Finite Group Databasebeta

← #28309495

Subgroups — #28309495

Order 13568.

408 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 80 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
2#20289957C211yesmaximal, characteristic
3#20289699C214yesmaximal
4#20279704C211yesmaximal, characteristic
5#16811640—21no—
6#16811626—21no—
7C2×#13133278—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811468—21no—
11C2×#13133215—21no—
12C2×C4×Dic106C2211yescharacteristic
13#16810554C412yes—
14#16810554C2212yes—
15#16810530—21no—
16#16810519—21no—
17#16810519—21no—
18C4×C4⋊Dic53C411yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20C2×#11719940—22no—
21#13133284—41no—
22#13133284—41no—
23C4⋊#88324—41no—
24#13133278—42no—
25C22⋊#88324—22no—
26C4×#88324—41no—
27C22×Dic106D411yescharacteristic
28#13133215—42no—
29#13133215—42no—
30C4×Dic106—41no—
31C4×Dic106—41no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53D411yescharacteristic
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919C2×C411yescharacteristic
37C2×#11719919C2311yescharacteristic
38C2×C4×Dic53—21no—
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—22no—
42#13133124—22no—
43#13133124C2×C412yes—
44C22×Q8×C53—21no—
45C4⋊Q8×C53—41no—
46C4⋊Q8×C53—41no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53C2×C412yes—
49C2×C42×C53C2×C411yescharacteristic
50C2×Q8×C53—42no—
51C2×Q8×C53—42no—
52C2×Q8×C53—42no—
53C22×C4×C53—22no—
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C22×C411yescharacteristic
57C22×Dic53—22no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C2×Dic106—41no—
63C2×Dic106—41no—
64C2×Dic106D812yes—
65C2×Dic106SD1612yes—
66#11719940—42no—
67#11719940—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C4⋊C4×C53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C42×C53—21no—
74C42×C53—21no—
75C4⋊Dic53—24no—
76C4⋊Dic53—41no—
77#11719919—41no—
78#11719919—41no—
79#11719919—44no—
80#11719919—42no—
81#11719919—42no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C4×Dic53—21no—
86C4×Dic53—42no—
87C23×C53C2×C22⋊C411yescharacteristic
88Q8×C53—84no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53#9911yescharacteristic
98C2×C4×C53#9911yescharacteristic
99C2×C4×C53#10212yes—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—24no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53#10212yes—
112C2×Dic53C2×D811yescharacteristic
113C2×Dic53C2×SD1611yescharacteristic
114Dic106—42no—
115Dic106—42no—
116#88324—82no—
117#88324—82no—
118#11883—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53C2×#9911yescharacteristic
122C22×C53C2×#10211yescharacteristic
123C22×C53#42111yescharacteristic
124C4×C53—21no—
125C4×C53—22no—
126C4×C53—41no—
127C4×C53—82no—
128C4×C53#35612yes—
129Dic53—21no—
130Dic53—22no—
131Dic53—21no—
132Dic53—44no—
133Dic53—41no—
134Dic53—41no—
135#2546—531no—
136#1801—531no—
137#1461—531no—
138#1346—534no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#151911yescharacteristic
142C2×C53#212311yescharacteristic
143C2×C53C2×#35611yescharacteristic
144C2×C4⋊Q8—1061no—
145C2×C4⋊Q8#615711yescharacteristic
146C2×C4×Q8—531no—
147#431—1061no—
148#423—1061no—
149C2×#101—1061no—
150#398—532no—
151#398—532no—
152#391—1061no—
153#389—1061no—
154#389—1061no—
155C4×C4⋊C4—531no—
156#343—532no—
157#324—532no—
158#324—1061no—
159C53#1188311yescharacteristic, complemented, Sylow
160C22×Q8—21no—
161C22×Q8—531no—
162C2×M16—1062no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—2122no—
166C4⋊Q8—2122no—
167C4×Q8—2121no—
168C4×Q8—2121no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—531no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1062no—
175C2×C4⋊C4—1062no—
176C2×C4⋊C4C2×#615712yes—
177C2×C42—531no—
178C2×C42—531no—
179C2×C42—1061no—
180C2×C42C2×#615711yescharacteristic
181C4⋊C8—2121no—
182C4⋊C8—2121no—
183C4⋊C8—2121no—
184#101—2122no—
185C22⋊C8—1062no—
186C4×C8—2121no—
187#95—532no—
188#95—1062no—
189#95—1062no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—532no—
194C2×Q8—532no—
195C2×Q8—2121no—
196C2×Q8—2121no—
197C22×C4—22no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—531no—
202C22×C4—1062no—
203C22×C4C22⋊#615711yescharacteristic
204C22×C4C22⋊#615711yescharacteristic
205C22×C4C22×#615711yescharacteristic
206M16—2122no—
207M16—2122no—
208C2×C8—2122no—
209C2×C8—2122no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—1064no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C4⋊C4—2121no—
217C4⋊C4—2124no—
218C4⋊C4—2122no—
219C4⋊C4—2122no—
220C42—21no—
221C42—21no—
222C42—1061no—
223C42—1061no—
224C42—1061no—
225C42—1061no—
226C42—2122no—
227C23C2×C22⋊#615711yescharacteristic
228Q8—84no—
229Q8—2122no—
230Q8—2122no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—532no—
240C2×C4—531no—
241C2×C4—531no—
242C2×C4—1061no—
243C2×C4—1061no—
244C2×C4—1061no—
245C2×C4—1061no—
246C2×C4—1064no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—2122no—
250C2×C4—2122no—
251C2×C4—2121no—
252C2×C4—2121no—
253C2×C4#1313327511yescharacteristic
254C2×C4#1313327511yescharacteristic
255C2×C4#1313327912yes—
256C8—4242no—
257C8—4242no—
258C22—21no—
259C22—21no—
260C22C2×#1313327511yescharacteristic
261C22C2×#1313327911yescharacteristic
262C22#1681162111yescharacteristic
263C4—22no—
264C4—21no—
265C4—41no—
266C4—82no—
267C4—1061no—
268C4—1062no—
269C4—1061no—
270C4—2124no—
271C4—2121no—
272C4—2121no—
273C4#1681153312yes—
274C2—21no—
275C2—21no—
276C2#2029015111yescharacteristic
277C2#2029097711yescharacteristic
278C2C2×#1681153311yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×#11719919C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11883
LCSG ▷ C22×C4×C53 ▷ C23×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
159. C53#11883semidirect

Maximal subgroups

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

subgroupindex
#202905812
#202899572
#202896992
#202797042
#1188353