Finite Group Databasebeta

← #27302780

Subgroups — #27302780

Order 13568.

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

normal subgroups only

This group has 420 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#20274711C211yesmaximal, characteristic
2C4×#13133136C211yesmaximal, characteristic
3#20272842C214yesmaximal
4C53×#2567C211yesmaximal, characteristic
5#16810647—21no—
6C2×#13133136C2212yes—
7C2×#13133136C2212yes—
8C4×#11719917—21no—
9#16810392—21no—
10#16810392—21no—
11C2×C4⋊Q8×C53C411yescharacteristic
12C2×C4×Q8×C53C2211yescharacteristic
13C53×#395—21no—
14C53×#395—21no—
15C53×#391C2212yes—
16C53×#389—21no—
17C53×#389—21no—
18C53×#384C412yes—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719917—22no—
21C2×#11719917—22no—
22C4⋊#88322—41no—
23C4⋊#88322—41no—
24#13133136—22no—
25#13133136—24no—
26#13133136Q814yescomplemented
27#13133128—41no—
28C4×#88322—41no—
29C22×Q8×C53—21no—
30C22×Q8×C53C2311yescharacteristic
31C4⋊Q8×C53—41no—
32C4⋊Q8×C53—41no—
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C42×C53C2×C412yes—
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—21no—
48#95×C53—22no—
49#95×C53—22no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53—42no—
53C2×Q8×C53—82no—
54C2×Q8×C53C2×Q812yes—
55C2×Q8×C53#4112yes—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C4⋊C4×C53—24no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—82no—
73C4⋊C4×C53—82no—
74C42×C53—21no—
75C42×C53—21no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—42no—
81#11719917—24no—
82#11719917—42no—
83#11719917—42no—
84#11719917—42no—
85C2×#88322—42no—
86C2×#88322—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88Q8×C53—42no—
89Q8×C53—42no—
90Q8×C53—84no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—24no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53#9911yescharacteristic
111C2×C4×C53#9911yescharacteristic
112C2×C4×C53C4×Q812yes—
113C2×C4×C53C22⋊Q812yes—
114C2×C4×C53#12411yescharacteristic
115C2×C4×C53C4⋊Q811yescharacteristic
116#88322—44no—
117#88322—82no—
118#13302—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53#39111yescharacteristic
122C22×C53C2×#9911yescharacteristic
123C22×C53#41011yescharacteristic
124C4×C53—21no—
125C4×C53—22no—
126C4×C53—21no—
127C4×C53—22no—
128C4×C53—21no—
129C4×C53—44no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—82no—
134C4×C53#35612yes—
135#2567D5311yescharacteristic
136#1801—531no—
137C4×#101—531no—
138#1394—534no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#199611yescharacteristic
142C2×C53C2×#35611yescharacteristic
143C2×C53#213411yescharacteristic
144C2×C4⋊Q8Dic5311yescharacteristic
145C2×C4×Q8C2×D5311yescharacteristic
146#423—1061no—
147C2×#101—532no—
148C2×#101—532no—
149C4×M16—1061no—
150#395—21no—
151#395—21no—
152#391C2×D5312yes—
153#389—21no—
154#389—21no—
155#384Dic5312yes—
156C4×C4⋊C4Dic5311yescharacteristic
157#343—1061no—
158#343—1061no—
159C53#1330211yescharacteristic, complemented, Sylow
160C22×Q8—21no—
161C22×Q8C22×D5311yescharacteristic
162C2×M16—1062no—
163C2×M16—1062no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C2×Dic5312yes—
173C2×C4⋊C4C2×Dic5311yescharacteristic
174C2×C4⋊C4C22⋊D5311yescharacteristic
175C2×C4⋊C4C22⋊D5311yescharacteristic
176C2×C42C2×Dic5311yescharacteristic
177C2×C42C2×Dic5312yes—
178C2×C42C22⋊D5311yescharacteristic
179C2×C42C22⋊D5311yescharacteristic
180C4⋊C8—2121no—
181C4⋊C8—2121no—
182#101—534no—
183#101—1062no—
184#101—1064no—
185#97—2121no—
186C4×C8—2121no—
187#95—22no—
188#95—22no—
189#95—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—82no—
194C2×Q8Q8×D5312yes—
195C2×Q8#1171994912yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4C22⋊Dic5311yescharacteristic
199C22×C4C22⋊Dic5311yescharacteristic
200C22×C4C22⋊Dic5311yescharacteristic
201C22×C4C22⋊Dic5311yescharacteristic
202C22×C4C22×Dic5311yescharacteristic
203C22×C4C2×C22⋊D5311yescharacteristic
204C22×C4C2×C22⋊D5311yescharacteristic
205M16—1064no—
206M16—2122no—
207M16—2122no—
208M16—2122no—
209C2×C8—2122no—
210C2×C8—2122no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C23C2×C22⋊Dic5311yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4#1313313211yescharacteristic
251C2×C4#1313313211yescharacteristic
252C2×C4Q8×Dic5312yes—
253C2×C4#1313319312yes—
254C2×C4#1313320611yescharacteristic
255C2×C4#1313321511yescharacteristic
256C8—4242no—
257C8—2124no—
258C22—21no—
259C22—21no—
260C22#1681053011yescharacteristic
261C22C2×#1313313211yescharacteristic
262C22#1681059711yescharacteristic
263C4—21no—
264C4—21no—
265C4—22no—
266C4—22no—
267C4—21no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#1681042212yes—
274C2—21no—
275C2—21no—
276C2#2027570111yescharacteristic
277C2C2×#1681042211yescharacteristic
278C2#2027640711yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC53×#2567C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Q8×C53C23
hypercenter#2567D53
hypocenterC53#13302
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
26. #13133136Q8semidirect
159. C53#13302semidirect

Maximal subgroups

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

subgroupindex
#202747112
C4×#131331362
#202728422
C53×#25672
#1330253