Finite Group Databasebeta

← #27293760

Subgroups — #27293760

Order 13568.

374 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279920C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272660C214yesmaximal
4C53×#1461C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6#16810680—21no—
7#16810543—21no—
8#16810543—21no—
9#16810530C412yes—
10#16810521—21no—
11#16810521—21no—
12#16810492C2212yes—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349—21no—
15#16810349C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#431—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×Dic106—21no—
21#13133217—41no—
22#13133217—41no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53C2×C412yes—
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×C4×Dic53C2×C412yes—
29C4⋊#88322—41no—
30C22⋊#88322—22no—
31#13133128—41no—
32#13133124—22no—
33#13133124—22no—
34#13133124—21no—
35C22×Q8×C53C2×C411yescharacteristic
36C4×Q8×C53—41no—
37C4×Q8×C53—41no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53D411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C4⋊C8×C53—41no—
45C22⋊C8×C53—22no—
46C4×C8×C53—41no—
47C2×Q8×C53—22no—
48C2×Q8×C53—22no—
49C2×Q8×C53—41no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53—22no—
59C2×Dic106—42no—
60C2×Dic106—82no—
61C2×C8×C53—42no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—41no—
65C42×C53—21no—
66C42×C53—21no—
67C42×C53—21no—
68C42×C53—21no—
69C42×C53—21no—
70C42×C53—21no—
71C4⋊Dic53—24no—
72C4⋊Dic53—41no—
73C4⋊Dic53—41no—
74C4⋊Dic53—42no—
75#11719919—82no—
76#11719919—82no—
77C4×Dic53—24no—
78C2×#88322—42no—
79C23×C53C2×C22⋊C411yescharacteristic
80Q8×C53—44no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53#9911yescharacteristic
96C2×C4×C53#9911yescharacteristic
97C2×C4×C53#10112yes—
98C2×C4×C53#13112yes—
99C2×Dic53—24no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105Dic106—84no—
106C8×C53—82no—
107#88322—82no—
108#11574—531nomaximal, Sylow
109C22×C53—21no—
110C22×C53—21no—
111C22×C53#40811yescharacteristic
112C22×C53C2×#9911yescharacteristic
113C22×C53C2×#10111yescharacteristic
114C4×C53—21no—
115C4×C53—22no—
116C4×C53—22no—
117C4×C53—41no—
118C4×C53—41no—
119C4×C53—41no—
120C4×C53—41no—
121C4×C53#35612yes—
122Dic53—44no—
123Dic53—82no—
124#2567—531no—
125#1467—531no—
126#1461D5311yescharacteristic
127#1325—534no—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53#147211yescharacteristic
131C2×C53#212811yescharacteristic
132C2×C53C2×#35611yescharacteristic
133C2×C4⋊Q8—531no—
134C2×C4×Q8C2×D5311yescharacteristic
135#432—1061no—
136#431—21no—
137#395—1061no—
138#395—1061no—
139#391—532no—
140#389—1061no—
141#389—1061no—
142#384—532no—
143C4×C4⋊C4—531no—
144#324—21no—
145#324—532no—
146#324—1061no—
147#324C2×D5312yes—
148C53#1157411yescharacteristic, complemented, Sylow
149C22×Q8—1061no—
150C22×Q8C4×D5311yescharacteristic
151C4⋊Q8—2121no—
152C4⋊Q8—2121no—
153C4×Q8—41no—
154C4×Q8—41no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—1061no—
157C2×C4⋊C4—1062no—
158C2×C4⋊C4—1062no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4D21211yescharacteristic
161C2×C4⋊C4C22⋊D5311yescharacteristic
162C2×C4⋊C4C22×D5311yescharacteristic
163C2×C42—532no—
164C2×C42C4×D5311yescharacteristic
165C2×C42D21211yescharacteristic
166C2×C42C22⋊D5311yescharacteristic
167C4⋊C8—41no—
168C4⋊C8—2121no—
169C22⋊C8—22no—
170C22⋊C8—1062no—
171#97—2121no—
172C4×C8—41no—
173#95—1061no—
174#95—1062no—
175#95—1062no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—41no—
179C2×Q8—4242no—
180C2×Q8—2122no—
181C22×C4—1062no—
182C22×C4—1062no—
183C22×C4#1171992111yescharacteristic
184C22×C4#1171992111yescharacteristic
185C22×C4#1171992111yescharacteristic
186C22×C4#1171992111yescharacteristic
187C22×C4C2×C4×D5311yescharacteristic
188C22×C4C2×D21211yescharacteristic
189C22×C4C2×C22⋊D5311yescharacteristic
190C2×C8—42no—
191C2×C8—2122no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—1064no—
196C4⋊C4—4242no—
197C4⋊C4—4242no—
198C4⋊C4—2122no—
199C4⋊C4—2121no—
200C4⋊C4—2121no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—1064no—
208C23C2×#1171992111yescharacteristic
209Q8—44no—
210Q8—4244no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1064no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4#1313313111yescharacteristic
232C2×C4#1313313111yescharacteristic
233C2×C4#1313313512yes—
234C2×C4#1313322312yes—
235C8—82no—
236C8—4242no—
237C22—21no—
238C22—21no—
239C22#1681058911yescharacteristic
240C22C2×#1313313111yescharacteristic
241C22C2×#1313313511yescharacteristic
242C4—21no—
243C4—22no—
244C4—22no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—41no—
249C4—4242no—
250C4—2124no—
251C4#1681042112yes—
252C2—21no—
253C2—21no—
254C2#2027307811yescharacteristic
255C2#2027638611yescharacteristic
256C2C2×#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#11574
LCSG ▷ C22×C4×C53 ▷ C22×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
148. C53#11574semidirect

Maximal subgroups

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

subgroupindex
#202799202
#202730592
#202726602
C53×#14612
#1157453