Finite Group Databasebeta

← #27285153

Subgroups — #27285153

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810999C211yesmaximal, characteristic
2#20279714C211yesmaximal, characteristic, complemented
3#20275643C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic, complemented
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×#460C211yesmaximal, characteristic, complemented
8C22×#11719922—22no—
9C2×#13133214C2211yescharacteristic
10C2×#13133206—21no—
11#16810999—22no—
12#16810999—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133138—22no—
15C2×C4×#88322C2211yescharacteristic
16#16810540—22no—
17#16810525—22no—
18#16810524—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810366—22no—
21Q8⋊#88322—22no—
22C22×SD16×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C2×C4×Q8×C53C2211yescharacteristic
25C53×#460—22no—
26C53×#460—22no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#102×C53—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212D412yes—
32C2×#11719922—28no—
33C2×#11719922—48no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133206—42no—
41#13133206—42no—
42C2×C4⋊Dic53D412yes—
43C2×#11719921—24no—
44C2×#11719921—24no—
45C2×C4×Dic53—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133138—44no—
49C4×#88322—21no—
50C4×#88322D412yes—
51#13133124—21no—
52C22×Q8×C53D411yescharacteristic
53C22×D4×C53—21no—
54C22×D4×C53—21no—
55C22×D4×C53—21no—
56C2×SD16×C53—24no—
57C2×SD16×C53—42no—
58C2×SD16×C53—42no—
59C22×C8×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53—21no—
63#127×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×C53D411yescharacteristic
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—21no—
73C4⋊C8×C53D412yes—
74#102×C53—24no—
75C24×C53—41no—
76C23×D53—22no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53D812yescomplemented
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—44no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—42no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—42no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×Dic53—22no—
102C2×D212—22no—
103C2×D212—22no—
104C2×D212—44no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—41no—
111C2×D212—42no—
112C2×D212—41no—
113C2×D212—41no—
114C2×D212—42no—
115SD16×C53—48no—
116C2×C8×C53—22no—
117C2×C8×C53—41no—
118C2×C8×C53—41no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C42×C53—41no—
124C42×C53—41no—
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129#11719922—416no—
130#11719921—48no—
131#11719921—84no—
132C4⋊Dic53—22no—
133C4⋊Dic53—22no—
134C4×Dic53—41no—
135C4×Dic53—41no—
136C2×#88322—24no—
137C2×#88322—22no—
138C2×#88322—41no—
139C2×#88322—42no—
140C23×C53—42no—
141C23×C53—81no—
142C23×C53—81no—
143C23×C53—82no—
144C23×C53—82no—
145C23×C53#12011yescharacteristic
146C22×D53—42no—
147C22×D53—42no—
148C22×D53—42no—
149C22×D53—44no—
150C22×D53—44no—
151Q8×C53—22no—
152Q8×C53—22no—
153Q8×C53—41no—
154Q8×C53—41no—
155D4×C53—44no—
156D4×C53—42no—
157D4×C53—42no—
158D4×C53—44no—
159D4×C53—82no—
160D4×C53—81no—
161D4×C53—81no—
162D4×C53—81no—
163D4×C53—81no—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12011yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12711yescharacteristic
176C2×C4×C53C2×D811yescharacteristic
177C2×C4×C53#13511yescharacteristic
178C2×C4×C53#13512yes—
179C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C53AGL(1,ℤ/8)12yes—
182C2×Dic53—42no—
183C2×Dic53—42no—
184C2×Dic53—42no—
185D212—44no—
186D212—82no—
187D212—81no—
188D212—81no—
189D212—82no—
190D212—81no—
191D212—81no—
192D212—81no—
193D212—81no—
194C8×C53—42no—
195#88322—24no—
196#88322—81no—
197#10047—531nomaximal, Sylow
198C22×C53—82no—
199C22×C53—82no—
200C22×C53—82no—
201C22×C53—162no—
202C22×C53—162no—
203C22×C53#39211yescharacteristic
204C22×C53#44711yescharacteristic
205C22×C53#44911yescharacteristic
206C22×C53#44912yes—
207C22×C53#45312yes—
208C2×D53—84no—
209C2×D53—84no—
210C2×D53—82no—
211C2×D53—82no—
212C2×D53—82no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—42no—
216C4×C53—81no—
217C4×C53#44711yescharacteristic
218C4×C53#44911yescharacteristic
219C4×C53#46312yes—
220C4×C53#46611yescharacteristic
221C4×C53#46911yescharacteristic
222C4×C53#49511yescharacteristic
223C4×C53#49611yescharacteristic
224Dic53—82no—
225C2×#493—531no—
226C2×#460D5311yescharacteristic, complemented
227#2548—531no—
228#1982—531no—
229C2×#331—531no—
230C2×Q8⋊C8—531no—
231#1279—531no—
232C2×C53—162no—
233C2×C53#162012yes—
234C2×C53#162112yes—
235C2×C53#199911yescharacteristic
236C2×C53#200211yescharacteristic
237C2×C53#201211yescharacteristic
238D53—162no—
239C22×SD16—21no—
240C22×D8—1062no—
