Finite Group Databasebeta

← #27360393

Subgroups — #27360393

Order 13568.

918 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 195 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#20285095C211yesmaximal, characteristic
2#20284845C212yesmaximal
3#20273936C218yesmaximal, complemented
4#20273441C211yesmaximal, characteristic
5#20273162C212yesmaximal
6C2×Q82×C53C211yesmaximal, characteristic
7#16811180C2211yescharacteristic
8#16810991—22no—
9#16810973C2214yes—
10#16810953—22no—
11#16810908C2214yes—
12#16810833C2214yes—
13#16810779—22no—
14#16810728—24no—
15#16810680C2212yes—
16#16810632—22no—
17C4×#11719917C2211yescharacteristic
18#16810372C2214yes—
19Q8⋊#88322C2218yes—
20Q82×C53—22no—
21Q82×C53—24no—
22Q82×C53C2214yes—
23C2×C4⋊Q8×C53—22no—
24C2×C4⋊Q8×C53—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4×Q8×C53—22no—
27C2×C4×Q8×C53C2212yes—
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#13133141—28no—
39#13133140—24no—
40C22⋊#88322—22no—
41#13133128—21no—
42#13133128—22no—
43C4×#88322C2312yes—
44C22×Q8×C53—22no—
45C22×Q8×C53D412yes—
46C22×Q8×C53D412yes—
47C4⋊Q8×C53—28no—
48C4⋊Q8×C53—28no—
49C4⋊Q8×C53—28no—
50C4⋊Q8×C53—22no—
51C4⋊Q8×C53—28no—
52C4⋊Q8×C53D412yes—
53C4⋊Q8×C53C2312yes—
54C4×Q8×C53—28no—
55C4×Q8×C53—22no—
56C4×Q8×C53—22no—
57C4×Q8×C53—28no—
58C4×Q8×C53D414yes—
59C4×Q8×C53C2314yes—
60C2×C4⋊C4×C53—24no—
61C2×C4⋊C4×C53—22no—
62C2×C4⋊C4×C53—22no—
63C2×C4⋊C4×C53D412yes—
64C2×C42×C53—22no—
65C2×C42×C53—22no—
66C2×C42×C53C2311yescharacteristic
67C2×Q8×C53—28no—
68C2×Q8×C53—28no—
69C2×Q8×C53—22no—
70C2×Q8×C53—22no—
71C2×Q8×C53—44no—
72C2×Q8×C53—84no—
73C2×Q8×C53C2×D414yes—
74C2×Q8×C53C2×D414yes—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53C2×D412yes—
79C22×C4×C53C2×D411yescharacteristic
80C2×Dic106—22no—
81C4⋊C4×C53—216no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—24no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—24no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53C2×D414yes—
88C42×C53—28no—
89C42×C53—24no—
90C42×C53—24no—
91C42×C53C2×D411yescharacteristic
92C42×C53C2×D412yes—
93C42×C53C2411yescharacteristic
94C22⋊Dic53—41no—
95#11719925—48no—
96C4⋊Dic53—22no—
97C4⋊Dic53—22no—
98#11719919—42no—
99C4×Dic53—41no—
100#11719917—42no—
101C2×#88322—22no—
102C2×#88322—24no—
103C23×C53#12011yescharacteristic
104Q8×C53—28no—
105Q8×C53—24no—
106Q8×C53—44no—
107Q8×C53—82no—
108Q8×C53—88no—
109Q8×C53#13718yes—
110C2×C4×C53—28no—
111C2×C4×C53—24no—
112C2×C4×C53—22no—
113C2×C4×C53—24no—
114C2×C4×C53—24no—
115C2×C4×C53—24no—
116C2×C4×C53—44no—
117C2×C4×C53—44no—
118C2×C4×C53#12011yescharacteristic
119C2×C4×C53#12012yes—
120C2×C4×C53C22×D411yescharacteristic
121C2×C4×C53C22×D412yes—
122C2×Dic53—42no—
123Dic106—44no—
124#88322—42no—
125#88322—44no—
126#18551—531nomaximal, Sylow
127C22×C53—22no—
128C22×C53#45512yes—
129C22×C53C2×#12011yescharacteristic
130C4×C53—28no—
131C4×C53—22no—
132C4×C53—21no—
133C4×C53—21no—
134C4×C53—44no—
