Finite Group Databasebeta

← #27286640

Subgroups — #27286640

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810970C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3C2×#16810818C211yesmaximal, characteristic
4C2×#16810371C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272550C211yesmaximal, characteristic, complemented
7C53×#1954C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133217C2211yescharacteristic
10C2×#13133214C2211yescharacteristic
11#16810970—22no—
12#16810970—22no—
13#16810913—22no—
14#16810913—22no—
15#16810818—22no—
16#16810818—22no—
17C2×#13133148—21no—
18C2×C4⋊#88322C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133139—22no—
21C2×#13133138—21no—
22C2×#13133137—21no—
23#16810371—22no—
24#16810366—22no—
25C2×C4×D4×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C22⋊C8×C53—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212—21no—
33C2×#11719922—24no—
34C2×#11719922—42no—
35C2×#11719922—42no—
36C22×#88322—21no—
37C22×#88322—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217—21no—
41#13133217D412yes—
42#13133214—21no—
43#13133214—21no—
44#13133214—21no—
45#13133214D412yes—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48#13133148—24no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322D412yes—
52C4⋊#88322D412yes—
53#13133139—28no—
54#13133138—24no—
55#13133137—24no—
56C22×D4×C53D411yescharacteristic
57C23×C4×C53—21no—
58C22×C8×C53—21no—
59C22×C8×C53D412yes—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C22⋊C8×C53—44no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C24×C53—21no—
73C23×D53—41no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53SD1612yes—
80C22×C4×C53—21no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×Dic53—41no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—41no—
93C2×D212—42no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—42no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—82no—
102C2×Dic106—22no—
103C2×Dic106—41no—
104C2×Dic106—82no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53—41no—
109C2×C8×C53—42no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118#11719922—48no—
119C4⋊Dic53—22no—
120C4⋊Dic53—22no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C23×C53—22no—
130C23×C53—41no—
131C23×C53—42no—
132C23×C53—41no—
133C23×C53—42no—
134C23×C53#12111yescharacteristic
135C22×D53—42no—
136C22×D53—82no—
137C22×D53—81no—
138C22×D53—81no—
139C22×D53—82no—
140D4×C53—22no—
141D4×C53—22no—
142D4×C53—41no—
143D4×C53—41no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—82no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12711yescharacteristic
156C2×C4×C53C2×SD1611yescharacteristic
157C2×C4×C53#13512yes—
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)12yes—
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53#13711yescharacteristic
162C2×Dic53—42no—
163C2×Dic53—82no—
164D212—44no—
165D212—42no—
166D212—42no—
167D212—82no—
168D212—82no—
169D212—81no—
170D212—81no—
171D212—81no—
172D212—81no—
173Dic106—82no—
174Dic106—82no—
175C8×C53—42no—
176C8×C53—81no—
177#88322—42no—
178#88322—42no—
179#10267—531nomaximal, Sylow
180C22×C53—44no—
181C22×C53—42no—
182C22×C53—82no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53#39211yescharacteristic
186C22×C53#46312yes—
187C22×C53#46511yescharacteristic
188C22×C53#46811yescharacteristic
189C22×C53#47212yes—
190C2×D53—82no—
191C2×D53—82no—
192C2×D53—82no—
193C2×D53—162no—
194C2×D53—162no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—81no—
198C4×C53#44811yescharacteristic
199C4×C53#44912yes—
200C4×C53#44911yescharacteristic
201C4×C53#45011yescharacteristic
202C4×C53#45211yescharacteristic
203C4×C53#49511yescharacteristic
204C4×C53#49611yescharacteristic
205Dic53—82no—
206C2×#486—531no—
207C2×#476—531no—
208C2×#459—531no—
209#1954D5311yescharacteristic, complemented
210C2×#333—531no—
211C2×#331—531no—
212#1277—531no—
213C2×C53—82no—
214C2×C53#167712yes—
215C2×C53#169712yes—
216C2×C53#200011yescharacteristic
217C2×C53#201211yescharacteristic
218C2×C53#201811yescharacteristic
219D53—162no—
220C22×D8—1061no—
221C2×C4⋊Q8—531no—
222C2×#127—531no—
223C2×C4×D4C2×D5311yescharacteristic
224#486—1062no—
225#486—1062no—
226#476—1062no—
227#476—1062no—
228#459—1062no—
