Finite Group Databasebeta

← #20283994

Subgroups — #20283994

Order 6784.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#13133263C211yesmaximal, characteristic
2C2×#13133250C211yesmaximal, characteristic, complemented
3C22×#11719913C211yesmaximal, characteristic, complemented
4#16811194C211yesmaximal, characteristic, complemented
5#16810787C214yesmaximal, complemented
6#16810776C214yesmaximal, complemented
7#16810629C211yesmaximal, characteristic, complemented
8#16810620C211yesmaximal, characteristic, complemented
9C53×#408C211yesmaximal, characteristic, complemented
10#13133263—24no—
11#13133263—24no—
12C2×#11719949—21no—
13C2×#11719945C2212yes—
14C2×#11719945C2211yescharacteristic
15C22×Dic106C2211yescharacteristic, complemented
16C22×D212C2212yes—
17C22×D212C2211yescharacteristic
18C2×D4×D53—22no—
19C2×D4×D53—21no—
20#13133250—24no—
21C2×Dic212C2212yescomplemented
22C2×#11719913—24no—
23C2×#11719913C2212yescomplemented
24C2×#11719913C2214yescomplemented
25#13133215—21no—
26#13133204—21no—
27#13133198—22no—
28#13133196C2214yescomplemented
29#13133176C2211yescharacteristic, complemented
30#13133142C2212yescomplemented
31#13133139C2212yescomplemented
32#13133139C2214yescomplemented
33C2×#41×C53C2211yescharacteristic
34C2×M16×C53C2211yescharacteristic, complemented
35C22×C8×C53C2211yescharacteristic, complemented
36C22⋊C8×C53C2214yescomplemented
37C23×D53—22no—
38C23×D53—21no—
39#41×C53—22no—
40#41×C53—22no—
41C2×Q8×C53D411yescharacteristic
42C2×D4×C53D412yes—
43C2×D4×C53D411yescharacteristic
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—22no—
47C2×C22⋊D53—21no—
48C22×Dic53—21no—
49#11719949—28no—
50D4×D53—216no—
51D4×D53—28no—
52#11719945—22no—
53#11719945—24no—
54#11719945—24no—
55#11719945D414yes—
56C2×D212—22no—
57C2×D212—21no—
58C2×D212—22no—
59C2×D212—24no—
60C2×D212—24no—
61C2×D212D414yes—
62C2×D212C2311yescharacteristic
63C2×D212C2312yes—
64C2×C4×D53—22no—
65C2×C4×D53—21no—
66C2×Dic106—22no—
67C2×Dic106—22no—
68C2×Dic106—21no—
69C2×Dic106C2312yes—
70C2×Dic106C2311yescharacteristic
71M16×C53—22no—
72C2×C8×C53—22no—
73C2×C8×C53C2312yes—
74C2×C8×C53C2312yes—
75C22⋊Dic53—21no—
76C22⋊Dic53—24no—
77C4⋊Dic53C2312yes—
78#11719919—24no—
79C4×Dic53—21no—
80Dic212—24no—
81#11719913—28no—
82#11719913—24no—
83C23×C53C2×D412yes—
84C23×C53C2×D411yescharacteristic
85C22×D53—22no—
86C22×D53—21no—
87C22×D53—28no—
88C22×D53—24no—
89C22×D53—82no—
90C22×D53—84no—
91Q8×C53—22no—
92D4×C53—24no—
93D4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53C2×D412yes—
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2×D412yes—
99C2×C4×C53C2×D412yes—
100C2×C4×C53C2411yescharacteristic
101C22⋊D53—28no—
102C22⋊D53—24no—
103C2×Dic53—21no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—42no—
107D212—28no—
108D212—24no—
109D212—21no—
110D212—22no—
111D212C2×D412yes—
112D212C2×D414yes—
113C4×D53—28no—
114C4×D53—24no—
115Dic106—21no—
116Dic106—22no—
117Dic106—22no—
118Dic106—42no—
119Dic106C2×D412yes—
120C8×C53—22no—
121C8×C53—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—24no—
125C22×C53#12012yes—
126C22×C53C22×D411yescharacteristic
127C2×D53—24no—
128C2×D53—22no—
129C2×D53—44no—
130C2×D53—42no—
131C2×D53—84no—
132C2×D53—88no—
133C4×C53—22no—
134C4×C53#12012yes—
135C4×C53C22×D411yescharacteristic
136C4×C53C22×D411yescharacteristic
137Dic53—22no—
138Dic53—42no—
139Dic53—42no—
140#3009—531nomaximal, Sylow
141C2×C53—22no—
142C2×C53—42no—
143C2×C53C2×#12011yescharacteristic
144C2×C53#57712yes—
145D53—44no—
146D53—42no—
147C2×H(5,𝔽2)—531no—
