Finite Group Databasebeta

← #27300095

Subgroups — #27300095

Order 13568.

468 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C2×#16810422C212yesmaximal
2#20274711C212yesmaximal
3#20272842C212yesmaximal
4C4×C4⋊Q8×C53C211yesmaximal, characteristic
5#16810647—22no—
6C2×#13133136C2214yes—
7#16810422—24no—
8#16810392—22no—
9C2×C4⋊Q8×C53C2211yescharacteristic
10C2×C4×Q8×C53C411yescharacteristic
11C2×C4×Q8×C53C2211yescharacteristic
12C53×#391C411yescharacteristic
13C53×#391C2211yescharacteristic
14C53×#389—22no—
15C53×#384C411yescharacteristic
16C4×C4⋊C4×C53—22no—
17C43×C53C411yescharacteristic
18C2×#11719917—24no—
19C4⋊#88322—42no—
20C4⋊#88322—42no—
21#13133136—24no—
22#13133136—24no—
23C22×Q8×C53C2×C411yescharacteristic
24C22×Q8×C53C2311yescharacteristic
25C4⋊Q8×C53—22no—
26C4⋊Q8×C53D412yes—
27C4⋊Q8×C53Q812yes—
28C4×Q8×C53—41no—
29C4×Q8×C53—41no—
30C4×Q8×C53—41no—
31C4×Q8×C53—41no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53C2×C411yescharacteristic
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53D412yes—
38C2×C42×C53—21no—
39C2×C42×C53—21no—
40C2×C42×C53—21no—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53D412yes—
45#95×C53—22no—
46#95×C53—22no—
47C2×Q8×C53—21no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53C4⋊C412yes—
53C2×Q8×C53C2×D411yescharacteristic
54C2×Q8×C53C2×Q811yescharacteristic
55C2×Q8×C53#4112yes—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C412yes—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D412yes—
65C4⋊C4×C53—24no—
66C4⋊C4×C53—24no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—44no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—41no—
72C42×C53—24no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—22no—
76C42×C53—22no—
77C42×C53—21no—
78C42×C53—42no—
79C42×C53—41no—
80C42×C53—41no—
81C42×C53—41no—
82C42×C53C4⋊C412yes—
83#11719917—44no—
84#11719917—44no—
85C2×#88322—44no—
86C23×C53C2×C22⋊C411yescharacteristic
87Q8×C53—41no—
88Q8×C53—41no—
89Q8×C53—42no—
90Q8×C53—42no—
91Q8×C53—42no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53#9911yescharacteristic
113C2×C4×C53#9911yescharacteristic
114C2×C4×C53C2×C4⋊C411yescharacteristic
115C2×C4×C53#11711yescharacteristic
116C2×C4×C53#12112yes—
117C2×C4×C53C22⋊Q812yes—
118#88322—84no—
119#12802—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53#38011yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53#41011yescharacteristic
125C4×C53—24no—
126C4×C53—22no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—22no—
130C4×C53—21no—
131C4×C53—42no—
132C4×C53—42no—
133C4×C53—42no—
134C4×C53—41no—
135C4×C53—81no—
136C4×C53#35612yescomplemented
137C4×C4⋊Q8D5311yescharacteristic
138C2×#356—532no—
139#1801—532no—
140#1394—532no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53#179911yescharacteristic
144C2×C53C2×#35611yescharacteristic
145C2×C53#213211yescharacteristic
146C2×C4⋊Q8C2×D5311yescharacteristic
147C2×C4×Q8Dic5311yescharacteristic
148C2×C4×Q8C2×D5311yescharacteristic
149#423—1062no—
150C2×#101—534no—
151#391Dic5311yescharacteristic
152#391C2×D5311yescharacteristic
153#389—22no—
154#384Dic5311yescharacteristic
155C4×C4⋊C4—22no—
156C43Dic5311yescharacteristic
157#356—1064no—
158#343—1062no—
159C53#1280211yescharacteristic, complemented, Sylow
160C22×Q8C2×Dic5311yescharacteristic
161C22×Q8C22×D5311yescharacteristic
162C2×M16—1064no—
163C4⋊Q8—22no—
164C4⋊Q8Dic10612yes—
165C4⋊Q8D21212yes—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—41no—
169C4×Q8—41no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C2×Dic5311yescharacteristic
174C2×C4⋊C4C2×Dic5311yescharacteristic
175C2×C4⋊C4C22⋊D5312yes—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C2×Dic5311yescharacteristic
180C2×C42C2×Dic5311yescharacteristic
181C2×C42C2×Dic5311yescharacteristic
182C2×C42C22⋊D5312yes—
183C4⋊C8—2122no—
184C4⋊C8—2122no—
185#101—1064no—
186#101—1064no—
187#95—22no—
188#95—22no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8C4⋊Dic5312yes—
195C2×Q8C2×Dic10611yescharacteristic
196C2×Q8C2×D21211yescharacteristic
197C2×Q8#1171994512yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4C22⋊Dic5312yes—
203C22×C4C22⋊Dic5311yescharacteristic
204C22×C4C22⋊Dic5311yescharacteristic
205C22×C4C22×Dic5311yescharacteristic
206C22×C4C2×C22⋊D5312yes—
207M16—2124no—
208M16—2124no—
209C2×C8—2124no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C42—24no—
218C42—22no—
219C42—22no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—41no—
224C42—42no—
225C42—41no—
226C42—41no—
227C42C4⋊Dic5312yes—
228C23C2×C22⋊Dic5311yescharacteristic
229Q8—41no—
230Q8—41no—
231Q8—42no—
232Q8—42no—
233Q8—42no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4#1313313211yescharacteristic
255C2×C4#1313313211yescharacteristic
256C2×C4C2×C4⋊Dic5311yescharacteristic
257C2×C4#1313317611yescharacteristic
258C2×C4#1313319112yes—
259C2×C4#1313319612yes—
260C8—4244no—
261C22—21no—
262C22—21no—
263C22#1681048311yescharacteristic
264C22C2×#1313313211yescharacteristic
265C22#1681059711yescharacteristic
266C4—24no—
267C4—22no—
268C4—21no—
269C4—21no—
270C4—22no—
271C4—21no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—42no—
276C4—81no—
277C4#1681042212yescomplemented
278C2—21no—
279C2—21no—
280C2#2027470311yescharacteristic
281C2C2×#1681042211yescharacteristic
282C2#2027640011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC4×C4⋊Q8×C53C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenterC4×C4⋊Q8D53
hypocenterC53#12802
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
136. C4×C53#356semidirect
159. C53#12802semidirect
277. C4#16810422semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202747112
#202728422
C4×C4⋊Q8×C532
#1280253