Finite Group Databasebeta

← #27369280

Subgroups — #27369280

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285665C211yesmaximal, characteristic, complemented
2#20284514C211yesmaximal, characteristic
3C2×#16810904C211yesmaximal, characteristic
4#20283738C211yesmaximal, characteristic
5#20274167C214yesmaximal, complemented
6#20274081C214yesmaximal, complemented
7#20273401C211yesmaximal, characteristic, complemented
8#20273373C211yesmaximal, characteristic, complemented
9C53×#1571C211yesmaximal, characteristic, complemented
10#16811180C2211yescharacteristic
11C2×#13133217C2211yescharacteristic
12C2×C4×D212C2211yescharacteristic
13#16811003C2212yescomplemented
14#16810974C2212yescomplemented
15#16810924—22no—
16#16810914C2212yes—
17#16810904—22no—
18#16810904C2214yes—
19#16810904C2212yes—
20#16810834—22no—
21#16810781—22no—
22C8×D212C2214yes—
23#16810688—22no—
24#16810676C2211yescharacteristic
25#16810665—21no—
26C2×#13133151—21no—
27#16810629—21no—
28C2×#13133139—21no—
29C22⋊C8×D53—22no—
30#16810371C2212yescomplemented
31#16810368C2212yescomplemented
32#16810368C2214yescomplemented
33C53×#431C2211yescharacteristic
34C53×#423C2211yescharacteristic
35C2×C4⋊C8×C53C2211yescharacteristic
36C53×#335C2214yescomplemented
37C22×Dic106—21no—
38C22×D212D411yescharacteristic
39C22×C4×D53—21no—
40C2×Dic212—24no—
41C2×C8×D53—24no—
42#13133217—21no—
43#13133217—21no—
44#13133217—21no—
45#13133217C2312yes—
46#13133217C2311yescharacteristic
47#13133217C2311yescharacteristic
48#13133209—21no—
49#13133196—21no—
50#13133196—21no—
51C4×Dic106—21no—
52C4×Dic106—21no—
53C4×D212—21no—
54C4×D212—21no—
55C4×D212D412yes—
56C4×D212C2312yes—
57#13133176—21no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53D411yescharacteristic
61C2×#11719921—21no—
62#13133151—24no—
63#13133151—22no—
64#13133149—22no—
65C4⋊#88322C2312yes—
66#13133142—24no—
67#13133139—24no—
68#13133139—22no—
69#13133139—22no—
70C22⋊#88322—21no—
71#13133129—24no—
72C4×#88322—21no—
73C2×M16×C53—21no—
74C22×C8×C53—21no—
75C2×C42×C53C2311yescharacteristic
76C4⋊C8×C53—21no—
77C4⋊C8×C53—21no—
78C4⋊C8×C53D412yes—
79C4⋊C8×C53C2312yes—
80C4⋊C8×C53C2312yes—
81C22⋊C8×C53D412yes—
82C4×C8×C53C2312yes—
83C23×D53—21no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×Dic53—21no—
88C22×Dic53—41no—
89C2×D212—21no—
90C2×D212—22no—
91C2×D212Q1614yescomplemented
92C2×D212C2×D412yes—
93C2×C4×D53—22no—
94C2×C4×D53—22no—
95C2×C4×D53—82no—
96C2×Dic106—22no—
97C2×Dic106—21no—
98C2×Dic106—21no—
99C2×Dic106—41no—
100C2×Dic106—82no—
101M16×C53—42no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—42no—
105C2×C8×C53C2×D414yes—
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53C2411yescharacteristic
110C22⋊Dic53—41no—
111#11719921—22no—
112#11719921—22no—
113C4⋊Dic53—21no—
114C4⋊Dic53—21no—
115C4⋊Dic53—22no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53—22no—
119C4⋊Dic53—21no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C4⋊Dic53C2×D412yes—
123#11719919—41no—
124#11719919—41no—
125C4×Dic53—41no—
126C2×#88322—22no—
127Dic212—84no—
128C8×D53—84no—
129C23×C53#12111yescharacteristic
130C22×D53—22no—
131C22×D53—24no—
132C22×D53—81no—
133C22×D53—81no—
134C2×C4×C53—22no—
135C2×C4×C53—41no—
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53C2×Q1612yes—
140C2×C4×C53#13512yes—
141C2×C4×C53C22×D411yescharacteristic
142C2×C4×C53C22×D411yescharacteristic
143C2×C4×C53C2×#4111yescharacteristic
144C2×Dic53—22no—
145C2×Dic53—42no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—81no—
149C2×Dic53—82no—
150D212—22no—
151D212—24no—
152D212C2×Q1614yes—
153C4×D53—44no—
154C4×D53—82no—
155Dic106—44no—
156Dic106—42no—
157Dic106—82no—
158Dic106—81no—
159C8×C53—24no—
160C8×C53—42no—
161C8×C53—42no—
162#88322—42no—
163#19387—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53#46712yes—
167C22×C53C2×#12111yescharacteristic
168C2×D53—24no—
169C2×D53—42no—
170C2×D53—42no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53#46712yes—
176C4×C53C22×Q1611yescharacteristic
177C4×C53C2×#13511yescharacteristic
178C4×C53#57612yes—
179Dic53—42no—
180Dic53—82no—
181Dic53—81no—
182Dic53—81no—
183#3209—531no—
184#3077—531no—
185C2×#474—531no—
186#2972—531no—
187#1683—534no—
188#1669—534no—
189#1571D5311yescharacteristic, complemented
190#1549—531no—
191#1542—531no—
192C2×C53—22no—
193C2×C53#304111yescharacteristic
194C2×C53C2×#46711yescharacteristic
