Finite Group Databasebeta

← #28308094

Subgroups — #28308094

Order 13568.

460 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 90 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#20289951C212yesmaximal
2#20289658C212yesmaximal, complemented
3#20289646C212yesmaximal, complemented
4#20277425C211yesmaximal, characteristic
5C2×C22⋊#88324—22no—
6#16811468C2214yes—
7#16811466—22no—
8C2×#13133204C411yescharacteristic
9C2×#13133182C411yescharacteristic
10C2×D4×Dic53C2211yescharacteristic
11#16810529C2211yescharacteristic
12#16810517—22no—
13#16810495—22no—
14#16810492C2211yescharacteristic
15#16810469C411yescharacteristic
16C53×#390C411yescharacteristic
17C22×#88324—42no—
18C22⋊#88324—24no—
19C22⋊#88324—42no—
20C22×Dic106C2×C411yescharacteristic
21C23×Dic53D412yes—
22#13133204—22no—
23#13133204C814yescomplemented
24#13133182—42no—
25D4×Dic53—42no—
26C2×C4⋊Dic53C2311yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53D412yes—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34#13133124—21no—
35#13133124—21no—
36#13133124—22no—
37C22×D4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C42×C53C2×C411yescharacteristic
41C24×C53C22⋊C412yes—
42C2×D4×C53—41no—
43C2×D4×C53—41no—
44C2×D4×C53C2×C812yes—
45C2×D4×C53M1612yes—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C22×C411yescharacteristic
49C22×Dic53—21no—
50C22×Dic53—21no—
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—82no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C2×D412yes—
57C2×Dic106—21no—
58C2×Dic106—41no—
59C2×Dic106—41no—
60C2×Dic106C2×C812yescomplemented
61C2×#88324—44no—
62C2×#88324—84no—
63C4⋊C4×C53—41no—
64C22⋊C4×C53—82no—
65C42×C53—41no—
66C22⋊Dic53—28no—
67C22⋊Dic53—42no—
68C4⋊Dic53—41no—
69#11719919—42no—
70#11719919—82no—
71C4×Dic53—21no—
72C4×Dic53—22no—
73C4×Dic53—42no—
74C4×Dic53C2×C812yescomplemented
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—42no—
79C23×C53C22⋊C814yes—
80C23×C53C2×C22⋊C411yescharacteristic
81D4×C53—42no—
82D4×C53—42no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53C22×C811yescharacteristic
91C2×C4×C53C2×M1611yescharacteristic
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—21no—
96C2×Dic53—41no—
97C2×Dic53—44no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—42no—
103C2×Dic53—41no—
104C2×Dic53—82no—
105C2×Dic53#9912yes—
106Dic106—42no—
107Dic106—42no—
108#88324—84no—
109#11481—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53—24no—
113C22×C53—44no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—82no—
117C22×C53C2×C22⋊C811yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53#41311yescharacteristic
120C4×C53—21no—
121C4×C53—21no—
122C4×C53—81no—
123C4×C53—81no—
124Dic53—22no—
125Dic53—42no—
126Dic53—42no—
127Dic53—41no—
128Dic53—42no—
129#2326—531no—
130#1459—532no—
131#1325—532no—
132#1319—532no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53—44no—
136C2×C53#145911yescharacteristic
137C2×C53#212711yescharacteristic
138C2×C53#213511yescharacteristic
139C2×#124—531no—
140C2×C4×Q8—531no—
141C2×C4×D4—531no—
142C2×C22⋊C8—1062no—
143#391—531no—
144#390#615711yescharacteristic
145#388—1062no—
146#385—1062no—
147#384—531no—
148#376—531no—
149#324—534no—
150#323—1062no—
151C53#1148111yescharacteristic, complemented, Sylow
152C22×Q8—531no—
153C22×D4C2×#615711yescharacteristic
154C23×C4—532no—
155C22×C8—2122no—
156#124—534no—
157#124—1062no—
158C4×Q8—2122no—
159C4×D4—2122no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4C2×#615711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—1062no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—1061no—
170C2×C42C2×#615711yescharacteristic
171C22⋊C8—1064no—
172C22⋊C8—2122no—
173#95—1061no—
174#95—1061no—
175#95—1062no—
176C24C22⋊#615712yes—
177C2×Q8—532no—
178C2×Q8—1061no—
179C2×Q8—2121no—
180C2×Q8—2121no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4D53⋊C812yes—
184C2×D4#1171993612yes—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—531no—
188C22×C4—531no—
189C22×C4—532no—
190C22×C4—1062no—
191C22×C4—1062no—
192C22×C4—1061no—
193C22×C4—1061no—
194C22×C4—4242no—
195C22×C4C22×#615711yescharacteristic
196C2×C8—4244no—
197C2×C8—2124no—
198C4⋊C4—41no—
199C4⋊C4—4242no—
200C4⋊C4—2121no—
201C4⋊C4—2122no—
202C22⋊C4—82no—
203C22⋊C4—1068no—
204C22⋊C4—2122no—
205C42—41no—
206C42—532no—
207C42—1062no—
208C42—1061no—
209C42—2122no—
210C23—22no—
211C23—22no—
212C23—42no—
213C23—42no—
214C23#1313327414yes—
215C23C2×C22⋊#615711yescharacteristic
216Q8—2122no—
217Q8—2122no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—532no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1061no—
231C2×C4—1062no—
232C2×C4—4242no—
233C2×C4—2121no—
234C2×C4—2124no—
235C2×C4—2121no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2122no—
240C2×C4—2121no—
241C2×C4C2×D53⋊C811yescharacteristic
242C2×C4C2×#1171993611yescharacteristic
243C8—4244no—
244C22—21no—
245C22—21no—
246C22—24no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22C2×#1313327411yescharacteristic
252C22D53⋊#9911yescharacteristic
253C22#1681161011yescharacteristic
254C4—21no—
255C4—21no—
256C4—81no—
257C4—81no—
258C4—1062no—
259C4—2122no—
260C4—2122no—
261C4—2122no—
262C4—2121no—
263C2—21no—
264C2—21no—
265C2—44no—
266C2#2028995311yescharacteristic
267C2#2029099111yescharacteristic
268C2#2029101811yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊Dic53C23
hypercenter#390#6157
hypocenterC53#11481
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20289658C2semidirect
3. #20289646C2semidirect
23. #13133204C8semidirect
60. C2×Dic106C2×C8semidirect
74. C4×Dic53C2×C8semidirect
151. C53#11481semidirect

Maximal subgroups

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

subgroupindex
#202899512
#202896582
#202896462
#202774252
#1148153