Finite Group Databasebeta

← #28308404

Subgroups — #28308404

Order 13568.

394 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 74 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#20289977C211yesmaximal, characteristic
2#20289957C211yesmaximal, characteristic
3#20289645C214yesmaximal
4#20279704C211yesmaximal, characteristic
5#16811644—21no—
6#16811640—21no—
7#16811469—21no—
8#16811469—21no—
9#16811468C2212yes—
10#16811468C2212yes—
11C2×#13133215C412yes—
12C2×C4×Dic106C2211yescharacteristic
13#16810554—21no—
14#16810554—21no—
15#16810530C2212yes—
16#16810519—21no—
17#16810519—21no—
18C4×C4⋊Dic53C411yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20#13133284—41no—
21C4⋊#88324—41no—
22C22⋊#88324—22no—
23C22⋊#88324—22no—
24#13133272—41no—
25C4×#88324—41no—
26C22×Dic106C2×C411yescharacteristic
27#13133215—44no—
28C4×Dic106—41no—
29C4×Dic106—41no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53C2311yescharacteristic
32C2×#11719919—22no—
33C2×#11719919—22no—
34C2×#11719919D411yescharacteristic
35C2×#11719919D411yescharacteristic
36C2×C4×Dic53C2×C412yes—
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39#13133124—21no—
40#13133124—22no—
41#13133124—22no—
42C22×Q8×C53C2×C412yes—
43C4⋊Q8×C53—41no—
44C4⋊Q8×C53—41no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C2×Q8×C53—24no—
49C2×Q8×C53—42no—
50C2×Q8×C53—42no—
51C22×C4×C53—22no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—22no—
56C22×Dic53C22⋊C411yescharacteristic
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C2×D411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C2×Dic106—22no—
61C2×Dic106—22no—
62C2×Dic106—41no—
63C2×#88324—42no—
64C2×#88324—42no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C42×C53—21no—
69C42×C53—21no—
70C4⋊Dic53—24no—
71C4⋊Dic53—41no—
72#11719919—41no—
73#11719919—41no—
74#11719919—44no—
75#11719919—44no—
76C4×Dic53—21no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C4×Dic53—42no—
81C23×C53C2×C22⋊C411yescharacteristic
82Q8×C53—84no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53#9911yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C2×C4×C53#13112yes—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—24no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53#10012yes—
107Dic106—44no—
108#88324—82no—
109#88324—82no—
110#11568—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53#40811yescharacteristic
114C22×C53C2×#9911yescharacteristic
115C22×C53C2×#10011yescharacteristic
116C4×C53—21no—
117C4×C53—22no—
118C4×C53—41no—
119C4×C53—82no—
120C4×C53#35612yes—
121Dic53—22no—
122Dic53—44no—
123Dic53—41no—
124Dic53—41no—
125Dic53—41no—
126#2546—531no—
127#1467—531no—
128#1461—531no—
129#1319—534no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53#147111yescharacteristic
133C2×C53#212911yescharacteristic
134C2×C53C2×#35611yescharacteristic
135C2×C4⋊Q8—532no—
136C2×C4⋊Q8#615711yescharacteristic
137C2×C4×Q8—531no—
138#432—1061no—
139#431—1061no—
140#398—1061no—
141#398—1061no—
142#391—532no—
143#389—1061no—
144#389—1061no—
145C4×C4⋊C4—531no—
146#324—532no—
147#324—532no—
148#324—1061no—
149#324—1061no—
150C53#1156811yescharacteristic, complemented, Sylow
151C22×Q8—531no—
152C22×Q8C2×#615712yes—
153C4⋊Q8—41no—
154C4⋊Q8—41no—
155C4⋊Q8—2124no—
156C4×Q8—2121no—
157C4×Q8—2121no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—531no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4—1062no—
165C2×C4⋊C4—1062no—
166C2×C42—531no—
167C2×C42—531no—
168C2×C42—532no—
169C2×C42C2×#615711yescharacteristic
170C4⋊C8—2121no—
171C4⋊C8—2121no—
172C22⋊C8—1062no—
173C22⋊C8—1062no—
174#97—2121no—
175C4×C8—2121no—
176#95—1062no—
177#95—1062no—
178#95—1061no—
179C2×Q8—24no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—1062no—
183C2×Q8—1062no—
184C2×Q8—2121no—
185C22×C4—22no—
186C22×C4—531no—
187C22×C4—531no—
188C22×C4—531no—
189C22×C4—531no—
190C22×C4—1062no—
191C22×C4C22⋊#615711yescharacteristic
192C22×C4C22⋊#615711yescharacteristic
193C22×C4C22×#615711yescharacteristic
194C2×C8—2122no—
195C2×C8—2122no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—1064no—
200C4⋊C4—2121no—
201C4⋊C4—2121no—
202C4⋊C4—2121no—
203C4⋊C4—2124no—
204C4⋊C4—2124no—
205C42—21no—
206C42—21no—
207C42—1061no—
208C42—1061no—
209C42—1061no—
210C42—1061no—
211C42—2122no—
212C23C2×C22⋊#615711yescharacteristic
213Q8—84no—
214Q8—2124no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—532no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1064no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—2122no—
233C2×C4—2122no—
234C2×C4—2121no—
235C2×C4—2121no—
236C2×C4#1313327511yescharacteristic
237C2×C4#1313327511yescharacteristic
238C2×C4#1313330412yes—
239C8—4242no—
240C8—4242no—
241C22—21no—
242C22—21no—
243C22#1681159711yescharacteristic
244C22C2×#1313327511yescharacteristic
245C22D53⋊#10011yescharacteristic
246C4—21no—
247C4—22no—
248C4—41no—
249C4—82no—
250C4—1062no—
251C4—2124no—
252C4—2121no—
253C4—2121no—
254C4—2121no—
255C4#1681153312yes—
256C2—21no—
257C2—21no—
258C2#2028999011yescharacteristic
259C2#2029099711yescharacteristic
260C2C2×#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×C4⋊Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11568
LCSG ▷ C22×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
150. C53#11568semidirect

Maximal subgroups

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

subgroupindex
#202899772
#202899572
#202896452
#202797042
#1156853