Finite Group Databasebeta

← #28308270

Subgroups — #28308270

Order 13568.

412 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289978C212yesmaximal
2#20289658C212yesmaximal
3#20289646C212yesmaximal
4#20277284C211yesmaximal, characteristic
5#16811643—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×#13133215C411yescharacteristic
9C2×Q8×Dic53C2211yescharacteristic
10C2×#13133182C411yescharacteristic
11#16810529C2211yescharacteristic
12#16810520—22no—
13#16810492C2211yescharacteristic
14#16810474—22no—
15C42×Dic53C411yescharacteristic
16C53×#391C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20C22×Dic106C2×C411yescharacteristic
21#13133215—22no—
22#13133215—22no—
23Q8×Dic53—41no—
24Q8×Dic53—41no—
25#13133182—41no—
26#13133182—41no—
27C2×C4⋊Dic53D412yes—
28C2×C4⋊Dic53C2311yescharacteristic
29C2×#11719919—22no—
30C2×#11719919—21no—
31C2×#11719919—22no—
32C2×C4×Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D412yes—
38#13133124—22no—
39C22×Q8×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42#95×C53—22no—
43C2×Q8×C53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—21no—
52C22×Dic53—21no—
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C2×D412yes—
56C2×Dic106—21no—
57C2×Dic106—41no—
58C2×Dic106—41no—
59C2×Dic106M1612yes—
60C2×#88324—44no—
61C4⋊C4×C53—41no—
62C42×C53—41no—
63C4⋊Dic53—42no—
64C4⋊Dic53—41no—
65#11719919—28no—
66#11719919—44no—
67#11719919—42no—
68C4×Dic53—24no—
69C4×Dic53—21no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—42no—
75C4×Dic53—41no—
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—22no—
86C2×C4×C53—22no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53C2×M1611yescharacteristic
92C2×C4×C53C2×M1611yescharacteristic
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—42no—
104C2×Dic53#9911yescharacteristic
105C2×Dic53#9911yescharacteristic
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—42no—
118C4×C53—42no—
119C4×C53—81no—
120Dic53—24no—
121Dic53—21no—
122Dic53—42no—
123Dic53—41no—
124Dic53—42no—
125Dic53#35612yes—
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—531no—
138#432—1062no—
139#391—531no—
140#391#615711yescharacteristic
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—2121no—
153C4×Q8—2121no—
154C4×Q8—2121no—
155C4×Q8—2121no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—531no—
158C2×C4⋊C4—1062no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4C2×#615711yescharacteristic
162C2×C42—531no—
163C2×C42—532no—
164C2×C42—531no—
165C2×C42—1061no—
166C2×C42—1061no—
167C2×C42—1061no—
168C2×C42C2×#615711yescharacteristic
169C4⋊C8—2122no—
170C22⋊C8—1064no—
171#97—2122no—
172#95—22no—
173#95—1062no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—532no—
179C2×Q8—1061no—
180C2×Q8—2121no—
181C2×Q8—2121no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—532no—
187C22×C4—1061no—
188C22×C4—1061no—
189C22×C4C22⋊#615712yes—
190C22×C4C22×#615711yescharacteristic
191C2×C8—2124no—
192C4⋊C4—41no—
193C4⋊C4—1068no—
194C4⋊C4—2121no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C4⋊C4—2124no—
198C42—41no—
199C42—532no—
200C42—1062no—
201C42—1062no—
202C42—1061no—
203C42—1061no—
204C42—1064no—
205C42—1061no—
206C42—2121no—
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—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—531no—
225C2×C4—531no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1062no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—2121no—
234C2×C4—2121no—
235C2×C4—2121no—
236C2×C4—2122no—
237C2×C4D53⋊M1611yescharacteristic
238C2×C4D53⋊M1611yescharacteristic
239C8—4244no—
240C22—21no—
241C22—21no—
242C22#1681159511yescharacteristic
243C22D53⋊#9911yescharacteristic
244C22#1681161011yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—532no—
251C4—1064no—
252C4—1061no—
253C4—2122no—
254C4—2121no—
255C4—2122no—
256C2—21no—
257C2—21no—
258C2#2028997311yescharacteristic
259C2#2029098711yescharacteristic
260C2D53⋊#35611yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenter#391#6157
hypocenterC53#11532
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

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
#202899782
#202896582
#202896462
#202772842
#1153253