229#459—1062no—
230C2×#106—1061no—
231C2×C4⋊C8—531no—
232C2×C4⋊C8—531no—
233C2×C4⋊C8C2×D5311yescharacteristic
234C2×#102—1061no—
235C2×#102—1062no—
236C2×#102—1061no—
237C2×C22⋊C8—22no—
238C2×C4×C8C2×D5311yescharacteristic
239#333—1062no—
240#331—1062no—
241C53#1026711yescharacteristic, complemented, Sylow
242C22×Q8—1061no—
243C22×D4—1061no—
244C22×D4—1061no—
245C22×D4—1061no—
246C22×D4D21211yescharacteristic
247C23×C4—21no—
248C2×D8—1064no—
249C2×D8—2122no—
250C2×D8—2122no—
251C22×C8—21no—
252C22×C8—1061no—
253C22×C8—1061no—
254C22×C8C22⋊D5312yes—
255C4⋊Q8—532no—
256C4⋊Q8—1061no—
257C4⋊Q8—1061no—
258C4⋊Q8—1061no—
259#127—532no—
260#127—1061no—
261#127—1061no—
262#127—1061no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4D21211yescharacteristic
270C2×C22⋊C4—21no—
271C2×C42C22×D5311yescharacteristic
272#106—1064no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—532no—
276C4⋊C8—532no—
277C4⋊C8—1061no—
278C4⋊C8—1061no—
279#102—1064no—
280#102—1068no—
281#102—1064no—
282C22⋊C8—44no—
283C4×C8—21no—
284C4×C8—21no—
285C24—21no—
286C24—2121no—
287C2×Q8—1062no—
288C2×Q8—4242no—
289C2×Q8—2121no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—4242no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2122no—
304C2×D4—2121no—
305C2×D4—2122no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4#1171991312yes—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—2121no—
315C22×C4C2×D21211yescharacteristic
316C22×C4#1171994511yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318D8—2128no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—1062no—
325C2×C8—1062no—
326C2×C8—2121no—
327C2×C8—2121no—
328C2×C8—2121no—
329C2×C8—2121no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—2121no—
335C4⋊C4—2121no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42D4×D5311yescharacteristic
339C42D4×D5311yescharacteristic
340C42D4×D5311yescharacteristic
341C42D4×D5311yescharacteristic
342C23—22no—
343C23—41no—
344C23—42no—
345C23—41no—
346C23—42no—
347C23—4242no—
348C23—4241no—
349C23—4241no—
350C23—4242no—
351C23—2122no—
352C23#1313319111yescharacteristic
353Q8—4242no—
354Q8—4242no—
355D4—22no—
356D4—22no—
357D4—41no—
358D4—41no—
359D4—4242no—
360D4—4242no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—4241no—
365D4—2124no—
366D4—2122no—
367D4—2122no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—4242no—
375C2×C4—2122no—
376C2×C4#1313318511yescharacteristic
377C2×C4#1313318711yescharacteristic
378C2×C4#1313318711yescharacteristic
379C2×C4#1313319011yescharacteristic
380C2×C4#1313319011yescharacteristic
381C2×C4#1313321311yescharacteristic
382C2×C4C2×#1171991311yescharacteristic
383C2×C4#1313323712yes—
384C2×C4#1313323911yescharacteristic
385C2×C4#1313324312yes—
386C2×C4#1313324511yescharacteristic
387C2×C4#1313325011yescharacteristic
388C8—42no—
389C8—81no—
390C8—2122no—
391C8—2122no—
392C22—42no—
393C22—44no—
394C22—82no—
395C22—81no—
396C22—81no—
397C22—8482no—
398C22—8482no—
399C22—4242no—
400C22—4242no—
401C22—4242no—
402C22#1681053211yescharacteristic
403C22#1681083812yes—
404C22#1681085211yescharacteristic
405C22#1681086911yescharacteristic
406C22#1681089312yes—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—4242no—
411C4#1681076411yescharacteristic
412C4#1681076612yes—
413C4#1681077011yescharacteristic
414C4#1681077611yescharacteristic
415C4#1681078711yescharacteristic
416C4#1681100511yescharacteristic
417C4#1681101011yescharacteristic
418C2—82no—
419C2—8482no—
420C2#2027412412yes—
421C2#2027424812yes—
422C2#2027572011yescharacteristic
423C2#2027579111yescharacteristic
424C2#2027582411yescharacteristic
subgroupG/H
centerC23#13133191
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1954C2
socleC23×C53#121
prime radicalC2×#16810818C2
hypercenter#1954D53
hypocenterC53#10267
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. C2×#16810970C2semidirect
2. C2×#16810913C2semidirect
4. C2×#16810371C2semidirect
5. C2×#16810366C2semidirect
6. #20272550C2semidirect
7. C53×#1954C2semidirect
209. #1954D53semidirect
241. C53#10267semidirect

Maximal subgroups

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

subgroupindex
C2×#168109702
C2×#168109132
C2×#168108182
C2×#168103712
C2×#168103662
#202725502
C53×#19542
#1026753