Finite Group Databasebeta

← #28308077

Subgroups — #28308077

Order 13568.

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

normal subgroups only

This group has 480 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#20289952C212yesmaximal
2#20289655C212yesmaximal, complemented
3#20289646C212yesmaximal, complemented
4C4×#13133205C211yesmaximal, characteristic
5D53⋊(C22⋊C8)—22no—
6#16811468C2214yes—
7#16811467—22no—
8C2×#13133205C411yescharacteristic
9C2×C4×Dic106C411yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810529C2211yescharacteristic
12#16810525C411yescharacteristic
13#16810514—22no—
14#16810489C2211yescharacteristic
15C4×#11719921—22no—
16C43×C53C411yescharacteristic
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—41no—
26C4×Dic106—41no—
27C4×D212—41no—
28C4×D212—41no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2311yescharacteristic
31C2×#11719919—21no—
32C2×#11719921—22no—
33C2×#11719921—22no—
34C2×#11719921D412yes—
35C2×C4×Dic53—21no—
36#13133124—22no—
37#13133124—22no—
38C2×C42×C53—21no—
39C2×C42×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C23×D53C22⋊C412yes—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C22⋊C411yescharacteristic
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22×C411yescharacteristic
49C22×Dic53—21no—
50C22×Dic53—21no—
51C22×Dic53C2×D412yes—
52C2×D212—41no—
53C2×D212—41no—
54C2×D212C2×C812yes—
55C2×D212M1612yes—
56C2×C4×D53—22no—
57C2×C4×D53—22no—
58C2×C4×D53—44no—
59C2×Dic106—21no—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×Dic106C2×C812yescomplemented
63C2×#88324—44no—
64D53⋊C8—84no—
65C42×C53—21no—
66C42×C53—21no—
67C42×C53—21no—
68C42×C53—42no—
69C42×C53—42no—
70C42×C53—41no—
71C42×C53C2×C812yescomplemented
72#11719921—28no—
73#11719921—42no—
74#11719921—44no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77#11719919—42no—
78C4×Dic53—41no—
79C4×Dic53—41no—
80C23×C53C2×C22⋊C411yescharacteristic
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—42no—
84C22×D53—42no—
85C22×D53C22⋊C814yes—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53#9911yescharacteristic
98C2×C4×C53#9911yescharacteristic
99C2×C4×C53C22×C811yescharacteristic
100C2×C4×C53C2×M1611yescharacteristic
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53—41no—
108D212—42no—
109D212—42no—
110C4×D53—44no—
111C4×D53—82no—
112Dic106—42no—
113Dic106—42no—
114#88324—84no—
115#11477—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53C2×C22⋊C811yescharacteristic
119C22×C53C2×#9911yescharacteristic
120C22×C53#41311yescharacteristic
121C2×D53—24no—
122C2×D53—42no—
123C2×D53—42no—
124C2×D53—44no—
125C2×D53—82no—
126C4×C53—21no—
127C4×C53—21no—
128C4×C53—21no—
129C4×C53—42no—
130C4×C53—42no—
131C4×C53—41no—
132C4×C53#35412yes—
133Dic53—42no—
134Dic53—42no—
135Dic53—81no—
136C4×#124—531no—
137#1459—532no—
138#1324—532no—
139#1319—532no—
140C2×C53—21no—
141C2×C53—21no—
142C2×C53#145911yescharacteristic
143C2×C53C2×#35411yescharacteristic
144C2×C53#213211yescharacteristic
145D53—44no—
146C2×#124—531no—
147C2×C4×Q8—531no—
148C2×C4×D4—531no—
149C2×C22⋊C8—1062no—
150#391—531no—
151#390—531no—
152#388—1062no—
153#382—531no—
154C4×C22⋊C4—1062no—
155C43#615711yescharacteristic
156#324—534no—
157#323—1062no—
158C53#1147711yescharacteristic, complemented, Sylow
159C22×Q8—531no—
160C22×D4—531no—
161C23×C4—532no—
162C22×C8—2122no—
163#124—534no—
164#124—1062no—
165C4×Q8—2121no—
166C4×Q8—2121no—
167C4×D4—2121no—
168C4×D4—2121no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—1061no—
172C2×C22⋊C4—532no—
173C2×C22⋊C4—1062no—
174C2×C22⋊C4—1062no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—1061no—
178C2×C42C2×#615711yescharacteristic
179C2×C42C2×#615711yescharacteristic
180C2×C42C2×#615711yescharacteristic
181C22⋊C8—1064no—
182C22⋊C8—2122no—
183#95—1062no—
184#95—1062no—
185C24—532no—
186C2×Q8—532no—
187C2×Q8—1061no—
188C2×Q8—2121no—
189C2×Q8—2121no—
190C2×D4—532no—
191C2×D4—532no—
192C2×D4—2121no—
193C2×D4—2121no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—532no—
197C22×C4—1061no—
198C22×C4—1062no—
199C22×C4—1062no—
200C22×C4—1061no—
201C22×C4—2124no—
202C22×C4C22⋊#615711yescharacteristic
203C22×C4C22⋊#615711yescharacteristic
204C22×C4C22×#615711yescharacteristic
205C2×C8—4244no—
206C2×C8—2124no—
207C4⋊C4—2121no—
208C4⋊C4—2121no—
209C4⋊C4—2122no—
210C22⋊C4—1068no—
211C22⋊C4—2122no—
212C22⋊C4—2124no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42—2121no—
220C42—2121no—
221C42D53⋊C812yescomplemented
222C23—534no—
223C23—1062no—
224C23—1062no—
225C23—2122no—
226C23—2122no—
227C23C2×C22⋊#615711yescharacteristic
228Q8—2122no—
229Q8—2122no—
230D4—2122no—
231D4—2122no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—4242no—
246C2×C4—2122no—
247C2×C4—2121no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2124no—
251C2×C4—2122no—
252C2×C4#1313327611yescharacteristic
253C2×C4#1313327611yescharacteristic
254C2×C4C2×D53⋊C811yescharacteristic
255C2×C4D53⋊M1611yescharacteristic
256C8—4244no—
257C22—21no—
258C22—21no—
259C22—1064no—
260C22—4242no—
261C22—2124no—
262C22—2122no—
263C22—2122no—
264C22D53⋊(C22⋊C8)11yescharacteristic
265C22C2×#1313327611yescharacteristic
266C22#1681160911yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—42no—
272C4—41no—
273C4—4241no—
274C4—2122no—
275C4—2122no—
276C4#1681153012yes—
277C2—21no—
278C2—21no—
279C2—2124no—
280C2#2028995211yescharacteristic
281C2C2×#1681153011yescharacteristic
282C2#2029100811yescharacteristic
subgroupG/H
centerC2×C4#13133276
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC43×C53C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC43#6157
hypocenterC53#11477
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C43

Complements

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

NH ≅ G/Ntype
2. #20289655C2semidirect
3. #20289646C2semidirect
24. #13133205C8semidirect
62. C2×Dic106C2×C8semidirect
71. C42×C53C2×C8semidirect
158. C53#11477semidirect
221. C42D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
#202899522
#202896552
#202896462
C4×#131332052
#1147753