Finite Group Databasebeta

← #27278763

Subgroups — #27278763

Order 13568.

510 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 131 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#20273110C212yesmaximal
3#20273039C211yesmaximal, characteristic
4#20272548C212yesmaximal
5C4×C4⋊Q8×C53C211yesmaximal, characteristic
6#16810754—24no—
7#16810680—22no—
8#16810676C2212yes—
9#16810676C2212yes—
10Q8⋊#88322—24no—
11#16810349—22no—
12C2×C4⋊Q8×C53C411yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C2×C4×Q8×C53C2211yescharacteristic
15C53×#391C411yescharacteristic
16C53×#391C411yescharacteristic
17C53×#389—22no—
18C53×#384C411yescharacteristic
19C4×C4⋊C4×C53—22no—
20C43×C53C2211yescharacteristic
21C4⋊#88322—24no—
22C4⋊#88322—22no—
23C4⋊#88322D414yes—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133128—24no—
27C4×#88322—22no—
28C4×#88322—22no—
29C22×Q8×C53C2×C411yescharacteristic
30C22×Q8×C53C2×C411yescharacteristic
31C4⋊Q8×C53—22no—
32C4⋊Q8×C53—22no—
33C4×Q8×C53—22no—
34C4×Q8×C53—22no—
35C4×Q8×C53—22no—
36C4×Q8×C53—22no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D412yes—
43C2×C42×C53—21no—
44C2×C42×C53—21no—
45C2×C42×C53—21no—
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D412yes—
49C2×C42×C53C2311yescharacteristic
50#95×C53—22no—
51#95×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—21no—
54C2×Q8×C53—21no—
55C2×Q8×C53—21no—
56C2×Q8×C53—22no—
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53M1612yes—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C4⋊C4×C53—28no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—44no—
74C4⋊C4×C53—42no—
75C42×C53—24no—
76C42×C53—22no—
77C42×C53—22no—
78C42×C53—21no—
79C42×C53—22no—
80C42×C53—42no—
81C42×C53—41no—
82C42×C53—41no—
83C42×C53M1612yes—
84C42×C53C2×D412yes—
85C42×C53#4112yes—
86C2×#88322—24no—
87C2×#88322—42no—
88C23×C53C2×C22⋊C411yescharacteristic
89Q8×C53—24no—
90Q8×C53—42no—
91Q8×C53—42no—
92Q8×C53—42no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53C4×D412yes—
112C2×C4×C53#12011yescharacteristic
113C2×C4×C53#12112yes—
114C2×C4×C53#12311yescharacteristic
115C2×C4×C53C2×M1611yescharacteristic
116C2×C4×C53C2×M1611yescharacteristic
117C2×C4×C53#13112yes—
118#88322—44no—
119#88322—82no—
120#8962—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53#38611yescharacteristic
124C22×C53#40711yescharacteristic
125C22×C53#40811yescharacteristic
126C4×C53—24no—
127C4×C53—22no—
128C4×C53—21no—
129C4×C53—42no—
130C4×C53—42no—
131C4×C53—42no—
132C4×C53—42no—
133C4×C53—81no—
134C4×C53#43512yes—
135C4×C53#43612yes—
136C4×C53#45612yes—
137#2983—531no—
138C4×C4⋊Q8D5311yescharacteristic
139#1479—532no—
140#1461—531no—
141#1276—532no—
142C2×C53—21no—
143C2×C53—21no—
144C2×C53#187011yescharacteristic
145C2×C53#188411yescharacteristic
146C2×C53#188611yescharacteristic
147C2×C4⋊Q8Dic5311yescharacteristic
148C2×C4×Q8C2×D5311yescharacteristic
149C2×C4×Q8C2×D5311yescharacteristic
150#446—1064no—
151#432—1062no—
152#431—532no—
153#431—532no—
154#391Dic5311yescharacteristic
155#391Dic5311yescharacteristic
156#389—22no—
157#384Dic5311yescharacteristic
158C4×C4⋊C4—22no—
159C43C2×D5311yescharacteristic
160Q8⋊C8—1064no—
161#324—1062no—
162C53#896211yescharacteristic, complemented, Sylow
163C22×Q8C2×Dic5311yescharacteristic
164C22×Q8C2×Dic5311yescharacteristic
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C2×Dic5311yescharacteristic
175C2×C4⋊C4C2×Dic5311yescharacteristic
176C2×C4⋊C4C22⋊D5312yes—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42C2×Dic5311yescharacteristic
181C2×C42C2×Dic5311yescharacteristic
182C2×C42C22⋊D5312yes—
183C2×C42C22×D5311yescharacteristic
184C4⋊C8—534no—
185C4⋊C8—1064no—
186C4⋊C8—1062no—
187C22⋊C8—1062no—
188C22⋊C8—1062no—
189#97—1064no—
190C4×C8—1062no—
191C4×C8—1062no—
192#95—22no—
193#95—22no—
194C2×Q8—22no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8#1171991712yes—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4C22⋊Dic5312yes—
207C22×C4C22⋊Dic5312yes—
208C22×C4C22×Dic5311yescharacteristic
209C22×C4C2×C22⋊D5311yescharacteristic
210C22×C4C2×C22⋊D5311yescharacteristic
211C2×C8—1064no—
212C2×C8—2122no—
213C4⋊C4—28no—
214C4⋊C4—22no—
215C4⋊C4—24no—
216C4⋊C4—22no—
217C4⋊C4—44no—
218C4⋊C4—42no—
219C42—24no—
220C42—22no—
221C42—22no—
222C42—21no—
223C42—22no—
224C42—42no—
225C42—41no—
226C42—41no—
227C42#1171991712yes—
228C42D4×D5312yes—
229C42#1171994712yes—
230C23C2×C22⋊Dic5311yescharacteristic
231Q8—24no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4D4×Dic5312yes—
254C2×C4#1313318511yescharacteristic
255C2×C4#1313318912yes—
256C2×C4#1313319811yescharacteristic
257C2×C4C2×#1171991711yescharacteristic
258C2×C4C2×#1171991711yescharacteristic
259C2×C4#1313322412yes—
260C8—4242no—
261C8—2124no—
262C22—21no—
263C22—21no—
264C22#1681050311yescharacteristic
265C22#1681058711yescharacteristic
266C22#1681059011yescharacteristic
267C4—24no—
268C4—22no—
269C4—21no—
270C4—42no—
271C4—42no—
272C4—42no—
273C4—42no—
274C4—81no—
275C4#1681069512yes—
276C4#1681070112yes—
277C4#1681080512yes—
278C2—21no—
279C2—21no—
280C2#2027501311yescharacteristic
281C2#2027509911yescharacteristic
282C2#2027510911yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×C4⋊Q8×C53C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenterC4×C4⋊Q8D53
hypocenterC53#8962
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
162. C53#8962semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202731102
#202730392
#202725482
C4×C4⋊Q8×C532
#896253