Finite Group Databasebeta

← #27286711

Subgroups — #27286711

Order 13568.

562 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811026C211yesmaximal, characteristic, complemented
2C2×#16810914C211yesmaximal, characteristic
3C2×#16810910C211yesmaximal, characteristic
4#20275508C211yesmaximal, characteristic
5C2×#16810371C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#333C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16811026—22no—
11#16811026—22no—
12#16810914—22no—
13#16810914—22no—
14#16810910—22no—
15#16810910—22no—
16C2×#13133150—22no—
17C2×#13133149—21no—
18C2×#13133148—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133139—21no—
21C2×#13133137—21no—
22C2×#13133129—22no—
23C2×C4×#88322C2211yescharacteristic
24#16810371—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#333—22no—
29C22×Dic106—21no—
30C22×D212D411yescharacteristic
31C22×C4×D53—21no—
32C22×#88322—21no—
33C22×#88322D412yes—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217D412yes—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53D411yescharacteristic
45C2×#11719921—21no—
46#13133150—28no—
47#13133149—24no—
48#13133148—24no—
49C4⋊#88322—21no—
50C4⋊#88322D412yes—
51#13133139—24no—
52#13133137—24no—
53#13133129—44no—
54C4×#88322—21no—
55C4×#88322—21no—
56C22×Q8×C53—21no—
57C22×C8×C53—21no—
58C22×C8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53D412yes—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53D412yes—
70C23×D53—21no—
71C2×Q8×C53—22no—
72C2×Q8×C53—41no—
73C2×Q8×C53—82no—
74C22×C4×C53—41no—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—41no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212Q1612yes—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—42no—
89C2×C4×D53—42no—
90C2×Dic106—22no—
91C2×Dic106—41no—
92C2×Dic106—82no—
93C2×C8×C53—22no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—42no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53C2×D411yescharacteristic
106#11719921—41no—
107#11719921—41no—
108C4⋊Dic53—22no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111C4⋊Dic53—22no—
112C4⋊Dic53—41no—
113C4⋊Dic53—41no—
114C2×#88322—22no—
115C2×#88322—22no—
116C2×#88322—22no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322SD1614yes—
120C23×C53#12111yescharacteristic
121C22×D53—22no—
122C22×D53—41no—
123C22×D53—42no—
124C22×D53—41no—
125C22×D53—42no—
126Q8×C53—82no—
127Q8×C53—82no—
128C2×C4×C53—22no—
129C2×C4×C53—42no—
130C2×C4×C53—82no—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12011yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12711yescharacteristic
137C2×C4×C53C2×SD1612yes—
138C2×C4×C53C2×Q1611yescharacteristic
139C2×C4×C53#13511yescharacteristic
140C2×C4×C53#13711yescharacteristic
141C2×C4×C53#13712yes—
142C2×C4×C53#13711yescharacteristic
143C2×Dic53—42no—
144C2×Dic53—41no—
145C2×Dic53—42no—
146C2×Dic53—82no—
147D212—22no—
148D212—22no—
149D212—41no—
150D212—41no—
151C4×D53—82no—
152C4×D53—81no—
153C4×D53—81no—
154Dic106—82no—
155Dic106—82no—
156C8×C53—42no—
157C8×C53—81no—
158#88322—24no—
159#88322—42no—
160#10278—531nomaximal, Sylow
161C22×C53#39211yescharacteristic
162C22×C53#46112yes—
163C22×C53#46711yescharacteristic
164C22×C53#47011yescharacteristic
165C22×C53#47212yes—
166C2×D53—44no—
167C2×D53—42no—
168C2×D53—82no—
169C2×D53—81no—
170C2×D53—81no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—82no—
174C4×C53#44812yes—
175C4×C53#45111yescharacteristic
176C4×C53#45111yescharacteristic
177C4×C53#45211yescharacteristic
178C4×C53#45211yescharacteristic
179C4×C53#49211yescharacteristic
180C4×C53#49711yescharacteristic
181Dic53—81no—
182Dic53—82no—
183C2×#500—531no—
184C2×#476—531no—
185C2×#476—531no—
186#1954—531no—
187C2×#333—531no—
188C2×#333D5311yescharacteristic, complemented
189#1277—531no—
190C2×C53#167512yes—
191C2×C53#169512yes—
192C2×C53#200111yescharacteristic
193C2×C53#201311yescharacteristic
194C2×C53#201811yescharacteristic
195D53—82no—
196C2×C4⋊Q8—531no—
197C2×C4⋊Q8C2×D5311yescharacteristic
198C2×C4×D4—531no—
199#500—1062no—
200#500—1062no—
201#476—1062no—
