Finite Group Databasebeta

← #27362240

Subgroups — #27362240

Order 13568.

1056 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285840C212yesmaximal, complemented
2#20285175C211yesmaximal, characteristic, complemented
3#20273860C214yesmaximal, complemented
4#20273831C214yesmaximal, complemented
5#20273377C211yesmaximal, characteristic, complemented
6#20273132C212yesmaximal, complemented
7C53×#3549C211yesmaximal, characteristic
8C2×#13133247—22no—
9C2×#13133217C2211yescharacteristic, complemented
10#16811019—22no—
11#16811018—22no—
12#16811002C2212yescomplemented
13#16810995C2212yes—
14#16810969—22no—
15#16810939—24no—
16#16810913C2218yescomplemented
17#16810676C2212yescomplemented
18C4×#11719917C2211yescharacteristic, complemented
19#16810372C2212yescomplemented
20#16810367C2212yes—
21D4⋊#88322C2218yescomplemented
22C53×H(3,𝔽4)—22no—
23C53×#555—22no—
24C53×#555C2212yes—
25D42×C53—22no—
26D42×C53C2212yes—
27C53×#535—22no—
28C53×#535—24no—
29C53×#532C2211yescharacteristic
30C53×#532C2212yes—
31C22×Dic106—21no—
32#13133247—48no—
33C2×#11719925—24no—
34C2×#11719923—24no—
35C2×#11719917—21no—
36#13133217—21no—
37#13133217—22no—
38#13133217C2312yes—
39C2×C4⋊Dic53—22no—
40#13133148—28no—
41C4⋊#88322C2314yes—
42#13133138—28no—
43C22⋊#88322—22no—
44#13133128—21no—
45C4×#88322—22no—
46C4×#88322C2312yes—
47C2×#41×C53—22no—
48C2×#41×C53D412yes—
49C22×D4×C53—22no—
50C22×D4×C53—24no—
51C4⋊Q8×C53D412yes—
52C4⋊Q8×C53C2311yescharacteristic
53#127×C53—24no—
54#127×C53D412yes—
55#127×C53C2311yescharacteristic
56#124×C53—28no—
57#124×C53—22no—
58#124×C53—24no—
59#124×C53D412yes—
60#121×C53—24no—
61#121×C53—24no—
62#121×C53—24no—
63#121×C53—22no—
64#121×C53—22no—
65#121×C53D414yes—
66#120×C53—24no—
67#120×C53—28no—
68#120×C53—24no—
69C4×D4×C53—22no—
70C4×D4×C53—22no—
71C4×D4×C53C2314yes—
72#117×C53—22no—
73#117×C53—24no—
74C2×C42×C53C2311yescharacteristic
75C24×C53—24no—
76#41×C53—44no—
77#41×C53—84no—
78C2×Q8×C53—22no—
79C2×Q8×C53C2×D412yes—
80C2×D4×C53—28no—
81C2×D4×C53—24no—
82C2×D4×C53—22no—
83C2×D4×C53—24no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—22no—
87C2×D4×C53—88no—
88C2×D4×C53—84no—
89C2×D4×C53C2×D414yes—
90C2×D4×C53C2×D412yes—
91C22×C4×C53—22no—
92C22×C4×C53—24no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D412yes—
95C22×Dic53—41no—
96C2×Dic106—22no—
97C2×Dic106—44no—
98C2×Dic106—41no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53C2×D414yes—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—28no—
104C22⋊C4×C53—24no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—24no—
107C42×C53—24no—
108C42×C53C2×D412yes—
109C42×C53C2×D411yescharacteristic
110C42×C53C2411yescharacteristic
111#11719925—48no—
112#11719923—48no—
113C4⋊Dic53—24no—
114C4⋊Dic53—42no—
115#11719917—24no—
116C2×#88322—22no—
117C2×#88322—24no—
118C23×C53—24no—
119C23×C53—22no—
120C23×C53—88no—
121C23×C53—84no—
122C23×C53#12011yescharacteristic
123Q8×C53—42no—
124Q8×C53—82no—
125D4×C53—28no—
126D4×C53—42no—
127D4×C53—44no—
128D4×C53—44no—
129D4×C53—82no—
130D4×C53—82no—
131D4×C53—84no—
132D4×C53—88no—
133D4×C53—84no—
134C2×C4×C53—24no—
135C2×C4×C53—22no—
136C2×C4×C53—24no—
137C2×C4×C53—22no—
138C2×C4×C53—48no—
139C2×C4×C53—44no—
140C2×C4×C53#12012yes—
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53C22×D411yescharacteristic
143C2×C4×C53C22×D412yes—
144C2×Dic53—42no—
145C2×Dic53—82no—
146Dic106—44no—
147Dic106—82no—
148#88322—24no—
149#88322—44no—
150#18715—531nomaximal, Sylow
151C22×C53—22no—
152C22×C53—21no—
153C22×C53—44no—
154C22×C53—44no—
155C22×C53—44no—
156C22×C53—42no—
157C22×C53—44no—
158C22×C53—168no—
159C22×C53C2×#12011yescharacteristic
160C4×C53—22no—
161C4×C53—21no—
162C4×C53—44no—
163C4×C53—42no—
164C4×C53#45312yes—
165C4×C53#57714yes—
166Dic53—82no—
167#3549D5311yescharacteristic
168#3227—532no—
169#3152—531no—