241C2×#127—531no—
242C2×#127C2×D5311yescharacteristic
243C2×#124—1061no—
244C2×C4×Q8C2×D5311yescharacteristic
245#493—1062no—
246#493—1062no—
247#460—22no—
248#460—22no—
249C2×C4⋊C8—531no—
250C2×C4⋊C8C2×D5311yescharacteristic
251C2×#102—21no—
252C2×#102—1062no—
253C2×C4×C8—531no—
254#394—1062no—
255#390—1062no—
256#390—1061no—
257C4×C4⋊C4—531no—
258#331—1062no—
259Q8⋊C8—1062no—
260C53#1004711yescharacteristic, complemented, Sylow
261C22×Q8C22⋊D5311yescharacteristic
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4—532no—
266C22×D4—1061no—
267C22×D4—1061no—
268C2×SD16—24no—
269C2×SD16—42no—
270C2×SD16—42no—
271C2×D8—1068no—
272C2×D8—2128no—
273C22×C8—21no—
274C22×C8—1061no—
275C22×C8—1061no—
276#127—21no—
277#127—21no—
278#127—21no—
279#127—1061no—
280#127—1061no—
281#127—1061no—
282#127—1061no—
283#127C22⋊D5312yes—
284#124—2122no—
285#124—2122no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—532no—
292C2×C4⋊C4C22⋊D5311yescharacteristic
293C2×C22⋊C4—1064no—
294C2×C22⋊C4—1064no—
295C2×C42—21no—
296C2×C42—1061no—
297C2×C42C22×D5311yescharacteristic
298C4⋊C8—21no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301C4⋊C8C22⋊D5312yes—
302#102—24no—
303#102—2124no—
304C4×C8—532no—
305C4×C8—1061no—
306#95—1061no—
307C24—41no—
308C24—1062no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8#1171992212yescomplemented
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—42no—
328C2×D4—1062no—
329C2×D4—1062no—
330C2×D4—2124no—
331C2×D4—2121no—
332C2×D4—2121no—
333C2×D4—2121no—
334C2×D4—2121no—
335C2×D4—2121no—
336C2×D4—2121no—
337C2×D4—2122no—
338C2×D4—2121no—
339C2×D4—2121no—
340C2×D4—2122no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—1062no—
344C22×C4D4×D5311yescharacteristic
345C22×C4D4×D5311yescharacteristic
346C22×C4C2×C22⋊D5311yescharacteristic
347SD16—48no—
348D8—21216no—
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—1064no—
353C2×C8—1062no—
354C2×C8—2121no—
355C2×C8—2122no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—1062no—
361C4⋊C4—1062no—
362C22⋊C4—4244no—
363C22⋊C4—2128no—
364C42—41no—
365C42—41no—
366C42—2121no—
367C42—2121no—
368C42D4×D5311yescharacteristic
369C42#1171994711yescharacteristic
370C42C2×C22⋊D5311yescharacteristic
371C42C2×C22⋊D5311yescharacteristic
372C23—42no—
373C23—81no—
374C23—81no—
375C23—82no—
376C23—82no—
377C23—2122no—
378C23—2122no—
379C23—2124no—
380C23—2122no—
381C23—2124no—
382C23#1313318511yescharacteristic
383Q8—22no—
384Q8—22no—
385Q8—41no—
386Q8—41no—
387D4—44no—
388D4—42no—
389D4—42no—
390D4—44no—
391D4—82no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—81no—
396D4—4242no—
397D4—4241no—
398D4—4241no—
399D4—4241no—
400D4—4242no—
401D4—4241no—
402D4—4241no—
403D4—4241no—
404D4—2124no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—2122no—
412C2×C4—2122no—
413C2×C4—2122no—
414C2×C4#1313318511yescharacteristic
415C2×C4#1313318611yescharacteristic
416C2×C4#1313318811yescharacteristic
417C2×C4#1313318911yescharacteristic
418C2×C4#1313318911yescharacteristic
419C2×C4#1313321311yescharacteristic
420C2×C4C2×#1171992211yescharacteristic
421C2×C4#1313323711yescharacteristic
422C2×C4#1313323812yes—
423C2×C4#1313324211yescharacteristic
424C2×C4#1313324311yescharacteristic
425C2×C4#1313324412yes—
426C8—42no—
427C8—1064no—
428C8—4241no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—162no—
433C22—162no—
434C22—4242no—
435C22—4242no—
436C22—4244no—
437C22—4244no—
438C22—4242no—
439C22#1681053111yescharacteristic
440C22#1681075611yescharacteristic
441C22#1681076611yescharacteristic
442C22#1681076712yes—
443C22#1681078912yes—
444C4—21no—
445C4—21no—
446C4—42no—
447C4—81no—
448C4—4242no—
449C4#1681075711yescharacteristic
450C4#1681076811yescharacteristic
451C4#1681084012yes—
452C4#1681085711yescharacteristic
453C4#1681087411yescharacteristic
454C4#1681100411yescharacteristic
455C4#1681100911yescharacteristic
456C2—162no—
457C2—8482no—
458C2#2027377312yes—
459C2#2027377812yes—
460C2#2027571111yescharacteristic
461C2#2027572711yescharacteristic
462C2#2027578911yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#460C2
socleC23×C53#120
prime radicalC2×#16810999C2
hypercenterC2×#460D53
hypocenterC53#10047
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #20279714C2semidirect
3. #20275643C2semidirect
4. C2×#16810366C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. #20272554C2semidirect
7. C2×C53×#460C2semidirect
82. C2×Q8×C53D8semidirect
226. C2×#460D53semidirect
260. C53#10047semidirect
314. C2×Q8#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109992
#202797142
#202756432
C2×#168103662
C2×Q8⋊#883222
#202725542
C2×C53×#4602
#1004753