202#476—1062no—
203#476—1062no—
204#476—1062no—
205C2×#107—1062no—
206C2×#106—1061no—
207C2×#106—1061no—
208C2×C4⋊C8—531no—
209C2×C4⋊C8C2×D5311yescharacteristic
210C2×C4⋊C8C2×D5311yescharacteristic
211C2×#102—1061no—
212C2×#102—1061no—
213C2×C22⋊C8—1062no—
214C2×C4×C8—531no—
215#333—22no—
216#333—1062no—
217C53#1027811yescharacteristic, complemented, Sylow
218C22×Q8—21no—
219C22×Q8—1061no—
220C22×D4—531no—
221C23×C4—1061no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—532no—
225C22×C8—1061no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—532no—
230C4⋊Q8—1061no—
231C4⋊Q8—1061no—
232C4⋊Q8—1061no—
233C4⋊Q8D21212yes—
234C4×D4—1061no—
235C4×D4—1061no—
236C4×D4—1061no—
237C4×D4—1061no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C22⋊C4—1061no—
244C2×C42C22×D5311yescharacteristic
245#107—1068no—
246#106—1064no—
247#106—1064no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—532no—
252C4⋊C8—1061no—
253C4⋊C8C22⋊D5312yes—
254#102—1064no—
255#102—1064no—
256C22⋊C8—2124no—
257C4×C8—1061no—
258C4×C8—1061no—
259C24—1061no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—82no—
263C2×Q8—1062no—
264C2×Q8—4242no—
265C2×Q8—2121no—
266C2×D4—532no—
267C2×D4—1061no—
268C2×D4—1061no—
269C2×D4—1061no—
270C2×D4—1061no—
271C2×D4—1061no—
272C22×C4—41no—
273C22×C4—1061no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4—2121no—
277C22×C4—2122no—
278C22×C4—2122no—
279C22×C4#1171994511yescharacteristic
280C22×C4D4×D5311yescharacteristic
281C22×C4D4×D5311yescharacteristic
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—534no—
288C2×C8—1062no—
289C2×C8—1062no—
290C2×C8—1062no—
291C2×C8—2121no—
292C2×C8—2121no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—1062no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1062no—
301C4⋊C4—2121no—
302C4⋊C4—2121no—
303C22⋊C4—2121no—
304C22⋊C4—2121no—
305C42C2×D21211yescharacteristic
306C42D4×D5311yescharacteristic
307C42D4×D5311yescharacteristic
308C42C2×C22⋊D5311yescharacteristic
309C23—1062no—
310C23—2121no—
311C23—2122no—
312C23—2121no—
313C23—2122no—
314C23#1313319011yescharacteristic
315Q8—82no—
316Q8—82no—
317Q8—4242no—
318Q8—4242no—
319D4—1062no—
320D4—1062no—
321D4—2121no—
322D4—2121no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—4242no—
327C2×C4—4241no—
328C2×C4—4242no—
329C2×C4—4241no—
330C2×C4—2121no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4#1313318511yescharacteristic
334C2×C4#1313318711yescharacteristic
335C2×C4#1313318711yescharacteristic
336C2×C4#1313319011yescharacteristic
337C2×C4#1313319111yescharacteristic
338C2×C4#1313321311yescharacteristic
339C2×C4SD16×D5312yes—
340C2×C4Q16×D5311yescharacteristic
341C2×C4#1313323811yescharacteristic
342C2×C4#1313324611yescharacteristic
343C2×C4#1313324912yes—
344C2×C4#1313324911yescharacteristic
345C8—42no—
346C8—81no—
347C8—1064no—
348C8—2122no—
349C22—4242no—
350C22—4241no—
351C22—4241no—
352C22—2122no—
353C22—2124no—
354C22#1681053211yescharacteristic
355C22#1681082712yes—
356C22#1681086311yescharacteristic
357C22#1681088011yescharacteristic
358C22#1681089212yes—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—4241no—
363C4—4242no—
364C4#1681076312yes—
365C4#1681077711yescharacteristic
366C4#1681078011yescharacteristic
367C4#1681078211yescharacteristic
368C4#1681078811yescharacteristic
369C4#1681099411yescharacteristic
370C4#1681101611yescharacteristic
371C2—4242no—
372C2#2027411312yes—
373C2#2027423712yes—
374C2#2027572411yescharacteristic
375C2#2027579911yescharacteristic
376C2#2027582311yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#333C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenterC2×#333D53
hypocenterC53#10278
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
1. C2×#16811026C2semidirect
5. C2×#16810371C2semidirect
6. #20272549C2semidirect
7. C2×C53×#333C2semidirect
188. C2×#333D53semidirect
217. C53#10278semidirect

Maximal subgroups

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

subgroupindex
C2×#168110262
C2×#168109142
C2×#168109102
#202755082
C2×#168103712
#202725492
C2×C53×#3332
#1027853