Finite Group Databasebeta

← #28310618

Subgroups — #28310618

Order 13568.

564 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#16811529C212yesmaximal
2C4×#13133276C212yesmaximal
3#20289810C212yesmaximal
4#20277422C211yesmaximal, characteristic
5C2×#13133276C2214yes—
6C4×C22⋊#6157—22no—
7#16811529C418yescomplemented
8#16811501—22no—
9C2×#13133206C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810530C411yescharacteristic, complemented
12#16810524C2211yescharacteristic
13#16810514—22no—
14#16810494—22no—
15#16810492C411yescharacteristic, complemented
16#16810469C411yescharacteristic, complemented
17C2×C4×Q8×C53C411yescharacteristic, complemented
18C2×C22⋊#6157—24no—
19C2×C4×#6157—42no—
20#13133276—24no—
21#13133276C2×C418yes—
22(C2×#6157)⋊C4—42no—
23C22×D212C2311yescharacteristic
24C22×C4×D53D412yes—
25#13133206—22no—
26#13133206C2×C414yes—
27C4×D212—42no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—22no—
30C2×#11719921—22no—
31C2×#11719921—22no—
32C2×#11719921D412yes—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35#13133124—21no—
36#13133124—21no—
37#13133124—22no—
38C22×Q8×C53C2×C411yescharacteristic
39C4×Q8×C53—42no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C42×C53—21no—
43C2×C42×C53C2×C411yescharacteristic
44C23×D53C2×D412yes—
45C22×#6157—44no—
46C2×Q8×C53—21no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C4212yescomplemented
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C412yes—
58C2×D212—41no—
59C2×D212—41no—
60C2×D212C22×C412yes—
61C2×D212#4112yes—
62C2×C4×D53—22no—
63C2×C4×D53—22no—
64C2×C4×D53—82no—
65C22⋊#6157—28no—
66C22⋊#6157—44no—
67C4×#6157—84no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—42no—
70C42×C53—22no—
71C42×C53—42no—
72#11719921—28no—
73#11719921—42no—
74#11719921—82no—
75C4⋊Dic53—41no—
76#11719919—82no—
77C4×Dic53—21no—
78C4×Dic53—41no—
79C4×Dic53C4212yescomplemented
80C23×C53C2×C22⋊C411yescharacteristic
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—42no—
84C22×D53—42no—
85C22×D53C4×D414yes—
86C2×#6157—48no—
87C2×#6157—84no—
88C2×#6157—84no—
89Q8×C53—42no—
90Q8×C53—42no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53#9912yes—
102C2×C4×C53C2×C4211yescharacteristic
103C2×C4×C53#11711yescharacteristic
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—42no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112D212—42no—
113D212—42no—
114C4×D53—44no—
115C4×D53—82no—
116#12627—531nomaximal, Sylow
117C22×C53—21no—
118C22×C53—21no—
119C22×C53C4×C22⋊C411yescharacteristic
120C22×C53C2×#9911yescharacteristic
121C22×C53#41011yescharacteristic
122C2×D53—24no—
123C2×D53—42no—
124C2×D53—42no—
125C2×D53—44no—
126C2×D53—82no—
127#6157—88no—
128C4×C53—22no—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—41no—
132C4×C53—42no—
133C4×C53—81no—
134Dic53—42no—
135Dic53—42no—
136Dic53—81no—
137#2326—531no—
138C2×#354—532no—
139C4×#99—532no—
140#1393—532no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53C4×#9911yescharacteristic
144C2×C53#212711yescharacteristic
145C2×C53#212911yescharacteristic
146D53—44no—
147C2×#124—531no—
148C2×C4×Q8#615711yescharacteristic, complemented
149C2×C4×D4—531no—
150C2×#99—534no—
151#391—531no—
152#390—531no—
153#388—1062no—
154#385—1062no—
155#384—531no—
156C4×C22⋊C4—1062no—
157#376—531no—
158#354—538no—
159#342—1062no—
160C53#1262711yescharacteristic, complemented, Sylow
161C22×Q8C2×#615711yescharacteristic
162C22×D4—531no—
163C23×C4—532no—
164#124—534no—
165#124—1062no—
166C4×Q8—42no—
167C4×D4—2122no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—1062no—
171C2×C4⋊C4C2×#615711yescharacteristic
172C2×C22⋊C4—532no—
173C2×C22⋊C4—1062no—
174C2×C22⋊C4—1064no—
175C2×C22⋊C4—1062no—
176C2×C42—21no—
177C2×C42—531no—
178C2×C42—531no—
179C2×C42—2122no—
180C2×C42C2×#615711yescharacteristic
181#99—538no—
182#99—1064no—
183#95—1062no—
184#95—1061no—
185#95—1061no—
186#95—2122no—
187C24—532no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C4×#615712yescomplemented
192C2×D4—532no—
193C2×D4—532no—
194C2×D4—2121no—
195C2×D4—2121no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—532no—
199C22×C4—1061no—
200C22×C4—1062no—
201C22×C4—1062no—
202C22×C4—1061no—
203C22×C4—4242no—
204C22×C4—2124no—
205C22×C4C22⋊#615711yescharacteristic
206C22×C4C22⋊#615711yescharacteristic
207C22×C4C22×#615711yescharacteristic
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—4242no—
211C4⋊C4—2121no—
212C22⋊C4—1068no—
213C22⋊C4—1068no—
214C22⋊C4—4242no—
215C22⋊C4—2122no—
216C22⋊C4—2124no—
217C42—22no—
218C42—42no—
219C42—532no—
220C42—1061no—
221C42—4244no—
222C42—2121no—
223C23—534no—
224C23—1062no—
225C23—1062no—
226C23—2122no—
227C23—2122no—
228C23C2×C22⋊#615711yescharacteristic
229Q8—42no—
230Q8—42no—
231D4—2122no—
232D4—2122no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—4242no—
246C2×C4—4244no—
247C2×C4—4244no—
248C2×C4—2124no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2128no—
252C2×C4—2121no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2122no—
256C2×C4#1313327512yes—
257C2×C4C2×C4×#615711yescharacteristic
258C2×C4#1313329611yescharacteristic
259C22—21no—
260C22—21no—
261C22—1064no—
262C22—4242no—
263C22—2124no—
264C22—2122no—
265C22—2122no—
266C22C4×C22⋊#615711yescharacteristic
267C22C2×#1313327511yescharacteristic
268C22#1681160211yescharacteristic
269C4—21no—
270C4—21no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—81no—
275C4—4241no—
276C4—4248no—
277C4—2122no—
278C4—2122no—
279C2—21no—
280C2—21no—
281C2—2124no—
282C2C4×#1313327511yescharacteristic
283C2#2029099011yescharacteristic
284C2#2029099711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C2×#99
prime radical, cosocleC22×D212C23
hypercenterC2×C4×Q8#6157
hypocenterC53#12627
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
7. #16811529C4semidirect
11. #16810530C4semidirect
15. #16810492C4semidirect
16. #16810469C4semidirect
17. C2×C4×Q8×C53C4semidirect
49. C2×Q8×C53C42semidirect
79. C4×Dic53C42semidirect
148. C2×C4×Q8#6157semidirect
160. C53#12627semidirect
191. C2×Q8C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115292
C4×#131332762
#202898102
#202774222
#1262753