Finite Group Databasebeta

← #27362265

Subgroups — #27362265

Order 13568.

1004 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1004 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#20285653C212yesmaximal, complemented
2#20285168C211yesmaximal, characteristic
3#20273793C214yesmaximal, complemented
4#20273774C214yesmaximal, complemented
5#20273377C211yesmaximal, characteristic, complemented
6#20273136C212yesmaximal, complemented
7C53×#3550C211yesmaximal, characteristic, complemented
8C2×#13133242—22no—
9C2×#13133214C2211yescharacteristic
10#16811013—22no—
11#16810999C2212yes—
12#16810997—22no—
13#16810997C2212yes—
14#16810971—22no—
15#16810823C2218yes—
16#16810773—24no—
17#16810676C2212yes—
18C4×#11719917C2211yescharacteristic
19#16810372C2212yescomplemented
20#16810367C2212yescomplemented
21Q8⋊#88322C2218yescomplemented
22C53×#563—22no—
23C53×#559C2212yescomplemented
24Q82×C53C2212yescomplemented
25C53×#556—22no—
26D4×Q8×C53—22no—
27C53×#542—24no—
28C53×#536—22no—
29C53×#533C2212yes—
30C53×#532C2211yescharacteristic
31C22×D212—21no—
32C22×D212—22no—
33#13133242—48no—
34C2×#11719924—24no—
35C2×#11719924—28no—
36C2×#11719922—24no—
37C2×#11719917—21no—
38#13133214—21no—
39#13133214—22no—
40#13133214C2312yes—
41C4⋊#88322C2314yes—
42#13133137—28no—
43C22⋊#88322—22no—
44#13133128—21no—
45C4×#88322—22no—
46C4×#88322C2312yes—
47C2×#41×C53D412yes—
48C22×Q8×C53—22no—
49C4⋊Q8×C53—24no—
50C4⋊Q8×C53—24no—
51C4⋊Q8×C53D412yes—
52C4⋊Q8×C53D412yes—
53C4⋊Q8×C53C2311yescharacteristic
54#127×C53C2311yescharacteristic
55#126×C53—28no—
56#125×C53—22no—
57#124×C53—24no—
58#124×C53—24no—
59#124×C53D412yes—
60#123×C53—24no—
61#123×C53—24no—
62C22⋊Q8×C53—24no—
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53D414yes—
66#121×C53—22no—
67C4×Q8×C53—22no—
68C4×Q8×C53—24no—
69C4×Q8×C53C2314yes—
70C4×D4×C53—24no—
71C4×D4×C53—22no—
72#117×C53—22no—
73#117×C53—22no—
74C2×C42×C53C2311yescharacteristic
75C23×D53—41no—
76#41×C53—44no—
77C2×Q8×C53—24no—
78C2×Q8×C53—22no—
79C2×Q8×C53—84no—
80C2×Q8×C53C2×D414yes—
81C2×Q8×C53C2×D412yes—
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53C2×D412yes—
85C22×C4×C53—24no—
86C22×C4×C53—22no—
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D411yescharacteristic
89C2×D212—24no—
90C2×D212—22no—
91C2×D212—44no—
92C2×D212—42no—
93C2×D212—41no—
94C2×D212—84no—
95C4⋊C4×C53—28no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—24no—
102C4⋊C4×C53C2×D414yes—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—28no—
105C22⋊C4×C53—24no—
106C22⋊C4×C53—24no—
107C42×C53—24no—
108C42×C53—24no—
109C42×C53C2×D412yes—
110C42×C53C2×D411yescharacteristic
111C42×C53C2411yescharacteristic
112#11719924—48no—
113#11719924—88no—
114#11719922—48no—
115#11719917—24no—
116C2×#88322—22no—
117C2×#88322—24no—
118C23×C53—22no—
119C23×C53#12011yescharacteristic
120C22×D53—42no—
121C22×D53—82no—
122C22×D53—84no—
123Q8×C53—28no—
124Q8×C53—42no—
125Q8×C53—44no—
126Q8×C53—84no—
127D4×C53—42no—
128D4×C53—44no—
129D4×C53—82no—
130C2×C4×C53—24no—
131C2×C4×C53—22no—
132C2×C4×C53—24no—
133C2×C4×C53—22no—
134C2×C4×C53—24no—
135C2×C4×C53—44no—
136C2×C4×C53—84no—
137C2×C4×C53#12012yes—
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53C22×D411yescharacteristic
140C2×C4×C53C22×D412yes—
141D212—44no—
142D212—48no—
143D212—84no—
144D212—82no—
145#88322—24no—
146#88322—44no—
147#18718—531nomaximal, Sylow
148C22×C53—22no—
149C22×C53—21no—
150C22×C53—44no—
151C22×C53—42no—
152C22×C53C2×#12011yescharacteristic
153C2×D53—84no—
154C2×D53—82no—
155C2×D53—164no—
156C4×C53—22no—
157C4×C53—21no—
158C4×C53—44no—
159C4×C53—42no—
160C4×C53—44no—
161C4×C53#45312yes—
162C4×C53#57714yes—
163#3550D5311yescharacteristic, complemented
164#3208—532no—
