Finite Group Databasebeta

← #27286659

Subgroups — #27286659

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810973C211yesmaximal, characteristic, complemented
2C2×#16810900C211yesmaximal, characteristic
3C2×#16810829C211yesmaximal, characteristic
4#20275508C211yesmaximal, characteristic
5C2×#16810368C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C22×#11719913—21no—
9C2×#13133217C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810973—22no—
12#16810973—22no—
13#16810900—22no—
14#16810900—22no—
15#16810829—22no—
16#16810829—22no—
17C2×#13133150—21no—
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133142—21no—
20C2×#13133141—22no—
21C2×#13133137—21no—
22C2×#13133129—22no—
23C2×C4×#88322C2211yescharacteristic
24#16810368—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#332—22no—
29C22×Dic106—21no—
30C22×D212D411yescharacteristic
31C22×C4×D53—21no—
32C2×#11719913—24no—
33C2×#11719913—42no—
34C2×#11719913—42no—
35C22×#88322—21no—
36C22×#88322D412yes—
37#13133217—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217D412yes—
41C4×D212—21no—
42C4×D212—21no—
43C4×D212—21no—
44C4×D212—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53D411yescharacteristic
48C2×#11719921—21no—
49#13133150—24no—
50C4⋊#88322—21no—
51C4⋊#88322D412yes—
52#13133142—24no—
53#13133141—28no—
54#13133137—24no—
55#13133129—44no—
56C4×#88322—21no—
57C4×#88322—21no—
58C22×Q8×C53—21no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53D412yes—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D412yes—
72C23×D53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—42no—
76C2×Q8×C53—42no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—41no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212SD1612yes—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—42no—
92C2×C4×D53—42no—
93C2×Dic106—22no—
94C2×Dic106—41no—
95C2×Dic106—41no—
96C2×Dic106—41no—
97C2×Dic106—41no—
98C2×Dic106—41no—
99C2×C8×C53—22no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53—42no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112#11719921—41no—
113#11719921—41no—
114C4⋊Dic53—22no—
115C4⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4⋊Dic53—22no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C2×#88322—22no—
121C2×#88322—22no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126#11719913—48no—
127C23×C53#12111yescharacteristic
128C22×D53—22no—
129C22×D53—41no—
130C22×D53—42no—
131C22×D53—41no—
132C22×D53—42no—
133Q8×C53—44no—
134Q8×C53—81no—
135Q8×C53—81no—
136C2×C4×C53—22no—
137C2×C4×C53—42no—
138C2×C4×C53—82no—
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12711yescharacteristic
145C2×C4×C53C2×SD1611yescharacteristic
146C2×C4×C53#13512yes—
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)12yes—
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53#13711yescharacteristic
151C2×Dic53—42no—
152C2×Dic53—41no—
153C2×Dic53—42no—
154C2×Dic53—81no—
155C2×Dic53—81no—
156D212—22no—
157D212—22no—
158D212—41no—
159D212—41no—
160C4×D53—82no—
161C4×D53—81no—
162C4×D53—81no—
163Dic106—42no—
164Dic106—42no—
165Dic106—81no—
166Dic106—81no—
167C8×C53—42no—
168C8×C53—81no—
169#88322—42no—
170#88322—42no—
171#10270—531nomaximal, Sylow
172C22×C53#39211yescharacteristic
173C22×C53#46312yes—
174C22×C53#46511yescharacteristic
175C22×C53#46811yescharacteristic
176C22×C53#47312yes—
177C2×D53—44no—
178C2×D53—42no—
179C2×D53—82no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—82no—
185C4×C53#44811yescharacteristic
186C4×C53#44912yes—
187C4×C53#44911yescharacteristic
188C4×C53#45011yescharacteristic
189C4×C53#45211yescharacteristic
190C4×C53#49511yescharacteristic
191C4×C53#49611yescharacteristic
192Dic53—82no—
193Dic53—81no—
194C2×#487—531no—
195C2×#474—531no—
196C2×#461—531no—
197#1954—531no—
198C2×#332—531no—
199C2×#332D5311yescharacteristic, complemented
200#1277—531no—
201C2×C53#167912yes—
202C2×C53#169912yes—
203C2×C53#200011yescharacteristic
204C2×C53#201211yescharacteristic
205C2×C53#202011yescharacteristic
206D53—82no—
207C22×SD16—1061no—
208C2×C4⋊Q8—531no—
209C2×C4⋊Q8C2×D5311yescharacteristic
210C2×C4×D4—531no—
211#487—1062no—
212#487—1062no—
213#474—1062no—
