Finite Group Databasebeta

← #20283977

Subgroups — #20283977

Order 6784.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#11719945C211yesmaximal, characteristic
2C2×#13133250C212yesmaximal, complemented
3C2×#13133196C211yesmaximal, characteristic, complemented
4#16810787C214yesmaximal, complemented
5#16810781C212yesmaximal, complemented
6#16810776C212yesmaximal, complemented
7#16810625C212yesmaximal, complemented
8C53×#407C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10C2×#11719945—21no—
11C2×#11719945—28no—
12C2×#11719945C2212yes—
13C2×#11719945C2212yes—
14C22×Dic106C2211yescharacteristic
15C22×Dic106C2211yescharacteristic, complemented
16C22×D212C2211yescharacteristic
17C22×C22⋊D53—21no—
18C22×C4×D53—21no—
19#13133250—28no—
20C2×Dic212C2214yescomplemented
21C2×#11719913C2214yescomplemented
22#13133196—21no—
23#13133196—21no—
24#13133196C2212yes—
25#13133196C2212yescomplemented
26C2×C4⋊Dic53C2211yescharacteristic, complemented
27C2×#11719919—21no—
28C2×C22⋊Dic53—21no—
29#13133142C2214yescomplemented
30#13133139C2214yescomplemented
31C23×C4×C53C2211yescharacteristic
32C2×M16×C53C2212yescomplemented
33C22⋊C8×C53C2212yes—
34C22⋊C8×C53C2212yescomplemented
35C24×C53D411yescharacteristic
36C23×D53—21no—
37C22×C4×C53—24no—
38C22×C4×C53D411yescharacteristic
39C22×C4×C53D411yescharacteristic
40C22×C4×C53D411yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C2×C22⋊D53—28no—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—22no—
47C22×Dic53—21no—
48C22×Dic53—21no—
49C22×Dic53—21no—
50#11719945—28no—
51#11719945—28no—
52#11719945—24no—
53#11719945—24no—
54#11719945—24no—
55#11719945D418yes—
56C2×D212—21no—
57C2×D212—24no—
58C2×D212C2312yes—
59C2×C4×D53—28no—
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×Dic106—21no—
63C2×Dic106—24no—
64C2×Dic106—24no—
65C2×Dic106—21no—
66C2×Dic106C2312yes—
67C2×Dic106C2312yes—
68M16×C53—24no—
69C2×C8×C53C2314yes—
70C22⋊Dic53—22no—
71C22⋊Dic53—22no—
72C4⋊Dic53—21no—
73C4⋊Dic53C2312yes—
74#11719919—22no—
75#11719919—22no—
76Dic212—28no—
77#11719913—28no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—41no—
81C23×C53C2×D411yescharacteristic
82C23×C53C2×D411yescharacteristic
83C23×C53C2×D411yescharacteristic
84C22×D53—24no—
85C22×D53—22no—
86C22×D53—42no—
87C22×D53—42no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2×D414yes—
97C2×C4×C53C2411yescharacteristic
98C22⋊D53—28no—
99C22⋊D53—28no—
100C2×Dic53—24no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106D212—24no—
107D212—22no—
108D212C2×D414yes—
109C4×D53—28no—
110C4×D53—28no—
111Dic106—24no—
112Dic106—22no—
113Dic106—24no—
114Dic106—42no—
115Dic106C2×D414yes—
116C8×C53—24no—
117C22×C53—21no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—44no—
125C22×C53#12012yes—
126C22×C53#13712yescomplemented
127C22×C53C22×D411yescharacteristic
128C2×D53—24no—
129C2×D53—48no—
130C2×D53—42no—
131C2×D53—42no—
132C4×C53—21no—
133C4×C53—21no—
134C4×C53#12012yes—
135C4×C53C22×D412yes—
136Dic53—24no—
137Dic53—42no—
138Dic53—42no—
139#3007—531nomaximal, Sylow
140C2×C53—22no—
141C2×C53—22no—
142C2×C53—41no—
143C2×C53C2×#12011yescharacteristic
144C2×C53C2×#13711yescharacteristic
145C2×C53#57511yescharacteristic
146D53—44no—
147C22×#41—531no—
148C2×#137—532no—
149C2×C22⋊Q8—531no—
150#452—534no—
151#451—532no—