148C2×#137—531no—
149C22×SD16—531no—
150#536—531no—
151#452—534no—
152#450—534no—
153#418—531no—
154#416—531no—
155#408D5311yescharacteristic, complemented
156C53#300911yescharacteristic, complemented, Sylow
157H(5,𝔽2)—1064no—
158H(5,𝔽2)—1064no—
159C2×#41—531no—
160C2×#41—532no—
161C2×#41—1061no—
162C2×#41C2×D5311yescharacteristic
163C22×Q8—531no—
164C22×D4—531no—
165C22×D4—532no—
166C22×D4—1062no—
167C22×D4—1061no—
168#137—1064no—
169C2×Q16—532no—
170C2×SD16—534no—
171C2×SD16—532no—
172C2×SD16—1064no—
173C2×M16C2×D5311yescharacteristic, complemented
174C22×C8C2×D5311yescharacteristic, complemented
175C4⋊Q8—1061no—
176#124—1061no—
177#123—1062no—
178C22⋊Q8—534no—
179#117—531no—
180#103—532no—
181#102—532no—
182#102—534no—
183C22⋊C8C2×D5314yescomplemented
184C24—1062no—
185C24—1061no—
186#41—22no—
187#41—22no—
188#41—534no—
189#41—1064no—
190#41—1064no—
191#41—1068no—
192#41—1062no—
193C2×Q8—531no—
194C2×Q8—532no—
195C2×Q8—1061no—
196C2×Q8—1062no—
197C2×Q8—1062no—
198C2×Q8D21211yescharacteristic
199C2×D4—534no—
200C2×D4—532no—
201C2×D4—531no—
202C2×D4—1064no—
203C2×D4—1064no—
204C2×D4—10616no—
205C2×D4—1068no—
206C2×D4—1062no—
207C2×D4—1061no—
208C2×D4—1062no—
209C2×D4—1061no—
210C2×D4—1062no—
211C2×D4D21212yes—
212C2×D4D21211yescharacteristic
213C22×C4—1062no—
214C22×C4—1061no—
215C22×C4—1061no—
216C22×C4C22×D5312yes—
217C22×C4C22×D5311yescharacteristic
218Q16—1064no—
219SD16—1068no—
220SD16—1064no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D5312yes—
224C2×C8C22×D5312yes—
225C4⋊C4—532no—
226C4⋊C4—1064no—
227C22⋊C4—1061no—
228C22⋊C4—1064no—
229C42—1061no—
230C23—1068no—
231C23—1064no—
232C23—1062no—
233C23—1061no—
234C23—4242no—
235C23—4244no—
236C23C2×D21212yes—
237C23C2×D21211yescharacteristic
238Q8—22no—
239Q8—532no—
240Q8—1061no—
241Q8—1062no—
242Q8—1062no—
243Q8—2122no—
244D4—24no—
245D4—22no—
246D4—534no—
247D4—532no—
248D4—1068no—
249D4—1068no—
250D4—1064no—
251D4—1064no—
252D4—1061no—
253D4—1062no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—1068no—
257C2×C4—1061no—
258C2×C4—1064no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—2122no—
262C2×C4C2×D21212yes—
263C2×C4C2×D21211yescharacteristic
264C2×C4D4×D5312yes—
265C2×C4D4×D5312yes—
266C2×C4C23×D5311yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—1064no—
273C22—1062no—
274C22—4244no—
275C22—4248no—
276C22—2124no—
277C22—2122no—
278C22#1313318712yes—
279C22C22×D21211yescharacteristic
280C4—22no—
281C4—1062no—
282C4—2122no—
283C4—2122no—
284C4#1313318712yes—
285C4C2×D4×D5311yescharacteristic
286C4C2×D4×D5311yescharacteristic
287C2—22no—
288C2—42no—
289C2—2124no—
290C2—2122no—
291C2C2×#1313318711yescharacteristic
292C2#1681143512yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#408C2
socleC22×C53C22×D4
hypercenter#408D53
hypocenterC53#3009
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#13133250C2semidirect
3. C22×#11719913C2semidirect
4. #16811194C2semidirect
5. #16810787C2semidirect
6. #16810776C2semidirect
7. #16810629C2semidirect
8. #16810620C2semidirect
9. C53×#408C2semidirect
15. C22×Dic106C22semidirect
21. C2×Dic212C22semidirect
23. C2×#11719913C22semidirect
24. C2×#11719913C22semidirect
28. #13133196C22semidirect
29. #13133176C22semidirect
30. #13133142C22semidirect
31. #13133139C22semidirect
32. #13133139C22semidirect
34. C2×M16×C53C22semidirect
35. C22×C8×C53C22semidirect
36. C22⋊C8×C53C22semidirect
155. #408D53semidirect
156. C53#3009semidirect
173. C2×M16C2×D53semidirect
174. C22×C8C2×D53semidirect
183. C22⋊C8C2×D53semidirect