214#474—1062no—
215#461—1062no—
216#461—1062no—
217C2×#107—1061no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8C2×D5311yescharacteristic
220C2×C4⋊C8C2×D5311yescharacteristic
221C2×#103—1061no—
222C2×#103—1062no—
223C2×#102—1061no—
224C2×C22⋊C8—1062no—
225C2×C4×C8—531no—
226#332—22no—
227#332—1062no—
228C53#1027011yescharacteristic, complemented, Sylow
229C22×Q8—21no—
230C22×Q8—1061no—
231C22×D4—531no—
232C23×C4—1061no—
233C2×SD16—1064no—
234C2×SD16—2122no—
235C2×SD16—2122no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—532no—
239C22×C8—1061no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—532no—
244C4⋊Q8—1061no—
245C4⋊Q8—1061no—
246C4⋊Q8—1061no—
247C4⋊Q8D21212yes—
248C4×D4—1061no—
249C4×D4—1061no—
250C4×D4—1061no—
251C4×D4—1061no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—531no—
255C2×C4⋊C4—1061no—
256C2×C4⋊C4—1061no—
257C2×C22⋊C4—1061no—
258C2×C42C22×D5311yescharacteristic
259#107—1064no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—532no—
264C4⋊C8—1061no—
265C4⋊C8C22⋊D5312yes—
266#103—1064no—
267#103—1068no—
268#102—1064no—
269C22⋊C8—2124no—
270C4×C8—1061no—
271C4×C8—1061no—
272C24—1061no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—1062no—
278C2×Q8—2121no—
279C2×Q8—2121no—
280C2×Q8—2121no—
281C2×Q8—2121no—
282C2×Q8—2121no—
283C2×D4—532no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C22×C4—41no—
290C22×C4—1061no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2122no—
295C22×C4—2122no—
296C22×C4#1171994511yescharacteristic
297C22×C4D4×D5311yescharacteristic
298C22×C4D4×D5311yescharacteristic
299SD16—2128no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—1062no—
306C2×C8—1062no—
307C2×C8—1062no—
308C2×C8—1062no—
309C2×C8—2121no—
310C2×C8—2121no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1061no—
316C4⋊C4—1062no—
317C4⋊C4—1061no—
318C4⋊C4—1062no—
319C4⋊C4—2121no—
320C4⋊C4—2121no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C42C2×D21211yescharacteristic
324C42D4×D5311yescharacteristic
325C42D4×D5311yescharacteristic
326C42C2×C22⋊D5311yescharacteristic
327C23—1062no—
328C23—2121no—
329C23—2122no—
330C23—2121no—
331C23—2122no—
332C23#1313319011yescharacteristic
333Q8—44no—
334Q8—81no—
335Q8—81no—
336Q8—4241no—
337Q8—4241no—
338Q8—2122no—
339Q8—2122no—
340D4—1062no—
341D4—1062no—
342D4—2121no—
343D4—2121no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—82no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4—4242no—
351C2×C4—4241no—
352C2×C4—2121no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4#1313318511yescharacteristic
356C2×C4#1313318711yescharacteristic
357C2×C4#1313318711yescharacteristic
358C2×C4#1313319011yescharacteristic
359C2×C4#1313319111yescharacteristic
360C2×C4#1313321311yescharacteristic
361C2×C4SD16×D5311yescharacteristic
362C2×C4#1313323812yes—
363C2×C4#1313324011yescharacteristic
364C2×C4#1313324112yes—
365C2×C4#1313324411yescharacteristic
366C2×C4#1313324911yescharacteristic
367C8—42no—
368C8—81no—
369C8—2122no—
370C8—2122no—
371C22—4241no—
372C22—4242no—
373C22—4241no—
374C22—2122no—
375C22—2124no—
376C22#1681053211yescharacteristic
377C22#1681083712yes—
378C22#1681085511yescharacteristic
379C22#1681087211yescharacteristic
380C22#1681089712yes—
381C4—21no—
382C4—21no—
383C4—82no—
384C4—4242no—
385C4—4241no—
386C4#1681076311yescharacteristic
387C4#1681076912yes—
388C4#1681077111yescharacteristic
389C4#1681077211yescharacteristic
390C4#1681078411yescharacteristic
391C4#1681100411yescharacteristic
392C4#1681101111yescharacteristic
393C2—4242no—
394C2#2027413712yes—
395C2#2027426112yes—
396C2#2027571711yescharacteristic
397C2#2027579411yescharacteristic
398C2#2027583511yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenterC2×#332D53
hypocenterC53#10270
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. C2×#16810973C2semidirect
5. C2×#16810368C2semidirect
6. #20272549C2semidirect
7. C2×C53×#332C2semidirect
199. C2×#332D53semidirect
228. C53#10270semidirect

Maximal subgroups

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

subgroupindex
C2×#168109732
C2×#168109002
C2×#168108292
#202755082
C2×#168103682
#202725492
C2×C53×#3322
#1027053