170#1634—534no—
171#1629—534no—
172#1543—531no—
173#1484—532no—
174C2×C53—41no—
175C2×C53—84no—
176C2×C53—82no—
177C2×C53#300912yes—
178C2×C53C2×#45311yescharacteristic
179C2×#137—1062no—
180H(3,𝔽4)—22no—
181#555—22no—
182#555C2×D5312yes—
183D42—22no—
184D42C2×D5312yes—
185#535—22no—
186#535—24no—
187#532C2×D5311yescharacteristic
188#532C2×D5312yes—
189C2×C4⋊Q8—531no—
190#498—1062no—
191#497—1062no—
192#494—532no—
193#492—532no—
194#486—1062no—
195#481—1064no—
196#476—538no—
197#431—532no—
198C4×M16—531no—
199#333—532no—
200#331—532no—
201D4⋊C8—538no—
202C53#1871511yescharacteristic, complemented, Sylow
203C2×#41—22no—
204C2×#41C22⋊D5312yes—
205C22×Q8—1061no—
206C22×D4—22no—
207C22×D4—24no—
208#137—2128no—
209C2×Q16—1064no—
210C2×SD16—1064no—
211C2×M16—1061no—
212C4⋊Q8—532no—
213C4⋊Q8—1062no—
214C4⋊Q8—1061no—
215C4⋊Q8C22⋊D5312yes—
216C4⋊Q8C22×D5311yescharacteristic
217#127—24no—
218#127C22⋊D5312yes—
219#127C22×D5311yescharacteristic
220#124—28no—
221#124—24no—
222#124—22no—
223#124C22⋊D5312yes—
224#121—24no—
225#121—24no—
226#121—22no—
227#121—22no—
228#121—24no—
229#121C22⋊D5314yes—
230#120—24no—
231#120—28no—
232#120—24no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4C22×D5314yes—
236#117—22no—
237#117—24no—
238C2×C4⋊C4—1062no—
239C2×C42C22×D5311yescharacteristic
240#106—1068no—
241C4⋊C8—534no—
242#102—1068no—
243C22⋊C8—1062no—
244#97—1061no—
245C4×C8—532no—
246C4×C8—1062no—
247C24—24no—
248#41—44no—
249#41—84no—
250C2×Q8—22no—
251C2×Q8—1062no—
252C2×Q8—2124no—
253C2×Q8—2121no—
254C2×Q8C2×C22⋊D5312yes—
255C2×D4—24no—
256C2×D4—28no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—88no—
263C2×D4—84no—
264C2×D4C2×C22⋊D5314yes—
265C2×D4C2×C22⋊D5312yes—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—2121no—
269C22×C4C2×C22⋊D5312yes—
270C22×C4C2×C22⋊D5311yescharacteristic
271Q16—2128no—
272SD16—2128no—
273M16—1064no—
274C2×C8—1062no—
275C2×C8—1064no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—1064no—
279C4⋊C4—2122no—
280C4⋊C4C2×C22⋊D5314yes—
281C22⋊C4—28no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C42—24no—
287C42C2×C22⋊D5312yes—
288C42C2×C22⋊D5311yescharacteristic
289C42C23×D5311yescharacteristic
290C23—24no—
291C23—22no—
292C23—88no—
293C23—84no—
294C23#1313318611yescharacteristic
295Q8—42no—
296Q8—82no—
297Q8—4242no—
298Q8—2124no—
299D4—28no—
300D4—42no—
301D4—44no—
302D4—44no—
303D4—82no—
304D4—82no—
305D4—84no—
306D4—88no—
307D4—84no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—4242no—
315C2×C4—2122no—
316C2×C4#1313318612yes—
317C2×C4#1313318611yescharacteristic
318C2×C4C22×C22⋊D5311yescharacteristic
319C2×C4C22×C22⋊D5312yes—
320C8—1064no—
321C8—2124no—
322C22—22no—
323C22—21no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—42no—
328C22—44no—
329C22—168no—
330C22C2×#1313318611yescharacteristic
331C4—22no—
332C4—21no—
333C4—44no—
334C4—42no—
335C4—4242no—
336C4#1681079112yes—
337C4#1681143414yes—
338C2—41no—
339C2—84no—
340C2—82no—
341C2#2028399212yes—
342C2C2×#1681079111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3549C2
socleC22×C53C2×#120
hypercenter#3549D53
hypocenterC53#18715
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3549

Complements

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

NH ≅ G/Ntype
1. #20285840C2semidirect
2. #20285175C2semidirect
3. #20273860C2semidirect
4. #20273831C2semidirect
5. #20273377C2semidirect
6. #20273132C2semidirect
9. C2×#13133217C22semidirect
12. #16811002C22semidirect
16. #16810913C22semidirect
17. #16810676C22semidirect
18. C4×#11719917C22semidirect
19. #16810372C22semidirect
21. D4⋊#88322C22semidirect
202. C53#18715semidirect

Maximal subgroups

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

subgroupindex
#202858402
#202851752
#202738602
#202738312
#202733772
#202731322
C53×#35492
#1871553