Finite Group Databasebeta

← #27295151

Subgroups — #27295151

Order 13568.

408 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 99 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#20279700C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic
3#20272722C214yesmaximal
4C53×#1461C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6C2×#13133215—21no—
7#16810647—21no—
8C2×#13133136—21no—
9#16810553C412yes—
10#16810553C2212yes—
11#16810528—21no—
12#16810521—21no—
13#16810521—21no—
14C4×C4⋊Dic53C411yescharacteristic
15#16810392C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#431—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×Dic106—21no—
21C2×#11719917—22no—
22#13133217—41no—
23#13133217—41no—
24#13133215—42no—
25#13133215—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719919—22no—
30C2×#11719919—22no—
31C2×C4×Dic53—21no—
32C4⋊#88322—41no—
33C4⋊#88322—41no—
34#13133136—42no—
35#13133124—22no—
36#13133124—22no—
37#13133124C2×C412yes—
38C22×Q8×C53D411yescharacteristic
39C4×Q8×C53—41no—
40C4×Q8×C53—41no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C4⋊C8×C53—41no—
48C22⋊C8×C53—22no—
49C4×C8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C2×Q8×C53D812yes—
53C2×Q8×C53SD1612yes—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C2×Dic106—42no—
64C2×Dic106—42no—
65C2×Dic106—42no—
66C2×C8×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—21no—
75C42×C53—21no—
76C4⋊Dic53—24no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79C4⋊Dic53—42no—
80#11719919—44no—
81#11719919—42no—
82#11719919—42no—
83C4×Dic53—42no—
84#11719917—42no—
85#11719917—42no—
86C2×#88322—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88Q8×C53—42no—
89Q8×C53—42no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53#9911yescharacteristic
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#10212yes—
106C2×C4×C53#10212yes—
107C2×C4×C53C2×D811yescharacteristic
108C2×C4×C53C2×SD1611yescharacteristic
109C2×Dic53—24no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—42no—
114C2×Dic53—42no—
115Dic106—84no—
116C8×C53—82no—
117#88322—82no—
118#11883—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53C2×#9911yescharacteristic
122C22×C53C2×#10211yescharacteristic
123C22×C53#42111yescharacteristic
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—22no—
127C4×C53—22no—
128C4×C53—21no—
129C4×C53—41no—
130C4×C53—41no—
131C4×C53—41no—
132C4×C53#35612yes—
133Dic53—44no—
134Dic53—82no—
135#2546—531no—
136#1801—531no—
137#1461D5311yescharacteristic
138#1346—534no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#151911yescharacteristic
142C2×C53#212311yescharacteristic
143C2×C53C2×#35611yescharacteristic
144C2×C4⋊Q8—531no—
145C2×C4⋊Q8—1061no—
146C2×C4×Q8C2×D5311yescharacteristic
147#431—21no—
148#423—1061no—
149C2×#101—1061no—
150#398—532no—
151#398—532no—
152#391—1061no—
153#389—1061no—
154#389—1061no—
155C4×C4⋊C4—531no—
156#343—532no—
157#324—21no—
158#324C2×D5312yes—
159C53#1188311yescharacteristic, complemented, Sylow
160C22×Q8—1061no—
161C22×Q8D21211yescharacteristic
162C2×M16—1062no—
163C4⋊Q8—2121no—
164C4⋊Q8—2121no—
165C4⋊Q8—2122no—
166C4⋊Q8—2122no—
167C4×Q8—41no—
168C4×Q8—41no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—1061no—
171C2×C4⋊C4—1062no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4C4×D5311yescharacteristic
175C2×C4⋊C4C22⋊D5311yescharacteristic
176C2×C4⋊C4C22×D5311yescharacteristic
177C2×C42—1061no—
178C2×C42C4×D5311yescharacteristic
179C2×C42D21211yescharacteristic
180C2×C42C22⋊D5311yescharacteristic
181C4⋊C8—41no—
182C4⋊C8—2121no—
183C4⋊C8—2121no—
184#101—2122no—
185C22⋊C8—22no—
186C4×C8—41no—
187#95—532no—
188#95—1062no—
189#95—1062no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—2122no—
193C2×Q8—2122no—
194C2×Q8—2122no—
195C2×Q8#1171991312yes—
196C2×Q8D42412yes—
197C22×C4—1062no—
198C22×C4—1062no—
199C22×C4#1171992111yescharacteristic
200C22×C4#1171992111yescharacteristic
201C22×C4#1171992111yescharacteristic
202C22×C4#1171992111yescharacteristic
203C22×C4C2×C4×D5311yescharacteristic
204C22×C4C2×D21211yescharacteristic
205C22×C4C2×C22⋊D5311yescharacteristic
206M16—2122no—
207M16—2122no—
208C2×C8—42no—
209C2×C8—2122no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—1064no—
214C4⋊C4—2122no—
215C4⋊C4—2121no—
216C4⋊C4—2121no—
217C4⋊C4—2124no—
218C4⋊C4—2122no—
219C4⋊C4—2122no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—2122no—
227C23C2×#1171992111yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—4244no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—1064no—
245C2×C4—2122no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4#1313313111yescharacteristic
251C2×C4#1313313111yescharacteristic
252C2×C4#1313313912yes—
253C2×C4#1313313912yes—
254C2×C4C2×D42411yescharacteristic
255C2×C4C2×#1171991311yescharacteristic
256C8—82no—
257C8—4242no—
258C22—21no—
259C22—21no—
260C22C2×#1313313111yescharacteristic
261C22C2×#1313313911yescharacteristic
262C22#1681063911yescharacteristic
263C4—21no—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—4242no—
272C4—2124no—
273C4#1681042112yes—
274C2—21no—
275C2—21no—
276C2#2027328411yescharacteristic
277C2#2027636811yescharacteristic
278C2C2×#1681042111yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1461C2
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1461D53
hypocenterC53#11883
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
159. C53#11883semidirect

Maximal subgroups

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

subgroupindex
#202797002
#202747112
#202727222
C53×#14612
#1188353