Finite Group Databasebeta

← #28308093

Subgroups — #28308093

Order 13568.

460 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 86 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#20289952C212yesmaximal
2#20289657C212yesmaximal, complemented
3#20289646C212yesmaximal, complemented
4#20277426C211yesmaximal, characteristic
5D53⋊(C22⋊C8)—22no—
6#16811468C2214yes—
7#16811467—22no—
8C2×#13133205C411yescharacteristic
9C2×C4×Dic106C411yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810529C2211yescharacteristic
12#16810525C411yescharacteristic
13#16810515—22no—
14#16810497—22no—
15#16810493C2211yescharacteristic
16C53×#376C411yescharacteristic
17C2×D53⋊C8—42no—
18#13133274—42no—
19C22⋊#88324—24no—
20C22×Dic106C2×C411yescharacteristic
21C22×D212C2×C411yescharacteristic
22C22×C4×D53D412yes—
23#13133205—22no—
24#13133205C814yescomplemented
25C4×Dic106—42no—
26C4×D212—42no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2311yescharacteristic
30C2×#11719919—21no—
31C2×#11719921—22no—
32C2×#11719921—22no—
33C2×#11719921D412yes—
34C2×C4×Dic53—21no—
35#13133124—22no—
36C2×C42×C53C2×C411yescharacteristic
37C2×C42×C53C2×C411yescharacteristic
38C2×C42×C53C2×C411yescharacteristic
39#95×C53—21no—
40#95×C53—21no—
41C23×D53C22⋊C412yes—
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53C22⋊C411yescharacteristic
45C22×C4×C53C22⋊C411yescharacteristic
46C22×C4×C53C22×C411yescharacteristic
47C22×Dic53—21no—
48C22×Dic53—21no—
49C22×Dic53C2×D412yes—
50C2×D212—41no—
51C2×D212—41no—
52C2×D212C2×C812yes—
53C2×D212M1612yes—
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—82no—
57C2×Dic106—21no—
58C2×Dic106—41no—
59C2×Dic106—41no—
60C2×Dic106C2×C812yescomplemented
61C2×#88324—44no—
62D53⋊C8—84no—
63C42×C53—22no—
64C42×C53—21no—
65C42×C53—41no—
66C42×C53C2×C812yescomplemented
67#11719921—28no—
68#11719921—42no—
69#11719921—82no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4⋊Dic53—82no—
73#11719919—42no—
74C4×Dic53—42no—
75C23×C53C2×C22⋊C411yescharacteristic
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—42no—
79C22×D53—42no—
80C22×D53C22⋊C814yes—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53#9912yes—
93C2×C4×C53C22×C811yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—41no—
98C2×Dic53—41no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—41no—
102D212—42no—
103D212—42no—
104C4×D53—44no—
105C4×D53—82no—
106Dic106—42no—
107Dic106—42no—
108#88324—84no—
109#11481—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53C2×C22⋊C811yescharacteristic
113C22×C53C2×#9911yescharacteristic
114C22×C53#41311yescharacteristic
115C2×D53—24no—
116C2×D53—42no—
117C2×D53—42no—
118C2×D53—44no—
119C2×D53—82no—
120C4×C53—22no—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—42no—
124C4×C53—41no—
125C4×C53—81no—
126Dic53—42no—
127Dic53—42no—
128Dic53—81no—
129#2326—531no—
130#1459—532no—
131#1325—532no—
132#1319—532no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53#145911yescharacteristic
136C2×C53#212711yescharacteristic
137C2×C53#213511yescharacteristic
138D53—44no—
139C2×#124—531no—
140C2×C4×Q8—531no—
141C2×C4×D4—531no—
142C2×C22⋊C8—1062no—
143#391—531no—
144#390—531no—
145#388—1062no—
146#385—1062no—
147#384—531no—
148#376#615711yescharacteristic
149#324—534no—
150#323—1062no—
151C53#1148111yescharacteristic, complemented, Sylow
152C22×Q8—531no—
153C22×D4—531no—
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—531no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4—1061no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—1062no—
166C2×C22⋊C4—1062no—
167C2×C42—1061no—
168C2×C42C2×#615711yescharacteristic
169C2×C42C2×#615711yescharacteristic
170C2×C42C2×#615711yescharacteristic
171C22⋊C8—1064no—
172C22⋊C8—2122no—
173#95—21no—
174#95—21no—
175#95—1062no—
176C24—532no—
177C2×Q8—532no—
178C2×Q8—1061no—
179C2×Q8—2121no—
180C2×Q8—2121no—
181C2×D4—532no—
182C2×D4—532no—
183C2×D4—2121no—
184C2×D4—2121no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—532no—
188C22×C4—1061no—
189C22×C4—1062no—
190C22×C4—1062no—
191C22×C4—1061no—
192C22×C4—4242no—
193C22×C4C22⋊#615711yescharacteristic
194C22×C4C22⋊#615711yescharacteristic
195C22×C4C22×#615711yescharacteristic
196C2×C8—4244no—
197C2×C8—2124no—
198C4⋊C4—4242no—
199C4⋊C4—2121no—
200C4⋊C4—2121no—
201C4⋊C4—2122no—
202C22⋊C4—1068no—
203C22⋊C4—4242no—
204C22⋊C4—2122no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—2122no—
209C42D53⋊C812yescomplemented
210C23—534no—
211C23—1062no—
212C23—1062no—
213C23—2122no—
214C23—2122no—
215C23C2×C22⋊#615711yescharacteristic
216Q8—2122no—
217Q8—2122no—
218D4—2122no—
219D4—2122no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—1062no—
232C2×C4—1062no—
233C2×C4—4242no—
234C2×C4—2124no—
235C2×C4—2122no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2122no—
240C2×C4#1313327612yes—
241C2×C4C2×D53⋊C811yescharacteristic
242C2×C4D53⋊M1611yescharacteristic
243C8—4244no—
244C22—21no—
245C22—21no—
246C22—1064no—
247C22—4242no—
248C22—2124no—
249C22—2122no—
250C22—2122no—
251C22D53⋊(C22⋊C8)11yescharacteristic
252C22C2×#1313327611yescharacteristic
253C22#1681160911yescharacteristic
254C4—21no—
255C4—21no—
256C4—22no—
257C4—42no—
258C4—41no—
259C4—81no—
260C4—4241no—
261C4—2122no—
262C4—2122no—
263C2—21no—
264C2—21no—
265C2—2124no—
266C2#2028995211yescharacteristic
267C2#2029099211yescharacteristic
268C2#2029101911yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#376C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊Dic53C23
hypercenter#376#6157
hypocenterC53#11481
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
2. #20289657C2semidirect
3. #20289646C2semidirect
24. #13133205C8semidirect
60. C2×Dic106C2×C8semidirect
66. C42×C53C2×C8semidirect
151. C53#11481semidirect
209. C42D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202899522
#202896572
#202896462
#202774262
#1148153