Finite Group Databasebeta

← #27281787

Subgroups — #27281787

Order 13568.

448 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283800C211yesmaximal, characteristic
2#20273432C212yesmaximal
3#20273059C211yesmaximal, characteristic
4#20272571C212yesmaximal
5C53×#2567C211yesmaximal, characteristic
6#16810754—24no—
7#16810680—21no—
8#16810680C2212yes—
9#16810680C2212yes—
10#16810676—21no—
11#16810369—24no—
12#16810349—21no—
13#16810349—21no—
14C2×C4⋊Q8×C53C2211yescharacteristic
15C2×C4×Q8×C53C2211yescharacteristic
16C53×#395—21no—
17C53×#395—21no—
18C53×#391C412yes—
19C53×#389—21no—
20C53×#389—21no—
21C53×#384C412yes—
22C4×C4⋊C4×C53C2211yescharacteristic
23C4⋊#88322—22no—
24C4⋊#88322—22no—
25C4⋊#88322—22no—
26C4⋊#88322D414yes—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133128—22no—
30#13133128—22no—
31#13133128—22no—
32C4×#88322—22no—
33C22×Q8×C53—21no—
34C22×Q8×C53C2×C411yescharacteristic
35C4⋊Q8×C53—22no—
36C4⋊Q8×C53—22no—
37C4×Q8×C53—22no—
38C4×Q8×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C4⋊C4×C53D411yescharacteristic
47C2×C42×C53C2×C412yes—
48C2×C42×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53C2311yescharacteristic
51#95×C53—22no—
52#95×C53—22no—
53#95×C53—21no—
54C2×Q8×C53—22no—
55C2×Q8×C53—22no—
56C2×Q8×C53—41no—
57C2×Q8×C53—42no—
58C2×Q8×C53—82no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22×C411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C4⋊C4×C53—24no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—22no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—82no—
76C4⋊C4×C53—82no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—42no—
82C42×C53C2×D412yes—
83C42×C53#4112yes—
84C2×#88322—24no—
85C2×#88322—42no—
86C23×C53C2×C22⋊C411yescharacteristic
87Q8×C53—44no—
88Q8×C53—84no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—24no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53#10112yes—
109C2×C4×C53C4×D412yes—
110C2×C4×C53#12011yescharacteristic
111C2×C4×C53#12112yes—
112C2×C4×C53#12311yescharacteristic
113#88322—44no—
114#88322—82no—
115#9454—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53#38611yescharacteristic
119C22×C53C2×#10111yescharacteristic
120C22×C53#41311yescharacteristic
121C4×C53—21no—
122C4×C53—22no—
123C4×C53—22no—
124C4×C53—22no—
125C4×C53—44no—
126C4×C53—41no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53—82no—
130C4×C53#45612yes—
131#2983—531no—
132#2567D5311yescharacteristic
133#1556—532no—
134#1467—531no—
135#1285—532no—
136C2×C53—21no—
137C2×C53—21no—
138C2×C53#188611yescharacteristic
139C2×C53#190111yescharacteristic
140C2×C53#191011yescharacteristic
141C2×C4⋊Q8C2×D5311yescharacteristic
142C2×C4×Q8C2×D5311yescharacteristic
143#446—1064no—
144#432—532no—
145#432—532no—
146#432—1061no—
147#431—1061no—
148#395—21no—
149#395—21no—
150#391Dic5312yes—
151#389—21no—
152#389—21no—
153#384Dic5312yes—
154C4×C4⋊C4C2×D5311yescharacteristic
155#332—1064no—
156#324—1061no—
157#324—1061no—
158C53#945411yescharacteristic, complemented, Sylow
159C22×Q8—21no—
160C22×Q8C2×Dic5311yescharacteristic
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C2×Dic5312yes—
170C2×C4⋊C4C2×Dic5311yescharacteristic
171C2×C4⋊C4C22⋊D5311yescharacteristic
172C2×C4⋊C4C22⋊D5311yescharacteristic
173C2×C42C2×Dic5312yes—
174C2×C42C22⋊D5311yescharacteristic
175C2×C42C22⋊D5311yescharacteristic
176C2×C42C22×D5311yescharacteristic
177C4⋊C8—534no—
178C4⋊C8—1062no—
179C4⋊C8—1062no—
180C4⋊C8—1062no—
181C22⋊C8—1062no—
182C22⋊C8—1062no—
183#97—1062no—
184#97—1062no—
185#97—1062no—
186C4×C8—1062no—
187#95—21no—
188#95—22no—
189#95—22no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×Q8—82no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4C22⋊Dic5311yescharacteristic
198C22×C4C22⋊Dic5311yescharacteristic
199C22×C4C22⋊Dic5311yescharacteristic
200C22×C4C22⋊Dic5311yescharacteristic
201C22×C4C22×Dic5311yescharacteristic
202C22×C4C2×C22⋊D5311yescharacteristic
203C22×C4C2×C22⋊D5311yescharacteristic
204C2×C8—1064no—
205C2×C8—2122no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—42no—
220C42D4×D5312yes—
221C42#1171994712yes—
222C23C2×C22⋊Dic5311yescharacteristic
223Q8—44no—
224Q8—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4#1313313612yes—
245C2×C4D4×Dic5312yes—
246C2×C4#1313318511yescharacteristic
247C2×C4#1313318912yes—
248C2×C4#1313319811yescharacteristic
249C8—4242no—
250C8—2124no—
251C22—21no—
252C22—21no—
253C22#1681050311yescharacteristic
254C22C2×#1313313611yescharacteristic
255C22#1681060711yescharacteristic
256C4—22no—
257C4—21no—
258C4—22no—
259C4—22no—
260C4—44no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4#1681080512yes—
266C2—21no—
267C2—21no—
268C2#2027510911yescharacteristic
269C2#2027518411yescharacteristic
270C2#2027524311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2567C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2567D53
hypocenterC53#9454
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
158. C53#9454semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202734322
#202730592
#202725712
C53×#25672
#945453