Finite Group Databasebeta

← #28309577

Subgroups — #28309577

Order 13568.

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

normal subgroups only

This group has 372 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#20289714C214yesmaximal
4#20279931C211yesmaximal, characteristic
5#16811640—21no—
6#16811626—21no—
7C2×#13133278—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811468—21no—
11C2×C4×Dic106C2211yescharacteristic
12#16810561C412yes—
13#16810561C2212yes—
14#16810530—21no—
15#16810530—21no—
16#16810528—21no—
17#16810488—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
28C4×Dic106—41no—
29C4×Dic106—41no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53D411yescharacteristic
32C2×#11719919C2×C411yescharacteristic
33C2×#11719919C2311yescharacteristic
34C2×C4×Dic53—21no—
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40#13133124—22no—
41#13133124C2×C412yes—
42C22×Q8×C53—21no—
43C4⋊Q8×C53—41no—
44C4⋊Q8×C53—41no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48C2×Q8×C53—44no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
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—41no—
61C2×Dic106—41no—
62C2×Dic106SD1612yes—
63C2×Dic106Q1612yes—
64#11719940—42no—
65#11719940—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C42×C53—21no—
71C42×C53—21no—
72C4⋊Dic53—41no—
73C4⋊Dic53—42no—
74#11719919—41no—
75#11719919—41no—
76C4×Dic53—21no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C4×Dic53—42no—
81C23×C53C2×C22⋊C411yescharacteristic
82Q8×C53—82no—
83Q8×C53—82no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—41no—
89C2×C4×C53—44no—
90C2×C4×C53—42no—
91C2×C4×C53#9911yescharacteristic
92C2×C4×C53#9911yescharacteristic
93C2×C4×C53#10312yes—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53#10312yes—
106C2×Dic53C2×SD1611yescharacteristic
107C2×Dic53C2×Q1611yescharacteristic
108Dic106—42no—
109Dic106—42no—
110#88324—82no—
111#88324—82no—
112#11907—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53C2×#9911yescharacteristic
116C22×C53C2×#10311yescharacteristic
117C22×C53#42111yescharacteristic
118C4×C53—21no—
119C4×C53—22no—
120C4×C53—41no—
121C4×C53—82no—
122C4×C53#35512yes—
123Dic53—21no—
124Dic53—22no—
125Dic53—21no—
126Dic53—41no—
127Dic53—41no—
128Dic53—82no—
129#2568—531no—
130#1801—531no—
131#1461—531no—
132#1353—534no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53#152011yescharacteristic
136C2×C53#212311yescharacteristic
137C2×C53C2×#35511yescharacteristic
138C2×C4⋊Q8#615711yescharacteristic
139C2×C4×Q8—531no—
140#431—1061no—
141#423—1061no—
142C2×#101—1061no—
143#400—532no—
144#400—532no—
145#391—1061no—
146#391—1061no—
147#391—1061no—
148#382—1061no—
149C4×C4⋊C4—531no—
150#343—532no—
151#324—532no—
152#324—1061no—
153C53#1190711yescharacteristic, complemented, Sylow
154C22×Q8—21no—
155C22×Q8—531no—
156C2×M16—1062no—
157C4⋊Q8—41no—
158C4⋊Q8—41no—
159C4×Q8—2121no—
160C4×Q8—2121no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—1061no—
166C2×C4⋊C4C2×#615712yes—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—1061no—
170C2×C42C2×#615711yescharacteristic
171C4⋊C8—2121no—
172C4⋊C8—2121no—
173C4⋊C8—2121no—
174#101—2122no—
175C22⋊C8—1062no—
176C4×C8—2121no—
177#95—532no—
178#95—1062no—
179#95—1062no—
180#95—1062no—
181#95—1062no—
182C2×Q8—44no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—532no—
186C2×Q8—532no—
187C2×Q8—2121no—
188C2×Q8—2121no—
189C22×C4—22no—
190C22×C4—531no—
191C22×C4—531no—
192C22×C4—531no—
193C22×C4—531no—
194C22×C4—1062no—
195C22×C4C22⋊#615711yescharacteristic
196C22×C4C22⋊#615711yescharacteristic
197C22×C4C22×#615711yescharacteristic
198M16—2122no—
199M16—2122no—
200C2×C8—2122no—
201C2×C8—2122no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—2121no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C4⋊C4—2122no—
208C42—21no—
209C42—21no—
210C42—1061no—
211C42—1061no—
212C42—1061no—
213C42—1061no—
214C42—2122no—
215C23C2×C22⋊#615711yescharacteristic
216Q8—82no—
217Q8—82no—
218Q8—2122no—
219Q8—2122no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—532no—
228C2×C4—531no—
229C2×C4—531no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1061no—
236C2×C4—2122no—
237C2×C4—2122no—
238C2×C4—2122no—
239C2×C4—2121no—
240C2×C4—2121no—
241C2×C4#1313327511yescharacteristic
242C2×C4#1313327511yescharacteristic
243C2×C4#1313328012yes—
244C8—4242no—
245C8—4242no—
246C22—21no—
247C22—21no—
248C22C2×#1313327511yescharacteristic
249C22C2×#1313328011yescharacteristic
250C22#1681162211yescharacteristic
251C4—22no—
252C4—21no—
253C4—41no—
254C4—82no—
255C4—1061no—
256C4—1062no—
257C4—1061no—
258C4—4242no—
259C4—2121no—
260C4—2121no—
261C4#1681153112yes—
262C2—21no—
263C2—21no—
264C2#2029015511yescharacteristic
265C2#2029097711yescharacteristic
266C2C2×#1681153111yescharacteristic
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#11907
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
153. C53#11907semidirect

Maximal subgroups

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

subgroupindex
#202905812
#202899572
#202897142
#202799312
#1190753