Finite Group Databasebeta

← #28308426

Subgroups — #28308426

Order 13568.

374 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 74 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#20289977C211yesmaximal, characteristic
2#20289957C211yesmaximal, characteristic
3#20289658C214yesmaximal
4#20279924C211yesmaximal, characteristic
5#16811644—21no—
6#16811640—21no—
7#16811469—21no—
8#16811469—21no—
9#16811468C2212yes—
10#16811468C2212yes—
11C2×C4×Dic106C2211yescharacteristic
12#16810542—21no—
13#16810542—21no—
14#16810528C412yes—
15#16810519—21no—
16#16810519—21no—
17#16810492C2212yes—
18C4×C4⋊Dic53C411yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20#13133284—41no—
21C4⋊#88324—41no—
22C22⋊#88324—22no—
23C22⋊#88324—22no—
24#13133272—41no—
25C4×#88324—41no—
26C22×Dic106C2×C411yescharacteristic
27C4×Dic106—41no—
28C4×Dic106—41no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2311yescharacteristic
31C2×#11719919—22no—
32C2×#11719919—22no—
33C2×#11719919D411yescharacteristic
34C2×#11719919D411yescharacteristic
35C2×C4×Dic53C2×C412yes—
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38#13133124—21no—
39#13133124—22no—
40#13133124—22no—
41C22×Q8×C53—21no—
42C4⋊Q8×C53—41no—
43C4⋊Q8×C53—41no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C42×C53C2×C411yescharacteristic
47C2×Q8×C53—42no—
48C2×Q8×C53—82no—
49C22×C4×C53—22no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—22no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C2×D411yescharacteristic
57C22×Dic53C2×D411yescharacteristic
58C2×Dic106—22no—
59C2×Dic106—22no—
60C2×Dic106—41no—
61C2×#88324—42no—
62C2×#88324—42no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C42×C53—21no—
67C42×C53—21no—
68C4⋊Dic53—24no—
69C4⋊Dic53—41no—
70#11719919—41no—
71#11719919—41no—
72#11719919—82no—
73#11719919—82no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53—21no—
77C4×Dic53—21no—
78C4×Dic53—24no—
79C23×C53C2×C22⋊C411yescharacteristic
80Q8×C53—84no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53#9911yescharacteristic
90C2×C4×C53#9911yescharacteristic
91C2×C4×C53#13112yes—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—24no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53#10112yes—
105Dic106—44no—
106#88324—82no—
107#88324—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—41no—
117C4×C53—82no—
118C4×C53#35612yes—
119Dic53—22no—
120Dic53—44no—
121Dic53—41no—
122Dic53—41no—
123Dic53—41no—
124#2567—531no—
125#1467—531no—
126#1461—531no—
127#1325—534no—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53#147211yescharacteristic
131C2×C53#212811yescharacteristic
132C2×C53C2×#35611yescharacteristic
133C2×C4⋊Q8#615711yescharacteristic
134C2×C4×Q8—531no—
135#432—1061no—
136#431—1061no—
137#395—1061no—
138#395—1061no—
139#391—532no—
140#389—1061no—
141#389—1061no—
142#384—532no—
143C4×C4⋊C4—531no—
144#324—532no—
145#324—532no—
146#324—1061no—
147#324—1061no—
148C53#1157411yescharacteristic, complemented, Sylow
149C22×Q8—21no—
150C22×Q8—531no—
151C4⋊Q8—41no—
152C4⋊Q8—41no—
153C4×Q8—2121no—
154C4×Q8—2121no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—531no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4C2×#615712yes—
163C2×C42—531no—
164C2×C42—531no—
165C2×C42—532no—
166C2×C42C2×#615711yescharacteristic
167C4⋊C8—2121no—
168C4⋊C8—2121no—
169C22⋊C8—1062no—
170C22⋊C8—1062no—
171#97—2121no—
172C4×C8—2121no—
173#95—1062no—
174#95—1062no—
175#95—1061no—
176C2×Q8—42no—
177C2×Q8—82no—
178C2×Q8—1062no—
179C2×Q8—1062no—
180C2×Q8—2121no—
181C22×C4—22no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—1062no—
187C22×C4C22⋊#615711yescharacteristic
188C22×C4C22⋊#615711yescharacteristic
189C22×C4C22×#615711yescharacteristic
190C2×C8—2122no—
191C2×C8—2122no—
192C4⋊C4—42no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—1064no—
196C4⋊C4—4242no—
197C4⋊C4—4242no—
198C4⋊C4—2121no—
199C4⋊C4—2121no—
200C4⋊C4—2121no—
201C42—21no—
202C42—21no—
203C42—1061no—
204C42—1061no—
205C42—1061no—
206C42—1061no—
207C42—1064no—
208C23C2×C22⋊#615711yescharacteristic
209Q8—84no—
210Q8—2124no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—532no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1064no—
223C2×C4—1061no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4#1313327511yescharacteristic
233C2×C4#1313327511yescharacteristic
234C2×C4#1313330512yes—
235C8—4242no—
236C8—4242no—
237C22—21no—
238C22—21no—
239C22#1681159811yescharacteristic
240C22C2×#1313327511yescharacteristic
241C22D53⋊#10111yescharacteristic
242C4—21no—
243C4—22no—
244C4—41no—
245C4—82no—
246C4—1062no—
247C4—2124no—
248C4—2121no—
249C4—2121no—
250C4—2121no—
251C4#1681153312yes—
252C2—21no—
253C2—21no—
254C2#2028999511yescharacteristic
255C2#2029099411yescharacteristic
256C2C2×#1681153311yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11574
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

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
#202899772
#202899572
#202896582
#202799242
#1157453