C23×C53—41no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—42no—
144C23×C53—81no—
145C23×C53—81no—
146C23×C53—81no—
147C23×C53—81no—
148C23×C53—81no—
149C23×C53—81no—
150C23×C53#12111yescharacteristic
151D4×C53—22no—
152D4×C53—22no—
153D4×C53—42no—
154D4×C53—42no—
155D4×C53—42no—
156D4×C53—42no—
157D4×C53—41no—
158D4×C53—41no—
159D4×C53—82no—
160D4×C53—82no—
161D4×C53—81no—
162D4×C53—81no—
163D4×C53—81no—
164D4×C53—81no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—22no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53C2×C22⋊C411yescharacteristic
175C2×C4×C53C4×D411yescharacteristic
176C2×C4×C53C4×D411yescharacteristic
177C2×C4×C53#12011yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12311yescharacteristic
180C2×C4×C53C2×D811yescharacteristic
181C2×C4×C53#13511yescharacteristic
182C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C53#13711yescharacteristic
188C2×Dic53—81no—
189C2×Dic53—81no—
190C2×Dic53—81no—
191C8×C53—42no—
192#88322—22no—
193#88322—42no—
194#88322—41no—
195#88322—81no—
196#9229—531nomaximal, Sylow
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—42no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—81no—
203C22×C53—81no—
204C22×C53—82no—
205C22×C53—82no—
206C22×C53—82no—
207C22×C53—161no—
208C22×C53—161no—
209C22×C53—161no—
210C22×C53—161no—
211C22×C53#38611yescharacteristic
212C22×C53#45911yescharacteristic
213C22×C53#46311yescharacteristic
214C22×C53#46811yescharacteristic
215C22×C53#46911yescharacteristic
216C22×C53#47111yescharacteristic
217C22×C53#47211yescharacteristic
218C4×C53—21no—
219C4×C53—21no—
220C4×C53—42no—
221C4×C53#41711yescharacteristic
222C4×C53#41811yescharacteristic
223C4×C53C4×D811yescharacteristic
224C4×C53#44211yescharacteristic
225C4×C53#44711yescharacteristic
226C4×C53#44911yescharacteristic
227C4×C53#46811yescharacteristic
228C4×C53#46911yescharacteristic
229Dic53—161no—
230C2×#459D5311yescharacteristic
231C2×#436—531no—
232#1957—531no—
233#1802—531no—
234C2×#331—531no—
235#1296—531no—
236#1277—531no—
237C2×C53—82no—
238C2×C53—162no—
239C2×C53#168511yescharacteristic
240C2×C53#168711yescharacteristic
241C2×C53#169711yescharacteristic
242C2×C53#170111yescharacteristic
243C2×C53#187911yescharacteristic
244C2×C53#189511yescharacteristic
245C2×C53#190711yescharacteristic
246C22×D8—21no—
247C22×M16—1061no—
248C2×#127C2×D5311yescharacteristic
249C2×C4×D4C2×D5311yescharacteristic
250#459—22no—
251#459Dic5314yescomplemented
252#436—1062no—
253#436—1062no—
254C2×C4⋊C8—531no—
255C2×C4⋊C8—531no—
256C2×C4⋊C8C2×D5311yescharacteristic
257C2×#102—21no—
258C2×#102—1061no—
259C2×#102—1061no—
260C2×#102—1061no—
261C2×C22⋊C8—1061no—
262C2×C4×C8—531no—
263#384—531no—
264#340—1061no—
265#331—1062no—
266C53#922911yescharacteristic, complemented, Sylow
267C22×D4—21no—
268C22×D4—21no—
269C22×D4—21no—
270C22×D4C22⋊D5311yescharacteristic
271C23×C4—21no—
272C2×D8—24no—
273C2×D8—42no—
274C2×D8—42no—
275C2×M16—1064no—
276C2×M16—2122no—
277C2×M16—2122no—
278C22×C8—21no—
279C22×C8—531no—
280C22×C8—531no—
281C22×C8—1061no—
282C22×C8—1061no—
283#127—21no—
284#127—21no—
285#127—21no—
286#127C2×Dic5312yes—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C2×Dic5312yes—
290C4×D4C22⋊D5312yes—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4—1061no—
293C2×C4⋊C4—1061no—
294C2×C4⋊C4C22⋊D5311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C42C22×D5311yescharacteristic
297C4⋊C8—21no—
298C4⋊C8—532no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301C4⋊C8—1061no—
302C4⋊C8C2×Dic5312yes—
303#102—24no—
304#102—2122no—
305#102—2122no—
306#102—2122no—
