Finite Group Databasebeta

← #28309182

Subgroups — #28309182

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
1C4×#13133278C211yesmaximal, characteristic
2#20289977C211yesmaximal, characteristic
3#20289716C214yesmaximal
4#20279924C211yesmaximal, characteristic
5#16811644—21no—
6C2×#13133278—21no—
7C4×#11719940—21no—
8#16811503C2212yes—
9#16811469C2212yes—
10#16811468—21no—
11C2×C4×Dic106C2211yescharacteristic
12#16810542C412yes—
13#16810542C2212yes—
14#16810528—21no—
15#16810519—21no—
16#16810519—21no—
17#16810492—21no—
18C4×C4⋊Dic53C411yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20C2×#11719940—22no—
21C4⋊#88324—41no—
22#13133278—24no—
23C22⋊#88324—22no—
24#13133272—41no—
25#13133272—41no—
26C4×#88324—41no—
27C22×Dic106D411yescharacteristic
28C4×Dic106—41no—
29C4×Dic106—41no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53D411yescharacteristic
32C2×#11719919—22no—
33C2×#11719919—22no—
34C2×#11719919C2×C411yescharacteristic
35C2×#11719919C2311yescharacteristic
36C2×C4×Dic53—21no—
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39#13133124—22no—
40#13133124—22no—
41#13133124C2×C412yes—
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—42no—
49C2×Q8×C53—82no—
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—22no—
60C2×Dic106—22no—
61C2×Dic106—41no—
62#11719940—24no—
63#11719940—42no—
64C2×#88324—42no—
65C2×#88324—42no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—41no—
69C42×C53—21no—
70C42×C53—21no—
71C4⋊Dic53—24no—
72C4⋊Dic53—41no—
73#11719919—41no—
74#11719919—41no—
75#11719919—82no—
76#11719919—82no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C4×Dic53—21no—
81C4×Dic53—42no—
82C23×C53C2×C22⋊C411yescharacteristic
83Q8×C53—84no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53#9911yescharacteristic
93C2×C4×C53#9911yescharacteristic
94C2×C4×C53#10412yes—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—24no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53#13712yes—
108Dic106—44no—
109#88324—44no—
110#88324—82no—
111#11795—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53C2×#9911yescharacteristic
115C22×C53#41911yescharacteristic
116C22×C53C2×#10411yescharacteristic
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—41no—
120C4×C53—82no—
121C4×C53#35612yes—
122Dic53—22no—
123Dic53—44no—
124Dic53—41no—
125Dic53—41no—
126Dic53—41no—
127#2567—531no—
128C4×#101—531no—
129#1467—531no—
130#1354—534no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53#150811yescharacteristic
134C2×C53#212211yescharacteristic
135C2×C53C2×#35611yescharacteristic
136C2×C4⋊Q8#615711yescharacteristic
137C2×C4×Q8—531no—
138#432—1061no—
139C2×#101—1061no—
140C4×M16—1061no—
141#395—532no—
142#395—532no—
143#391—1061no—
144#389—1061no—
145#389—1061no—
146#384—1061no—
147C4×C4⋊C4—531no—
148#343—532no—
149#324—532no—
150#324—1061no—
151C53#1179511yescharacteristic, complemented, Sylow
152C22×Q8—531no—
153C22×Q8C2×#615712yes—
154C2×M16—1062no—
155C4⋊Q8—41no—
156C4⋊Q8—41no—
157C4×Q8—2121no—
158C4×Q8—2121no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—531no—
162C2×C4⋊C4—531no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—1061no—
165C2×C4⋊C4—1062no—
166C2×C4⋊C4—1062no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—1061no—
170C2×C42C2×#615711yescharacteristic
171C4⋊C8—2121no—
172#101—1064no—
173C22⋊C8—1062no—
174#97—2121no—
175#97—2121no—
176C4×C8—2121no—
177#95—532no—
178#95—1062no—
179#95—1062no—
180C2×Q8—42no—
181C2×Q8—82no—
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
194M16—1064no—
195M16—2122no—
196C2×C8—2122no—
197C2×C8—2122no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—1064no—
202C4⋊C4—4242no—
203C4⋊C4—4242no—
204C4⋊C4—2121no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C42—21no—
208C42—21no—
209C42—1061no—
210C42—1061no—
211C42—1061no—
212C42—1061no—
213C42—2122no—
214C23C2×C22⋊#615711yescharacteristic
215Q8—84no—
216Q8—2124no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—532no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1064no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2121no—
237C2×C4—2121no—
238C2×C4#1313327511yescharacteristic
239C2×C4#1313327511yescharacteristic
240C2×C4#1313328212yes—
241C8—4242no—
242C8—2124no—
243C22—21no—
244C22—21no—
245C22C2×#1313327511yescharacteristic
246C22#1681161611yescharacteristic
247C22C2×#1313328211yescharacteristic
248C4—22no—
249C4—21no—
250C4—41no—
251C4—82no—
252C4—1062no—
253C4—2124no—
254C4—2121no—
255C4—2121no—
256C4—2121no—
257C4#1681153312yes—
258C2—21no—
259C2—21no—
260C2#2029011511yescharacteristic
261C2#2029097411yescharacteristic
262C2C2×#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#11795
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
151. C53#11795semidirect

Maximal subgroups

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

subgroupindex
C4×#131332782
#202899772
#202897162
#202799242
#1179553