135C4×C53—44no—
136C4×C53C2×#13714yes—
137Dic53—82no—
138C2×Q82D5311yescharacteristic
139#3139—531no—
140#3113—532no—
141#1645—538no—
142#1558—531no—
143#1491—532no—
144C2×C53—22no—
145C2×C53#300412yes—
146C2×C53C2×#45511yescharacteristic
147Q82—22no—
148Q82—24no—
149Q82C2×D5314yes—
150#533—531no—
151C2×C4⋊Q8—22no—
152C2×C4⋊Q8—22no—
153C2×C4⋊Q8C2×D5311yescharacteristic
154C2×C4×Q8—22no—
155C2×C4×Q8C2×D5312yes—
156#491—1062no—
157#487—534no—
158#483—1062no—
159#475—534no—
160#462—534no—
161#451—1062no—
162#441—1064no—
163#432—532no—
164#419—1062no—
165C4×M16—531no—
166#333—534no—
167Q8⋊C8—538no—
168C53#1855111yescharacteristic, complemented, Sylow
169C22×Q8—22no—
170C22×Q8C22⋊D5312yes—
171C22×Q8C22⋊D5312yes—
172C2×Q16—1064no—
173C2×M16—1061no—
174C4⋊Q8—28no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—22no—
179C4⋊Q8—532no—
180C4⋊Q8C22⋊D5312yes—
181C4⋊Q8C22×D5312yes—
182#125—1061no—
183C22⋊Q8—1062no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—1062no—
189C4×Q8C22⋊D5314yes—
190C4×Q8C22×D5314yes—
191#117—1061no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C22⋊D5312yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C22×D5311yescharacteristic
199#106—1064no—
200C4⋊C8—534no—
201#103—1068no—
202#103—1064no—
203#103—1064no—
204C22⋊C8—1062no—
205#97—1061no—
206#97—1062no—
207C4×C8—532no—
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—84no—
214C2×Q8—1062no—
215C2×Q8C2×C22⋊D5314yes—
216C2×Q8C2×C22⋊D5314yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4C2×C22⋊D5312yes—
221C22×C4C2×C22⋊D5311yescharacteristic
222Q16—2128no—
223M16—2122no—
224C2×C8—1062no—
225C2×C8—1064no—
226C4⋊C4—216no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—1062no—
233C4⋊C4—1062no—
234C4⋊C4—2122no—
235C4⋊C4C2×C22⋊D5314yes—
236C22⋊C4—2121no—
237C42—28no—
238C42—24no—
239C42—24no—
240C42—2121no—
241C42C2×C22⋊D5311yescharacteristic
242C42C2×C22⋊D5312yes—
243C42C23×D5311yescharacteristic
244C23#1313318611yescharacteristic
245Q8—28no—
246Q8—24no—
247Q8—44no—
248Q8—82no—
249Q8—88no—
250Q8—2124no—
251Q8#1313324818yes—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—2122no—
261C2×C4#1313318612yes—
262C2×C4#1313318611yescharacteristic
263C2×C4C22×C22⋊D5311yescharacteristic
264C2×C4C22×C22⋊D5312yes—
265C8—2122no—
266C8—2124no—
267C22—22no—
268C22#1681080112yes—
269C22C2×#1313318611yescharacteristic
270C4—28no—
271C4—22no—
272C4—21no—
273C4—21no—
274C4—44no—
275C4—44no—
276C4—4242no—
277C4C2×#1313324814yes—
278C2—22no—
279C2#2028394912yes—
280C2C2×#1681080111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×Q82×C53C2
socleC22×C53C2×#120
prime radicalC23×C53#120
hypercenterC2×Q82D53
hypocenterC53#18551
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #20273936C2semidirect
168. C53#18551semidirect

Maximal subgroups

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

subgroupindex
#202850952
#202848452
#202739362
#202734412
#202731622
C2×Q82×C532
#1855153