Finite Group Databasebeta

← #27286666

Subgroups — #27286666

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811019C211yesmaximal, characteristic, complemented
2C2×#16810910C211yesmaximal, characteristic
3C2×#16810829C211yesmaximal, characteristic
4#20275508C211yesmaximal, characteristic
5C2×#16810368C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#333C211yesmaximal, characteristic, complemented
8C22×#11719913—21no—
9C2×#13133217C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16811019—22no—
12#16811019—22no—
13#16810910—22no—
14#16810910—22no—
15#16810829—22no—
16#16810829—22no—
17C2×#13133148—22no—
18C2×#13133148—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133142—21no—
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×#333—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#13133148—28no—
50#13133148—24no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53#13133142—24no—
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—82no—
76C22×C4×C53—41no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—41no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212SD1612yes—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—42no—
91C2×C4×D53—42no—
92C2×Dic106—22no—
93C2×Dic106—41no—
94C2×Dic106—41no—
95C2×Dic106—41no—
96C2×Dic106—41no—
97C2×Dic106—41no—
98C2×C8×C53—22no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—42no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C42×C53C2×D411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111#11719921—41no—
112#11719921—41no—
113C4⋊Dic53—21no—
114C4⋊Dic53—22no—
115C4⋊Dic53—21no—
116C4⋊Dic53—22no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C2×#88322—22no—
120C2×#88322—22no—
121C2×#88322—22no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322SD1614yescomplemented
125#11719913—48no—
126C23×C53#12111yescharacteristic
127C22×D53—22no—
128C22×D53—41no—
129C22×D53—42no—
130C22×D53—41no—
131C22×D53—42no—
132Q8×C53—82no—
133Q8×C53—82no—
134C2×C4×C53—22no—
135C2×C4×C53—42no—
136C2×C4×C53—82no—
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12711yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53C2×SD1612yes—
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53#13711yescharacteristic
148C2×C4×C53#13712yes—
149C2×Dic53—42no—
150C2×Dic53—41no—
151C2×Dic53—42no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154D212—22no—
155D212—22no—
156D212—41no—
157D212—41no—
158C4×D53—82no—
159C4×D53—81no—
160C4×D53—81no—
161Dic106—42no—
162Dic106—42no—
163Dic106—81no—
164Dic106—81no—
165C8×C53—42no—
166C8×C53—81no—
167#88322—24no—
168#88322—42no—
169#10271—531nomaximal, Sylow
170C22×C53#39211yescharacteristic
171C22×C53#46112yes—
172C22×C53#46511yescharacteristic
173C22×C53#46811yescharacteristic
174C22×C53#47112yes—
175C2×D53—44no—
176C2×D53—42no—
177C2×D53—82no—
178C2×D53—81no—
179C2×D53—81no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—82no—
183C4×C53#44811yescharacteristic
184C4×C53#44812yes—
185C4×C53#44911yescharacteristic
186C4×C53#45011yescharacteristic
187C4×C53#45211yescharacteristic
188C4×C53#49211yescharacteristic
189C4×C53#49711yescharacteristic
190Dic53—82no—
191Dic53—81no—
192C2×#498—531no—
193C2×#476—531no—
194C2×#461—531no—
195#1954—531no—
196C2×#333D5311yescharacteristic, complemented
197C2×#332—531no—
198#1277—531no—
199C2×C53#167112yes—
200C2×C53#169112yes—
201C2×C53#200011yescharacteristic
202C2×C53#201311yescharacteristic
203C2×C53#201911yescharacteristic
204D53—82no—
205C22×SD16—1061no—
206C2×C4⋊Q8—531no—
207C2×C4⋊Q8C2×D5311yescharacteristic
208C2×C4×D4—531no—
209#498—1062no—
210#498—1062no—
211#476—1062no—
212#476—1062no—
213#461—1062no—
214#461—1062no—
215C2×#106—1061no—
216C2×#106—1062no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8C2×D5311yescharacteristic
219C2×C4⋊C8C2×D5311yescharacteristic
220C2×#103—1061no—
221C2×#102—1061no—
222C2×C22⋊C8—1062no—