152#450—532no—
153#417—532no—
154#407D5311yescharacteristic, complemented
155C53#300711yescharacteristic, complemented, Sylow
156C2×#41—532no—
157C2×#41—532no—
158C2×#41—1068no—
159C2×#41—1061no—
160C2×#41—1061no—
161C22×Q8—531no—
162C22×Q8—531no—
163C22×D4—531no—
164C22×D4—1061no—
165C23×C4—1061no—
166C23×C4C2×D5311yescharacteristic
167#137—1068no—
168C2×Q16—534no—
169C2×SD16—534no—
170C2×M16C2×D5312yescomplemented
171C22⋊Q8—532no—
172C22⋊Q8—532no—
173C22⋊Q8—1061no—
174C22⋊Q8—1061no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—1061no—
177C2×C22⋊C4—1061no—
178#103—534no—
179#102—534no—
180C22⋊C8C2×D5312yes—
181C22⋊C8C2×D5312yescomplemented
182C24—1061no—
183C24D21211yescharacteristic
184#41—538no—
185#41—1064no—
186#41—1064no—
187#41—1068no—
188#41—1068no—
189#41—1064no—
190C2×Q8—532no—
191C2×Q8—532no—
192C2×Q8—1064no—
193C2×Q8—1061no—
194C2×Q8—1061no—
195C2×Q8—1064no—
196C2×D4—532no—
197C2×D4—1064no—
198C2×D4—1068no—
199C2×D4—1062no—
200C2×D4—1062no—
201C2×D4—1061no—
202C22×C4—24no—
203C22×C4—1068no—
204C22×C4—1061no—
205C22×C4—1062no—
206C22×C4—1062no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4D21211yescharacteristic
210C22×C4D21211yescharacteristic
211C22×C4D21211yescharacteristic
212C22×C4C22×D5311yescharacteristic
213C22×C4C22×D5311yescharacteristic
214C22×C4C22×D5311yescharacteristic
215Q16—1068no—
216SD16—1068no—
217M16—24no—
218C2×C8C22×D5314yes—
219C4⋊C4—532no—
220C4⋊C4—1061no—
221C4⋊C4—1062no—
222C4⋊C4—1062no—
223C22⋊C4—1062no—
224C22⋊C4—1062no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—1064no—
229C23—1062no—
230C23—2122no—
231C23—2122no—
232C23C2×D21211yescharacteristic
233C23C2×D21211yescharacteristic
234C23C2×D21211yescharacteristic
235Q8—534no—
236Q8—1064no—
237Q8—1062no—
238Q8—1064no—
239Q8—2122no—
240D4—534no—
241D4—1064no—
242D4—1068no—
243D4—1068no—
244D4—1062no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—1068no—
251C2×C4—1064no—
252C2×C4—1062no—
253C2×C4—1068no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4C2×D21211yescharacteristic
259C2×C4C2×D21211yescharacteristic
260C2×C4C2×D21211yescharacteristic
261C2×C4D4×D5314yes—
262C2×C4C23×D5311yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—1064no—
273C22—2128no—
274C22—2122no—
275C22—2122no—
276C22#1313318712yes—
277C22#1313325012yescomplemented
278C22C22×D21211yescharacteristic
279C4—21no—
280C4—21no—
281C4—1064no—
282C4—2122no—
283C4—2122no—
284C4#1313318712yes—
285C4C2×D4×D5312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—2124no—
290C2C2×#1313318711yescharacteristic
291C2C2×#1313325011yescharacteristic
292C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3007
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#13133250C2semidirect
3. C2×#13133196C2semidirect
4. #16810787C2semidirect
5. #16810781C2semidirect
6. #16810776C2semidirect
7. #16810625C2semidirect
8. C53×#407C2semidirect
15. C22×Dic106C22semidirect
20. C2×Dic212C22semidirect
21. C2×#11719913C22semidirect
25. #13133196C22semidirect
26. C2×C4⋊Dic53C22semidirect
29. #13133142C22semidirect
30. #13133139C22semidirect
32. C2×M16×C53C22semidirect
34. C22⋊C8×C53C22semidirect
126. C22×C53#137semidirect
154. #407D53semidirect
155. C53#3007semidirect
170. C2×M16C2×D53semidirect
181. C22⋊C8C2×D53semidirect
277. C22#13133250semidirect