Finite Group Databasebeta

← #28308267

Subgroups — #28308267

Order 13568.

412 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 78 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#20289977C212yesmaximal
2#20289658C212yesmaximal
3#20289645C212yesmaximal
4C4×#13133215C211yesmaximal, characteristic
5#16811644—22no—
6#16811469—22no—
7#16811468C2214yes—
8C2×#13133215C411yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10#16810530C2211yescharacteristic
11#16810528C411yescharacteristic
12#16810518—22no—
13#16810492C2211yescharacteristic
14C4×#11719919—22no—
15C42×Dic53C411yescharacteristic
16C2×C4×Q8×C53C411yescharacteristic
17C4⋊#88324—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20C22×Dic106C2×C411yescharacteristic
21#13133215—22no—
22#13133215—22no—
23C4×Dic106—41no—
24C4×Dic106—41no—
25C2×C4⋊Dic53C2311yescharacteristic
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×#11719919D412yes—
29C2×C4×Dic53—21no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34#13133124—22no—
35#13133124—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C4×Q8×C53—41no—
38C4×Q8×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53—21no—
42C2×C42×C53C2×C411yescharacteristic
43C2×Q8×C53—21no—
44C2×Q8×C53—41no—
45C2×Q8×C53—41no—
46C2×Q8×C53M1612yes—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C411yescharacteristic
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53—21no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C2×Dic106—21no—
57C2×Dic106—21no—
58C2×Dic106—41no—
59C2×Dic106—41no—
60C2×#88324—44no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—41no—
63C42×C53—21no—
64C42×C53—21no—
65C42×C53—41no—
66C42×C53—41no—
67C4⋊Dic53—41no—
68#11719919—28no—
69#11719919—42no—
70#11719919—44no—
71C4×Dic53—24no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—21no—
75C4×Dic53—42no—
76C4×Dic53—41no—
77C4×Dic53M1612yes—
78C23×C53C2×C22⋊C411yescharacteristic
79Q8×C53—42no—
80Q8×C53—42no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53#9911yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106Dic106—42no—
107Dic106—42no—
108#88324—84no—
109#11532—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53#40711yescharacteristic
113C22×C53C2×#9911yescharacteristic
114C22×C53#41311yescharacteristic
115C4×C53—21no—
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—41no—
119C4×C53—42no—
120C4×C53—81no—
121C4×C53#35612yes—
122Dic53—24no—
123Dic53—42no—
124Dic53—42no—
125Dic53—42no—
126C4×C4⋊Q8—531no—
127#1467—532no—
128#1325—532no—
129#1319—532no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53#146511yescharacteristic
133C2×C53#212611yescharacteristic
134C2×C53C2×#35611yescharacteristic
135C2×C4⋊Q8—531no—
136C2×C4×Q8—531no—
137C2×C4×Q8#615711yescharacteristic
138#432—1062no—
139#391—531no—
140#391—531no—
141#389—1062no—
142#384—531no—
143C4×C4⋊C4—1062no—
144C43—531no—
145#324—534no—
146#324—1062no—
147C53#1153211yescharacteristic, complemented, Sylow
148C22×Q8—531no—
149C22×Q8C2×#615711yescharacteristic
150C4⋊Q8—1062no—
151C4⋊Q8—1062no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—2121no—
155C4×Q8—2121no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4C2×#615711yescharacteristic
162C2×C42—21no—
163C2×C42—531no—
164C2×C42—532no—
165C2×C42—531no—
166C2×C42—1061no—
167C2×C42—1061no—
168C2×C42C2×#615711yescharacteristic
169C4⋊C8—2122no—
170C22⋊C8—1064no—
171#97—2122no—
172#95—1062no—
173#95—1062no—
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—1061no—
178C2×Q8—1061no—
179C2×Q8—2121no—
180C2×Q8—2121no—
181C2×Q8#1171993612yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—532no—
185C22×C4—532no—
186C22×C4—1061no—
187C22×C4—1061no—
188C22×C4C22⋊#615711yescharacteristic
189C22×C4C22⋊#615711yescharacteristic
190C22×C4C22×#615711yescharacteristic
191C2×C8—2124no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—1068no—
195C4⋊C4—2122no—
196C4⋊C4—2121no—
197C4⋊C4—2124no—
198C42—21no—
199C42—21no—
200C42—41no—
201C42—41no—
202C42—532no—
203C42—1064no—
204C42—1062no—
205C42—1062no—
206C42—1061no—
207C42—2121no—
208C42—2122no—
209C23C2×C22⋊#615711yescharacteristic
210Q8—42no—
211Q8—42no—
212Q8—2122no—
213Q8—2122no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—2121no—
233C2×C4—2121no—
234C2×C4—2122no—
235C2×C4#1313327511yescharacteristic
236C2×C4#1313327511yescharacteristic
237C2×C4D53⋊M1611yescharacteristic
238C2×C4C2×#1171993611yescharacteristic
239C8—4244no—
240C22—21no—
241C22—21no—
242C22#1681159411yescharacteristic
243C22C2×#1313327511yescharacteristic
244C22#1681160911yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—41no—
249C4—42no—
250C4—81no—
251C4—1064no—
252C4—2122no—
253C4—2122no—
254C4—2122no—
255C4#1681153312yes—
256C2—21no—
257C2—21no—
258C2#2028997011yescharacteristic
259C2#2029098611yescharacteristic
260C2C2×#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#11532
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
147. C53#11532semidirect

Maximal subgroups

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

subgroupindex
#202899772
#202896582
#202896452
C4×#131332152
#1153253