Finite Group Databasebeta

← #28308199

Subgroups — #28308199

Order 13568.

446 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 76 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#20289966C212yesmaximal
2#20289658C212yesmaximal, complemented
3#20289656C212yesmaximal, complemented
4#20277118C211yesmaximal, characteristic
5#16811643—22no—
6#16811468C2214yes—
7#16811466—22no—
8C2×#13133176C2211yescharacteristic
9C22×C4×Dic53C411yescharacteristic
10#16810492C411yescharacteristic
11#16810492C2211yescharacteristic
12#16810488C411yescharacteristic
13#16810488C2211yescharacteristic
14#16810481—22no—
15C4×#11719919—22no—
16C2×#117×C53C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20C23×Dic53—21no—
21#13133176—41no—
22#13133176—41no—
23C2×C4⋊Dic53C2×C411yescharacteristic
24C2×C4⋊Dic53C2311yescharacteristic
25C2×#11719919—22no—
26C2×#11719919—22no—
27C2×C22⋊Dic53D412yes—
28C2×C4×Dic53—22no—
29C2×C4×Dic53—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53D412yes—
34#13133124—22no—
35#13133124—22no—
36C23×C4×C53C2×C411yescharacteristic
37#117×C53—41no—
38#117×C53—41no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C42×C53—21no—
43C24×C53C22⋊C411yescharacteristic
44C22×C4×C53—22no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53—24no—
55C22×Dic53—21no—
56C22×Dic53—42no—
57C22×Dic53—42no—
58C22×Dic53C22⋊C412yes—
59C22×Dic53C2×D412yes—
60C2×#88324—44no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—41no—
63C22⋊C4×C53—42no—
64C42×C53—41no—
65C42×C53—41no—
66C22⋊Dic53—42no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69#11719919—44no—
70#11719919—44no—
71C4×Dic53—24no—
72C4×Dic53—22no—
73C4×Dic53—22no—
74C4×Dic53—21no—
75C4×Dic53—21no—
76C4×Dic53M1612yes—
77C4×Dic53M1612yes—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—41no—
81C23×C53—41no—
82C23×C53#10112yes—
83C23×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53#9911yescharacteristic
97C2×C4×C53#9911yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C2×C4×C53C2×M1611yescharacteristic
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—48no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53—42no—
111#88324—84no—
112#11513—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53—22no—
116C22×C53—42no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53#40711yescharacteristic
121C22×C53C2×#9911yescharacteristic
122C22×C53C2×#10111yescharacteristic
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—81no—
128C4×C53—81no—
129Dic53—24no—
130Dic53—42no—
131Dic53—42no—
132Dic53—42no—
133#2285—531no—
134#1464—532no—
135#1325—532no—
136#1324—532no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—42no—
140C2×C53#146311yescharacteristic
141C2×C53#212611yescharacteristic
142C2×C53#213211yescharacteristic
143C2×#117—531no—
144C2×#117#615711yescharacteristic
145C22×C42—531no—
146#432—1062no—
147#384—531no—
148#384—531no—
149#382—531no—
150#382—531no—
151#380—1062no—
152C4×C4⋊C4—1062no—
153#324—534no—
154#323—1062no—
155C53#1151311yescharacteristic, complemented, Sylow
156C23×C4—1061no—
157C23×C4C2×#615711yescharacteristic
158#117—41no—
159#117—41no—
160#117—2121no—
161#117—2121no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—1062no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4C2×#615711yescharacteristic
167C2×C4⋊C4C2×#615711yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—532no—
170C2×C42—21no—
171C2×C42—531no—
172C2×C42—531no—
173C2×C42—532no—
174C2×C42—1062no—
175C2×C42—1062no—
176C2×C42—1061no—
177C4⋊C8—2122no—
178C22⋊C8—1064no—
179#97—2122no—
180#95—1062no—
181#95—1062no—
182C24C22⋊#615711yescharacteristic
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—532no—
191C22×C4—532no—
192C22×C4—1061no—
193C22×C4—1064no—
194C22×C4—1061no—
195C22×C4—2122no—
196C22×C4—2122no—
197C22×C4C22⋊#615711yescharacteristic
198C22×C4C22×#615711yescharacteristic
199C2×C8—2124no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—2121no—
203C4⋊C4—2121no—
204C4⋊C4—2124no—
205C4⋊C4—2124no—
206C22⋊C4—42no—
207C22⋊C4—2122no—
208C42—41no—
209C42—41no—
210C42—532no—
211C42—532no—
212C42—1064no—
213C42—1062no—
214C42—1062no—
215C42—1061no—
216C42—1061no—
217C23—21no—
218C23—21no—
219C23—41no—
220C23—41no—
221C23#1313327812yes—
222C23C2×C22⋊#615711yescharacteristic
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—2128no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2122no—
245C2×C4—2122no—
246C2×C4#1313327511yescharacteristic
247C2×C4#1313327511yescharacteristic
248C2×C4D53⋊M1611yescharacteristic
249C2×C4D53⋊M1611yescharacteristic
250C8—4244no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—42no—
256C22—41no—
257C22—41no—
258C22#1681159511yescharacteristic
259C22C2×#1313327511yescharacteristic
260C22C2×#1313327811yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—81no—
266C4—81no—
267C4—1064no—
268C4—2122no—
269C4—2122no—
270C4—2122no—
271C2—21no—
272C2—21no—
273C2—42no—
274C2#2028996511yescharacteristic
275C2#2029098611yescharacteristic
276C2#2029100611yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11513
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20289658C2semidirect
3. #20289656C2semidirect
155. C53#11513semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202896582
#202896562
#202771182
#1151353