Finite Group Databasebeta

← #27360212

Subgroups — #27360212

Order 13568.

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

normal subgroups only

This group has 982 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#20285089C211yesmaximal, characteristic
2#20284845C212yesmaximal, complemented
3#20273901C218yesmaximal, complemented
4#20273429C211yesmaximal, characteristic
5#20273162C212yesmaximal, complemented
6C2×C53×#559C211yesmaximal, characteristic
7#16811180C2211yescharacteristic
8#16810987—22no—
9#16810969C2214yes—
10#16810953—22no—
11#16810908C2214yescomplemented
12#16810833C2214yescomplemented
13#16810779—22no—
14#16810728—24no—
15#16810680C2212yes—
16#16810628—22no—
17C4×#11719917C2211yescharacteristic
18#16810367C2214yes—
19Q8⋊#88322C2218yescomplemented
20C53×#559—22no—
21C53×#559—24no—
22C53×#559C2214yes—
23C2×#127×C53C2211yescharacteristic
24C2×#124×C53—22no—
25C2×#123×C53—22no—
26C2×C4×Q8×C53C2212yes—
27C2×C4×D4×C53—22no—
28C2×#11719925—24no—
29C2×#11719917—21no—
30#13133217C2312yes—
31#13133209—21no—
32#13133196—22no—
33C4×Dic106—22no—
34#13133176—21no—
35#13133148—24no—
36C4⋊#88322C2314yes—
37#13133141—24no—
38#13133140—24no—
39#13133138—28no—
40C22⋊#88322—22no—
41#13133128—21no—
42#13133128—22no—
43C4×#88322C2312yes—
44C22×Q8×C53D412yes—
45C22×D4×C53—22no—
46C22×D4×C53D412yes—
47C23×C4×C53—22no—
48#127×C53—22no—
49#127×C53D412yes—
50#127×C53C2312yes—
51#124×C53—28no—
52#124×C53—28no—
53#123×C53—28no—
54#123×C53—28no—
55C4×Q8×C53—22no—
56C4×Q8×C53—22no—
57C4×Q8×C53D414yes—
58C4×Q8×C53C2314yes—
59C4×D4×C53—28no—
60C4×D4×C53—28no—
61C2×C4⋊C4×C53—22no—
62C2×C4⋊C4×C53D412yes—
63C2×C22⋊C4×C53—24no—
64C2×C22⋊C4×C53—22no—
65C2×C42×C53—22no—
66C2×C42×C53C2311yescharacteristic
67C24×C53—22no—
68C2×Q8×C53—22no—
69C2×Q8×C53—44no—
70C2×Q8×C53C2×D414yes—
71C2×D4×C53—24no—
72C2×D4×C53—22no—
73C2×D4×C53—24no—
74C2×D4×C53—48no—
75C2×D4×C53—44no—
76C2×D4×C53C2×D414yes—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—28no—
80C22×C4×C53—84no—
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D411yescharacteristic
83C2×Dic106—22no—
84C4⋊C4×C53—24no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53C2×D414yes—
88C22⋊C4×C53—216no—
89C22⋊C4×C53—28no—
90C42×C53—24no—
91C42×C53—24no—
92C42×C53C2×D411yescharacteristic
93C42×C53C2×D412yes—
94C42×C53C2411yescharacteristic
95C22⋊Dic53—41no—
96#11719925—48no—
97C4⋊Dic53—22no—
98C4⋊Dic53—22no—
99#11719919—42no—
100C4×Dic53—41no—
101#11719917—42no—
102C2×#88322—22no—
103C2×#88322—24no—
104C23×C53—24no—
105C23×C53—48no—
106C23×C53—44no—
107C23×C53#12011yescharacteristic
108Q8×C53—24no—
109Q8×C53—24no—
110Q8×C53—44no—
111D4×C53—44no—
112D4×C53—48no—
113D4×C53—48no—
114D4×C53—82no—
115C2×C4×C53—22no—
116C2×C4×C53—24no—
117C2×C4×C53—24no—
118C2×C4×C53—24no—
119C2×C4×C53—44no—
120C2×C4×C53—44no—
121C2×C4×C53—88no—
122C2×C4×C53#12012yes—
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53C22×D411yescharacteristic
125C2×C4×C53C22×D412yes—
126C2×Dic53—42no—
127Dic106—44no—
128#88322—42no—
129#88322—44no—
130#18531—531nomaximal, Sylow
131C22×C53—22no—
132C22×C53—48no—
133C22×C53—44no—
134C22×C53—88no—
135C22×C53#45512yes—
136C22×C53C2×#12011yescharacteristic
137C4×C53—22no—
138C4×C53—21no—
139C4×C53—21no—
140C4×C53—44no—
141C4×C53—44no—
142C4×C53#57514yes—