165#3151—531no—
166#1623—534no—
167#1620—534no—
168#1543—531no—
169#1485—532no—
170C2×C53—41no—
171C2×C53—82no—
172C2×C53#301012yes—
173C2×C53C2×#45311yescharacteristic
174D53—162no—
175C2×AGL(1,ℤ/8)—1062no—
176#563—22no—
177#559C2×D5312yescomplemented
178Q82C2×D5312yescomplemented
179#556—22no—
180D4×Q8—22no—
181#542—24no—
182#536—22no—
183#533C2×D5312yes—
184#532C2×D5311yescharacteristic
185C2×#127—531no—
186#496—1062no—
187#493—532no—
188#492—532no—
189#492—1062no—
190#486—1062no—
191#460—538no—
192#450—1064no—
193#431—532no—
194C4×M16—531no—
195#333—532no—
196#331—532no—
197Q8⋊C8—538no—
198C53#1871811yescharacteristic, complemented, Sylow
199C2×#41C22⋊D5312yes—
200C22×Q8—22no—
201C22×D4—1062no—
202C22×D4—1061no—
203AGL(1,ℤ/8)—2128no—
204C2×SD16—1068no—
205C2×SD16—1064no—
206C2×D8—1064no—
207C2×M16—1061no—
208C4⋊Q8—24no—
209C4⋊Q8—24no—
210C4⋊Q8C22⋊D5312yes—
211C4⋊Q8C22⋊D5312yes—
212C4⋊Q8C22×D5311yescharacteristic
213#127—532no—
214#127—1062no—
215#127—1061no—
216#127C22×D5311yescharacteristic
217#126—28no—
218#125—22no—
219#124—24no—
220#124—24no—
221#124C22⋊D5312yes—
222#123—24no—
223#123—24no—
224C22⋊Q8—24no—
225C22⋊Q8—28no—
226C22⋊Q8—22no—
227C22⋊Q8C22⋊D5314yes—
228#121—22no—
229C4×Q8—24no—
230C4×Q8—22no—
231C4×Q8C22×D5314yes—
232C4×D4—24no—
233C4×D4—22no—
234#117—22no—
235#117—22no—
236C2×C42C22×D5311yescharacteristic
237C4⋊C8—534no—
238#102—1068no—
239C22⋊C8—1062no—
240#97—1061no—
241C4×C8—532no—
242C4×C8—1062no—
243C24—2121no—
244#41—44no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—84no—
248C2×Q8C2×C22⋊D5314yes—
249C2×Q8C2×C22⋊D5312yes—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—1064no—
253C2×D4—1062no—
254C2×D4—4244no—
255C2×D4—2124no—
256C2×D4—2122no—
257C2×D4—2121no—
258C2×D4C2×C22⋊D5312yes—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4C2×C22⋊D5312yes—
262C22×C4C2×C22⋊D5311yescharacteristic
263SD16—4248no—
264SD16—2128no—
265D8—2128no—
266M16—1064no—
267C2×C8—1062no—
268C2×C8—1064no—
269C4⋊C4—28no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4C2×C22⋊D5314yes—
277C22⋊C4—24no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C42—24no—
282C42—24no—
283C42C2×C22⋊D5312yes—
284C42C2×C22⋊D5311yescharacteristic
285C42C23×D5311yescharacteristic
286C23—22no—
287C23—4244no—
288C23—4242no—
289C23—2122no—
290C23#1313318611yescharacteristic
291Q8—28no—
292Q8—42no—
293Q8—44no—
294Q8—84no—
295D4—42no—
296D4—44no—
297D4—82no—
298D4—4244no—
299D4—4242no—
300D4—2124no—
301D4—2128no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—44no—
308C2×C4—84no—
309C2×C4#1313318612yes—
310C2×C4#1313318611yescharacteristic
311C2×C4C22×C22⋊D5311yescharacteristic
312C2×C4C22×C22⋊D5312yes—
313C8—1064no—
314C8—2124no—
315C22—22no—
316C22—21no—
317C22—44no—
318C22—42no—
319C22—8484no—
320C22—4244no—
321C22—4242no—
322C22C2×#1313318611yescharacteristic
323C4—22no—
324C4—21no—
325C4—44no—
326C4—42no—
327C4—44no—
328C4#1681079112yes—
329C4#1681143114yes—
330C2—41no—
331C2—82no—
332C2—8482no—
333C2#2028400112yes—
334C2C2×#1681079111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3550C2
socleC22×C53C2×#120
prime radical#20285168C2
hypercenter#3550D53
hypocenterC53#18718
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #20285653C2semidirect
3. #20273793C2semidirect
4. #20273774C2semidirect
5. #20273377C2semidirect
6. #20273136C2semidirect
7. C53×#3550C2semidirect
19. #16810372C22semidirect
20. #16810367C22semidirect
21. Q8⋊#88322C22semidirect
23. C53×#559C22semidirect
24. Q82×C53C22semidirect
163. #3550D53semidirect
177. #559C2×D53semidirect
178. Q82C2×D53semidirect
198. C53#18718semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202851682
#202737932
#202737742
#202733772
#202731362
C53×#35502
#1871853