307C22⋊C8—1064no—
308C4×C8—1061no—
309C4×C8—1061no—
310C24—21no—
311C24—41no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—41no—
321C2×D4—41no—
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—41no—
331C2×D4—81no—
332C2×D4—81no—
333C2×D4C22⋊Dic5312yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—2121no—
342C22×C4D4×D5311yescharacteristic
343C22×C4#1171994711yescharacteristic
344C22×C4C2×C22⋊D5311yescharacteristic
345D8—48no—
346M16—2128no—
347C2×C8—22no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—532no—
351C2×C8—1061no—
352C2×C8—1061no—
353C2×C8—1061no—
354C2×C8—1062no—
355C2×C8—1061no—
356C2×C8—1061no—
357C2×C8—2121no—
358C2×C8—2121no—
359C2×C8—2121no—
360C2×C8—2121no—
361C2×C8—2121no—
362C4⋊C4—21no—
363C4⋊C4—4241no—
364C4⋊C4—2121no—
365C4⋊C4—2121no—
366C4⋊C4—2121no—
367C4⋊C4—2121no—
368C4⋊C4C22⋊Dic5312yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42D4×D5311yescharacteristic
372C42#1171994711yescharacteristic
373C42C22×Dic5311yescharacteristic
374C42C2×C22⋊D5311yescharacteristic
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23#1313318911yescharacteristic
390D4—22no—
391D4—22no—
392D4—41no—
393D4—41no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—42no—
398D4—82no—
399D4—82no—
400D4—81no—
401D4—81no—
402D4—81no—
403D4—81no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—4241no—
414C2×C4—4241no—
415C2×C4—4241no—
416C2×C4C2×C22⋊Dic5311yescharacteristic
417C2×C4D4×Dic5311yescharacteristic
418C2×C4D4×Dic5311yescharacteristic
419C2×C4#1313318511yescharacteristic
420C2×C4#1313318911yescharacteristic
421C2×C4#1313319811yescharacteristic
422C2×C4D8×D5311yescharacteristic
423C2×C4#1313323611yescharacteristic
424C2×C4#1313324111yescharacteristic
425C2×C4#1313324111yescharacteristic
426C2×C4#1313324211yescharacteristic
427C2×C4#1313324311yescharacteristic
428C2×C4#1313324311yescharacteristic
429C2×C4#1313324711yescharacteristic
430C8—42no—
431C8—1062no—
432C8—4241no—
433C8—2122no—
434C8—2121no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—82no—
443C22—82no—
444C22—82no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22#1681050311yescharacteristic
450C22#1681081611yescharacteristic
451C22#1681083611yescharacteristic
452C22#1681086711yescharacteristic
453C22#1681087411yescharacteristic
454C22#1681088411yescharacteristic
455C22#1681089111yescharacteristic
456C4—21no—
457C4—21no—
458C4—42no—
459C4—8481no—
460C4#1681062411yescharacteristic
461C4#1681062811yescharacteristic
462C4D8×Dic5311yescharacteristic
463C4#1681073511yescharacteristic
464C4#1681075511yescharacteristic
465C4#1681076511yescharacteristic
466C4#1681086711yescharacteristic
467C4#1681087411yescharacteristic
468C2—82no—
469C2—162no—
470C2#2027417811yescharacteristic
471C2#2027419211yescharacteristic
472C2#2027424611yescharacteristic
473C2#2027427411yescharacteristic
474C2#2027506511yescharacteristic
475C2#2027515411yescharacteristic
476C2#2027522511yescharacteristic
subgroupG/H
centerC23#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#459C2
socleC23×C53#121
cosocleC2×C42×C53C23
hypercenterC2×#459D53
hypocenterC53#9229
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#16810701C2semidirect
2. #20275522C2semidirect
3. #20274715C2semidirect
4. C2×#16810367C2semidirect
5. #20272599C2semidirect
6. #20272549C2semidirect
25. C53×#459C4semidirect
135. C2×#88322D8semidirect
251. #459Dic53semidirect
266. C53#9229semidirect

Maximal subgroups

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

subgroupindex
C2×#168107012
#202755222
#202747152
C2×#168103672
#202725992
#202725492
C2×C53×#4592
#922953