195C2×C53#306911yescharacteristic
196D53—44no—
197#533—531no—
198C2×C4⋊Q8—531no—
199C2×C4×D4—531no—
200#494—532no—
201#487—532no—
202#478—1062no—
203#476—532no—
204#474—532no—
205#474—534no—
206#474—1062no—
207#462—1062no—
208#451—1062no—
209C8×D4—534no—
210C8×D4—1062no—
211#431—531no—
212#431C2×D5311yescharacteristic
213#428—1061no—
214C2×#107—1061no—
215#423C2×D5311yescharacteristic
216C2×C4⋊C8C2×D5311yescharacteristic
217#418—1061no—
218C2×#102—1061no—
219C2×C22⋊C8—1062no—
220#335C2×D5314yescomplemented
221#333—532no—
222#332—532no—
223#332—534no—
224C53#1938711yescharacteristic, complemented, Sylow
225C22×Q8—1061no—
226C22×D4—531no—
227C23×C4—1061no—
228C2×Q16—1064no—
229C2×M16—21no—
230C22×C8—21no—
231C22×C8—1064no—
232C4⋊Q8—532no—
233C4⋊Q8—531no—
234C4⋊Q8—531no—
235C4⋊Q8—1061no—
236C4⋊Q8—1061no—
237C4⋊Q8—1061no—
238#125—1061no—
239C22⋊Q8—1061no—
240C22⋊Q8—1061no—
241C4×Q8—1061no—
242C4×Q8—1061no—
243C4×D4—532no—
244C4×D4—532no—
245C4×D4—1061no—
246C4×D4—1061no—
247#117—1061no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C22⋊C4—1061no—
252C2×C42C22×D5311yescharacteristic
253#107—1064no—
254#107—1062no—
255#106—1062no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—532no—
259C4⋊C8D21212yes—
260C4⋊C8C22×D5312yes—
261C4⋊C8C22×D5312yes—
262#103—1064no—
263#102—1064no—
264#102—1062no—
265#102—1062no—
266C22⋊C8—1061no—
267C22⋊C8—1064no—
268C22⋊C8D21212yes—
269C4×C8—1061no—
270C4×C8C22×D5312yes—
271C24—1061no—
272C2×Q8—1062no—
273C2×Q8—1061no—
274C2×Q8—1061no—
275C2×Q8—4242no—
276C2×Q8—2121no—
277C2×D4—534no—
278C2×D4—532no—
279C2×D4—1061no—
280C2×D4—1062no—
281C22×C4—1061no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—4242no—
285C22×C4—2121no—
286C22×C4C2×D21211yescharacteristic
287C22×C4D4×D5311yescharacteristic
288C22×C4#1171994911yescharacteristic
289Q16—4244no—
290M16—42no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—1062no—
295C2×C8—4244no—
296C2×C8C2×D21214yes—
297C4⋊C4—532no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1062no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1062no—
304C4⋊C4—1061no—
305C4⋊C4—2121no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C22⋊C4—1062no—
310C22⋊C4—1062no—
311C22⋊C4—2121no—
312C42—2121no—
313C42C2×D21211yescharacteristic
314C42D4×D5311yescharacteristic
315C42#1171994911yescharacteristic
316C42C23×D5311yescharacteristic
317C23—1064no—
318C23—1062no—
319C23—4241no—
320C23—4241no—
321C23#1313319211yescharacteristic
322Q8—4241no—
323Q8—4242no—
324Q8—2122no—
325Q8—2124no—
326D4—534no—
327D4—1064no—
328D4—1062no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—1062no—
332C2×C4—4242no—
333C2×C4—4242no—
334C2×C4—4241no—
335C2×C4—2121no—
336C2×C4—2121no—
337C2×C4—2122no—
338C2×C4—2124no—
339C2×C4#1313319211yescharacteristic
340C2×C4#1313319211yescharacteristic
341C2×C4#1313319211yescharacteristic
342C2×C4Q16×D5312yes—
343C2×C4#1313323612yes—
344C2×C4C2×D4×D5311yescharacteristic
345C2×C4C22×D21211yescharacteristic
346C2×C4C2×#1171994911yescharacteristic
347C8—24no—
348C8—42no—
349C8—42no—
350C8—2122no—
351C22—21no—
352C22—21no—
353C22—1064no—
354C22—4244no—
355C22—2122no—
356C22—2122no—
357C22#1681086512yes—
358C22C2×#1313319211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—4241no—
363C4—4242no—
364C4—4241no—
365C4—2122no—
366C4#1681086512yes—
367C4C2×Q16×D5311yescharacteristic
368C4C2×#1313323611yescharacteristic
369C4#1681142812yes—
370C2—22no—
371C2—2124no—
372C2#2028421911yescharacteristic
373C2C2×#1681086511yescharacteristic
374C2#2028445311yescharacteristic
subgroupG/H
centerC22C2×#13133192
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1571C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#1571D53
hypocenterC53#19387
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
1. #20285665C2semidirect
5. #20274167C2semidirect
6. #20274081C2semidirect
7. #20273401C2semidirect
8. #20273373C2semidirect
9. C53×#1571C2semidirect
13. #16811003C22semidirect
14. #16810974C22semidirect
30. #16810371C22semidirect
31. #16810368C22semidirect
32. #16810368C22semidirect
36. C53×#335C22semidirect
91. C2×D212Q16semidirect
189. #1571D53semidirect
220. #335C2×D53semidirect
224. C53#19387semidirect

Maximal subgroups

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

subgroupindex
#202856652
#202845142
C2×#168109042
#202837382
#202741672
#202740812
#202734012
#202733732
C53×#15712
#1938753