143Dic53—82no—
144C2×#559D5311yescharacteristic
145#3138—531no—
146#3113—532no—
147#1640—538no—
148#1555—531no—
149#1491—532no—
150C2×C53—22no—
151C2×C53—84no—
152C2×C53#300512yes—
153C2×C53C2×#45511yescharacteristic
154#559—22no—
155#559—24no—
156#559C2×D5314yes—
157#533—531no—
158C2×#127C2×D5311yescharacteristic
159C2×#124—22no—
160C2×#123—22no—
161C2×C4×Q8C2×D5312yes—
162C2×C4×D4—22no—
163#490—1062no—
164#486—534no—
165#483—1062no—
166#475—534no—
167#462—534no—
168#451—1062no—
169#441—1064no—
170#432—532no—
171#418—1062no—
172C4×M16—531no—
173#331—534no—
174Q8⋊C8—538no—
175C53#1853111yescharacteristic, complemented, Sylow
176C22×Q8C22⋊D5312yes—
177C22×D4—22no—
178C22×D4C22⋊D5312yes—
179C23×C4—22no—
180C2×Q16—1064no—
181C2×M16—1061no—
182C4⋊Q8—532no—
183#127—22no—
184#127C22⋊D5312yes—
185#127C22×D5312yes—
186#125—1061no—
187#124—28no—
188#124—28no—
189#123—28no—
190#123—28no—
191C22⋊Q8—1062no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—1062no—
195C4×Q8C22⋊D5314yes—
196C4×Q8C22×D5314yes—
197C4×D4—28no—
198C4×D4—28no—
199#117—1061no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22⋊D5312yes—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C42—22no—
205C2×C42C22×D5311yescharacteristic
206#106—1064no—
207C4⋊C8—534no—
208#103—1064no—
209#103—1064no—
210#102—1068no—
211C22⋊C8—1062no—
212#97—1061no—
213#97—1062no—
214C4×C8—532no—
215C24—22no—
216C2×Q8—22no—
217C2×Q8—44no—
218C2×Q8—1062no—
219C2×Q8C2×C22⋊D5314yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—48no—
224C2×D4—44no—
225C2×D4C2×C22⋊D5314yes—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—84no—
230C22×C4C2×C22⋊D5312yes—
231C22×C4C2×C22⋊D5311yescharacteristic
232Q16—2128no—
233M16—2122no—
234C2×C8—1062no—
235C2×C8—1064no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—1062no—
240C4⋊C4—1062no—
241C4⋊C4—2122no—
242C4⋊C4C2×C22⋊D5314yes—
243C22⋊C4—216no—
244C22⋊C4—28no—
245C22⋊C4—2121no—
246C42—24no—
247C42—24no—
248C42—2121no—
249C42C2×C22⋊D5311yescharacteristic
250C42C2×C22⋊D5312yes—
251C42C23×D5311yescharacteristic
252C23—24no—
253C23—48no—
254C23—44no—
255C23#1313318611yescharacteristic
256Q8—24no—
257Q8—24no—
258Q8—44no—
259Q8—2124no—
260D4—44no—
261D4—48no—
262D4—48no—
263D4—82no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—88no—
271C2×C4—2122no—
272C2×C4#1313318612yes—
273C2×C4#1313318611yescharacteristic
274C2×C4C22×C22⋊D5311yescharacteristic
275C2×C4C22×C22⋊D5312yes—
276C8—2122no—
277C8—2124no—
278C22—22no—
279C22—48no—
280C22—44no—
281C22—88no—
282C22#1681080112yes—
283C22C2×#1313318611yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—44no—
288C4—44no—
289C4—4242no—
290C4#1681142114yes—
291C2—22no—
292C2—84no—
293C2#2028395712yes—
294C2C2×#1681080111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#559C2
socleC22×C53C2×#120
prime radicalC2×#127×C53C22
hypercenterC2×#559D53
hypocenterC53#18531
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #20284845C2semidirect
3. #20273901C2semidirect
5. #20273162C2semidirect
11. #16810908C22semidirect
12. #16810833C22semidirect
19. Q8⋊#88322C22semidirect
175. C53#18531semidirect

Maximal subgroups

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

subgroupindex
#202850892
#202848452
#202739012
#202734292
#202731622
C2×C53×#5592
#1853153