Finite Group Databasebeta

← #28312299

Subgroups — #28312299

Order 13568.

420 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 84 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#20290581C211yesmaximal, characteristic
2C4×#13133278C211yesmaximal, characteristic
3#20289812C214yesmaximal
4#20279923C211yesmaximal, characteristic
5#16811626—21no—
6C2×#13133278C2212yes—
7C2×#13133278C2212yes—
8C4×#11719940—21no—
9#16811503—21no—
10#16811503—21no—
11C2×#13133217C411yescharacteristic
12C2×C4×Dic106C2211yescharacteristic
13#16810544—21no—
14#16810544—21no—
15#16810529C2212yes—
16#16810522—21no—
17#16810522—21no—
18#16810493C412yes—
19C4×C4⋊C4×C53C411yescharacteristic
20C2×#11719940—22no—
21C2×#11719940—22no—
22#13133284—41no—
23C4⋊#88324—41no—
24#13133278—22no—
25#13133278—24no—
26#13133278Q814yescomplemented
27#13133272—41no—
28C4×#88324—41no—
29C22×Dic106—21no—
30C22×Dic106C2311yescharacteristic
31#13133217—41no—
32#13133217—41no—
33C4×Dic106—41no—
34C4×Dic106—41no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—22no—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53C2×C412yes—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919D411yescharacteristic
41C2×#11719919D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44#13133124—22no—
45#13133124—22no—
46C2×C4⋊C4×C53—21no—
47C2×C42×C53C2×C412yes—
48C2×C42×C53C2×C411yescharacteristic
49#95×C53—21no—
50C22×C4×C53—22no—
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22×C411yescharacteristic
54C22×Dic53—22no—
55C22×Dic53C22⋊C411yescharacteristic
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C2×D411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C2×Dic106—41no—
60C2×Dic106—41no—
61C2×Dic106—42no—
62C2×Dic106—82no—
63C2×Dic106C2×Q812yes—
64C2×Dic106#4112yes—
65#11719940—24no—
66#11719940—42no—
67#11719940—42no—
68#11719940—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C4⋊C4×C53—24no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—42no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77C4⋊Dic53—41no—
78C4⋊Dic53—42no—
79C4⋊Dic53—82no—
80C4⋊Dic53—82no—
81#11719919—41no—
82#11719919—41no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C23×C53C2×C22⋊C411yescharacteristic
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—22no—
91C2×C4×C53—24no—
92C2×C4×C53—21no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—41no—
96C2×C4×C53#9911yescharacteristic
97C2×C4×C53#9911yescharacteristic
98C2×C4×C53C4×Q812yes—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—42no—
110C2×Dic53C22⋊Q812yes—
111C2×Dic53#12411yescharacteristic
112C2×Dic53C4⋊Q811yescharacteristic
113Dic106—42no—
114Dic106—42no—
115Dic106—84no—
116#88324—44no—
117#88324—82no—
118#13302—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53#39111yescharacteristic
122C22×C53C2×#9911yescharacteristic
123C22×C53#41011yescharacteristic
124C4×C53—21no—
125C4×C53—22no—
126C4×C53—44no—
127C4×C53—41no—
128C4×C53#35612yes—
129Dic53—22no—
130Dic53—21no—
131Dic53—21no—
132Dic53—41no—
133Dic53—41no—
134Dic53—82no—
135#2567—531no—
136#1801—531no—
137C4×#101—531no—
138#1394—534no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#199611yescharacteristic
142C2×C53C2×#35611yescharacteristic
143C2×C53#213411yescharacteristic
144C2×C4⋊Q8—531no—
145C2×C4×Q8—531no—
146#423—1061no—
147C2×#101—532no—
148C2×#101—532no—
149C4×M16—1061no—
150#395—1061no—
151#395—1061no—
152#391—532no—
153#389—1061no—
154#389—1061no—
155#384—532no—
156C4×C4⋊C4#615711yescharacteristic
157#343—1061no—
158#343—1061no—
159C53#1330211yescharacteristic, complemented, Sylow
160C22×Q8—531no—
161C22×Q8—1061no—
162C2×M16—1062no—
163C2×M16—1062no—
164C4⋊Q8—2121no—
165C4⋊Q8—2121no—
166C4×Q8—2121no—
167C4×Q8—2121no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—532no—
173C2×C4⋊C4—1062no—
174C2×C4⋊C4—1062no—
175C2×C4⋊C4—1061no—
176C2×C42—531no—
177C2×C42—531no—
178C2×C42C2×#615712yes—
179C2×C42C2×#615711yescharacteristic
180C4⋊C8—2121no—
181C4⋊C8—2121no—
182#101—534no—
183#101—1062no—
184#101—1064no—
185#97—2121no—
186C4×C8—2121no—
187#95—21no—
188#95—1062no—
189#95—1062no—
190C2×Q8—532no—
191C2×Q8—532no—
192C2×Q8—4242no—
193C2×Q8—2121no—
194C2×Q8—2121no—
195C2×Q8—2122no—
196C22×C4—22no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—1062no—
202C22×C4C22⋊#615711yescharacteristic
203C22×C4C22⋊#615711yescharacteristic
204C22×C4C22×#615711yescharacteristic
205M16—1064no—
206M16—2122no—
207M16—2122no—
208M16—2122no—
209C2×C8—2122no—
210C2×C8—2122no—
211C4⋊C4—24no—
212C4⋊C4—4242no—
213C4⋊C4—4242no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C4⋊C4—2121no—
217C4⋊C4—2122no—
218C4⋊C4—2121no—
219C4⋊C4—2121no—
220C42—21no—
221C42—21no—
222C42—42no—
223C42—1061no—
224C42—1061no—
225C42—1061no—
226C42—1061no—
227C23C2×C22⋊#615711yescharacteristic
228Q8—4244no—
229Q8—2122no—
230Q8—2122no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—532no—
240C2×C4—531no—
241C2×C4—531no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1061no—
245C2×C4—1061no—
246C2×C4—1061no—
247C2×C4—1061no—
248C2×C4—2122no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4#1313327611yescharacteristic
254C2×C4#1313327611yescharacteristic
255C2×C4Q8×#615712yes—
256C8—4242no—
257C8—2124no—
258C22—21no—
259C22—21no—
260C22(C2×#6157)⋊Q811yescharacteristic
261C22C2×#1313327611yescharacteristic
262C22#1681160311yescharacteristic
263C4—22no—
264C4—21no—
265C4—44no—
266C4—41no—
267C4—1061no—
268C4—1062no—
269C4—1061no—
270C4—4242no—
271C4—2121no—
272C4—2121no—
273C4#1681153412yes—
274C2—21no—
275C2—21no—
276C2#2029090511yescharacteristic
277C2C2×#1681153411yescharacteristic
278C2#2029101611yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Dic106C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#13302
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
26. #13133278Q8semidirect
159. C53#13302semidirect

Maximal subgroups

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

subgroupindex
#202905812
C4×#131332782
#202898122
#202799232
#1330253