223C2×C4×C8—531no—
224#333—22no—
225#332—1062no—
226C53#1027111yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—1061no—
229C22×D4—531no—
230C23×C4—1061no—
231C2×SD16—1064no—
232C2×SD16—2122no—
233C2×SD16—2122no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—532no—
237C22×C8—1061no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—532no—
242C4⋊Q8—1061no—
243C4⋊Q8—1061no—
244C4⋊Q8—1061no—
245C4⋊Q8D21212yes—
246C4×D4—1061no—
247C4×D4—1061no—
248C4×D4—1061no—
249C4×D4—1061no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—1061no—
254C2×C4⋊C4—1061no—
255C2×C22⋊C4—1061no—
256C2×C42C22×D5311yescharacteristic
257#106—1064no—
258#106—1068no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—532no—
263C4⋊C8—1061no—
264C4⋊C8C22⋊D5312yes—
265#103—1064no—
266#102—1064no—
267C22⋊C8—2124no—
268C4×C8—1061no—
269C4×C8—1061no—
270C24—1061no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—82no—
274C2×Q8—1062no—
275C2×Q8—2121no—
276C2×Q8—2121no—
277C2×Q8—2121no—
278C2×Q8—2121no—
279C2×Q8—2121no—
280C2×D4—532no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C22×C4—41no—
287C22×C4—1061no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4—2122no—
292C22×C4—2122no—
293C22×C4#1171994511yescharacteristic
294C22×C4D4×D5311yescharacteristic
295C22×C4D4×D5311yescharacteristic
296SD16—2128no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—534no—
303C2×C8—1062no—
304C2×C8—1062no—
305C2×C8—1062no—
306C2×C8—2121no—
307C2×C8—2121no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—1062no—
315C4⋊C4—1062no—
316C4⋊C4—2121no—
317C4⋊C4—2121no—
318C22⋊C4—2121no—
319C22⋊C4—2121no—
320C42C2×D21211yescharacteristic
321C42D4×D5311yescharacteristic
322C42D4×D5311yescharacteristic
323C42C2×C22⋊D5311yescharacteristic
324C23—1062no—
325C23—2121no—
326C23—2122no—
327C23—2121no—
328C23—2122no—
329C23#1313319011yescharacteristic
330Q8—82no—
331Q8—82no—
332Q8—4241no—
333Q8—4241no—
334Q8—2122no—
335Q8—2122no—
336D4—1062no—
337D4—1062no—
338D4—2121no—
339D4—2121no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—4241no—
344C2×C4—4241no—
345C2×C4—4241no—
346C2×C4—4242no—
347C2×C4—4241no—
348C2×C4—2121no—
349C2×C4—2122no—
350C2×C4—2122no—
351C2×C4#1313318511yescharacteristic
352C2×C4#1313318711yescharacteristic
353C2×C4#1313318711yescharacteristic
354C2×C4#1313319011yescharacteristic
355C2×C4#1313319111yescharacteristic
356C2×C4#1313321311yescharacteristic
357C2×C4SD16×D5311yescharacteristic
358C2×C4SD16×D5312yes—
359C2×C4#1313324011yescharacteristic
360C2×C4#1313324411yescharacteristic
361C2×C4#1313324912yes—
362C2×C4#1313324911yescharacteristic
363C8—42no—
364C8—81no—
365C8—1064no—
366C8—2122no—
367C22—4242no—
368C22—4241no—
369C22—4241no—
370C22—2122no—
371C22—2124no—
372C22#1681053211yescharacteristic
373C22#1681082712yes—
374C22#1681085511yescharacteristic
375C22#1681087211yescharacteristic
376C22#1681088512yes—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—4242no—
381C4—4241no—
382C4#1681076311yescharacteristic
383C4#1681076312yes—
384C4#1681077111yescharacteristic
385C4#1681077211yescharacteristic
386C4#1681078411yescharacteristic
387C4#1681099411yescharacteristic
388C4#1681101611yescharacteristic
389C2—4242no—
390C2#2027409312yes—
391C2#2027421712yes—
392C2#2027571711yescharacteristic
393C2#2027579911yescharacteristic
394C2#2027583111yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#333C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenterC2×#333D53
hypocenterC53#10271
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×#16811019C2semidirect
5. C2×#16810368C2semidirect
6. #20272549C2semidirect
7. C2×C53×#333C2semidirect
124. C2×#88322SD16semidirect
196. C2×#333D53semidirect
226. C53#10271semidirect

Maximal subgroups

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

subgroupindex
C2×#168110192
C2×#168109102
C2×#168108292
#202755082
C2×#168103682
#202725492
C2×C53×#3332
#1027153