Finite Group Databasebeta

← #27293734

Subgroups — #27293734

Order 13568.

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

normal subgroups only

This group has 394 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#20279700C211yesmaximal, characteristic
2#20273059C211yesmaximal, characteristic
3#20272648C214yesmaximal
4C53×#1461C211yesmaximal, characteristic
5C2×#13133217C411yescharacteristic
6C2×#13133215C412yes—
7#16810680—21no—
8#16810553—21no—
9#16810553—21no—
10#16810528C2212yes—
11#16810521—21no—
12#16810521—21no—
13C4×C4⋊Dic53C411yescharacteristic
14#16810349—21no—
15#16810349C2212yes—
16C2×C4×Q8×C53C2211yescharacteristic
17C53×#431—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C22×Dic106C2×C412yes—
21#13133217—41no—
22#13133217—41no—
23#13133215—44no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×#11719919—22no—
28C2×#11719919—22no—
29C2×C4×Dic53C2×C412yes—
30C4⋊#88322—41no—
31C22⋊#88322—22no—
32#13133128—41no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C4×Q8×C53—41no—
38C4×Q8×C53—41no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53D411yescharacteristic
44C2×C42×C53D411yescharacteristic
45C4⋊C8×C53—41no—
46C22⋊C8×C53—22no—
47C4×C8×C53—41no—
48C2×Q8×C53—22no—
49C2×Q8×C53—22no—
50C2×Q8×C53—41no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C2×Dic106—24no—
61C2×Dic106—42no—
62C2×Dic106—42no—
63C2×C8×C53—42no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C42×C53—21no—
68C42×C53—21no—
69C42×C53—21no—
70C42×C53—21no—
71C42×C53—21no—
72C42×C53—21no—
73C4⋊Dic53—24no—
74C4⋊Dic53—42no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77#11719919—44no—
78#11719919—44no—
79C4×Dic53—42no—
80C2×#88322—42no—
81C23×C53C2×C22⋊C411yescharacteristic
82Q8×C53—44no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53#9911yescharacteristic
98C2×C4×C53#9911yescharacteristic
99C2×C4×C53#10012yes—
100C2×C4×C53#13112yes—
101C2×Dic53—24no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107Dic106—84no—
108C8×C53—82no—
109#88322—82no—
110#11568—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53#40811yescharacteristic
114C22×C53C2×#9911yescharacteristic
115C22×C53C2×#10011yescharacteristic
116C4×C53—21no—
117C4×C53—22no—
118C4×C53—22no—
119C4×C53—41no—
120C4×C53—41no—
121C4×C53—41no—
122C4×C53—41no—
123C4×C53#35612yes—
124Dic53—44no—
125Dic53—82no—
126#2546—531no—
127#1467—531no—
128#1461D5311yescharacteristic
129#1319—534no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53#147111yescharacteristic
133C2×C53#212911yescharacteristic
134C2×C53C2×#35611yescharacteristic
135C2×C4⋊Q8—532no—
136C2×C4⋊Q8—531no—
137C2×C4×Q8C2×D5311yescharacteristic
138#432—1061no—
139#431—21no—
140#398—1061no—
141#398—1061no—
142#391—532no—
143#389—1061no—
144#389—1061no—
145C4×C4⋊C4—531no—
146#324—21no—
147#324—532no—
148#324—1061no—
149#324C2×D5312yes—
150C53#1156811yescharacteristic, complemented, Sylow
151C22×Q8—532no—
152C22×Q8C4×D5311yescharacteristic
153C4⋊Q8—2121no—
154C4⋊Q8—2121no—
155C4⋊Q8—2124no—
156C4×Q8—41no—
157C4×Q8—41no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4D21211yescharacteristic
164C2×C4⋊C4C22⋊D5311yescharacteristic
165C2×C4⋊C4C22×D5311yescharacteristic
166C2×C42—532no—
167C2×C42C4×D5311yescharacteristic
168C2×C42D21211yescharacteristic
169C2×C42C22⋊D5311yescharacteristic
170C4⋊C8—41no—
171C4⋊C8—2121no—
172C22⋊C8—22no—
173C22⋊C8—1062no—
174#97—2121no—
175C4×C8—41no—
176#95—1062no—
177#95—1062no—
178#95—1061no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×Q8—1064no—
183C2×Q8—2122no—
184C2×Q8—2122no—
185C22×C4—1062no—
186C22×C4—1062no—
187C22×C4#1171992111yescharacteristic
188C22×C4#1171992111yescharacteristic
189C22×C4#1171992111yescharacteristic
190C22×C4#1171992111yescharacteristic
191C22×C4C2×C4×D5311yescharacteristic
192C22×C4C2×D21211yescharacteristic
193C22×C4C2×C22⋊D5311yescharacteristic
194C2×C8—42no—
195C2×C8—2122no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—1064no—
200C4⋊C4—2124no—
201C4⋊C4—2122no—
202C4⋊C4—2121no—
203C4⋊C4—2121no—
204C4⋊C4—2124no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—2122no—
212C23C2×#1171992111yescharacteristic
213Q8—44no—
214Q8—4244no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—1064no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2122no—
235C2×C4#1313313111yescharacteristic
236C2×C4#1313313111yescharacteristic
237C2×C4#1313313312yes—
238C2×C4#1313322312yes—
239C8—82no—
240C8—4242no—
241C22—21no—
242C22—21no—
243C22#1681058911yescharacteristic
244C22C2×#1313313111yescharacteristic
245C22C2×#1313313311yescharacteristic
246C4—21no—
247C4—22no—
248C4—22no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—4242no—
254C4—2124no—
255C4#1681042112yes—
256C2—21no—
257C2—21no—
258C2#2027307411yescharacteristic
259C2#2027638911yescharacteristic
260C2C2×#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#11568
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
150. C53#11568semidirect

Maximal subgroups

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

subgroupindex
#202797002
#202730592
#202726482
C53×#14612
#1156853