Finite Group Databasebeta

← #20283973

Subgroups — #20283973

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×#13133248C212yesmaximal, complemented
3#16810783C214yesmaximal, complemented
4#16810778C212yesmaximal, complemented
5#16810773C212yesmaximal, complemented
6#16810622C212yesmaximal, complemented
7#16810587C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C53C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10C2×#11719945—21no—
11C2×#11719945—28no—
12C2×#11719945C2212yes—
13C2×#11719945C2212yes—
14C22×Dic106C2211yescharacteristic
15C22×D212C2211yescharacteristic
16C22×C22⋊D53—21no—
17C22×C4×D53—21no—
18#13133248—28no—
19C2×#11719925C2214yescomplemented
20C2×#11719924C2214yescomplemented
21C2×#11719917C2212yescomplemented
22#13133140C2214yescomplemented
23#13133137C2214yescomplemented
24C22⋊#88322C2212yes—
25C22⋊#88322C2212yescomplemented
26C22×Q8×C53C2211yescharacteristic, complemented
27C23×C4×C53C2211yescharacteristic
28C22⋊Q8×C53—21no—
29C22⋊Q8×C53—21no—
30C22⋊Q8×C53C2212yes—
31C22⋊Q8×C53C2212yescomplemented
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53C2211yescharacteristic, complemented
34C2×C22⋊C4×C53—21no—
35C24×C53D411yescharacteristic
36C23×D53—21no—
37C2×Q8×C53—21no—
38C2×Q8×C53—24no—
39C2×Q8×C53C2312yes—
40C22×C4×C53—24no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53D411yescharacteristic
44C22×C4×C53D411yescharacteristic
45C22×C4×C53D411yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—28no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C22×Dic53—21no—
53#11719945—28no—
54#11719945—28no—
55#11719945—24no—
56#11719945—24no—
57#11719945—24no—
58#11719945D418yes—
59C2×D212—21no—
60C2×D212—24no—
61C2×D212C2312yes—
62C2×C4×D53—28no—
63C2×C4×D53—22no—
64C2×C4×D53—22no—
65C2×Dic106—21no—
66C2×Dic106—24no—
67C2×Dic106C2312yes—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—21no—
71C4⋊C4×C53C2312yes—
72C22⋊C4×C53—22no—
73C22⋊C4×C53—22no—
74#11719925—28no—
75#11719924—28no—
76#11719917—24no—
77C2×#88322C2314yes—
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—
88Q8×C53—24no—
89Q8×C53—42no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53C2×D414yes—
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2411yescharacteristic
104C22⋊D53—28no—
105C22⋊D53—28no—
106C2×Dic53—24no—
107C2×Dic53—22no—
108D212—24no—
109D212—22no—
110D212C2×D414yes—
111C4×D53—28no—
112C4×D53—28no—
113Dic106—24no—
114Dic106—22no—
115Dic106C2×D414yes—
116#88322—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—42no—
135C4×C53—42no—
136C4×C53#12012yes—
137C4×C53C22×D412yes—
138Dic53—24no—
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⋊Q8D5311yescharacteristic, complemented
150#452—534no—
151#451—532no—
152#450—532no—
153#417—532no—
154#407—531no—
155C53#300711yescharacteristic, complemented, Sylow
156C2×#41—532no—
157C2×#41—532no—
158C2×#41—1068no—
159C2×#41—1061no—
160C2×#41—1061no—
161C22×Q8—531no—
162C22×Q8C2×D5311yescharacteristic, complemented
163C22×D4—531no—
164C22×D4—1061no—
165C23×C4—1061no—
166C23×C4C2×D5311yescharacteristic
167#137—1068no—
168C2×Q16—534no—
169C2×SD16—534no—
170C2×M16—532no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D5312yes—
174C22⋊Q8C2×D5312yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D5311yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—534no—
179#102—534no—
180C22⋊C8—532no—
181C22⋊C8—532no—
182C24—1061no—
183C24C22⋊D5311yescharacteristic
184#41—538no—
185#41—1064no—
186#41—1064no—
187#41—1068no—
188#41—1068no—
189#41—1064no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—532no—
193C2×Q8—1064no—
194C2×Q8—1061no—
195C2×Q8C22×D5312yes—
196C2×D4—532no—
197C2×D4—1064no—
198C2×D4—1068no—
199C2×D4—1062no—
200C2×D4—1062no—
201C2×D4—1061no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—1068no—
206C22×C4—1061no—
207C22×C4—1062no—
208C22×C4—1062no—
209C22×C4C22⋊D5311yescharacteristic
210C22×C4C22⋊D5311yescharacteristic
211C22×C4C22⋊D5311yescharacteristic
212C22×C4C22×D5311yescharacteristic
213C22×C4C22×D5311yescharacteristic
214C22×C4C22×D5311yescharacteristic
215Q16—1068no—
216SD16—1068no—
217M16—1064no—
218C2×C8—534no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D5312yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—1064no—
229C23—1062no—
230C23—2122no—
231C23—2122no—
232C23C2×C22⋊D5311yescharacteristic
233C23C2×C22⋊D5311yescharacteristic
234C23C2×C22⋊D5311yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—534no—
238Q8—1064no—
239Q8—1062no—
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—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—1068no—
255C2×C4—1064no—
256C2×C4—1062no—
257C2×C4—1068no—
258C2×C4D4×D5314yes—
259C2×C4C2×C22⋊D5311yescharacteristic
260C2×C4C2×C22⋊D5311yescharacteristic
261C2×C4C2×C22⋊D5311yescharacteristic
262C2×C4C23×D5311yescharacteristic
263C8—1064no—
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#1313318512yes—
277C22#1313324812yescomplemented
278C22C22×C22⋊D5311yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—1064no—
284C4#1313318512yes—
285C4C2×D4×D5312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—2124no—
290C2C2×#1313318511yescharacteristic
291C2C2×#1313324811yescharacteristic
292C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×C22⋊Q8×C53C2
socleC22×C53C22×D4
hypercenterC2×C22⋊Q8D53
hypocenterC53#3007
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#13133248C2semidirect
3. #16810783C2semidirect
4. #16810778C2semidirect
5. #16810773C2semidirect
6. #16810622C2semidirect
7. #16810587C2semidirect
8. C2×C22⋊Q8×C53C2semidirect
19. C2×#11719925C22semidirect
20. C2×#11719924C22semidirect
21. C2×#11719917C22semidirect
22. #13133140C22semidirect
23. #13133137C22semidirect
25. C22⋊#88322C22semidirect
26. C22×Q8×C53C22semidirect
31. C22⋊Q8×C53C22semidirect
33. C2×C4⋊C4×C53C22semidirect
126. C22×C53#137semidirect
149. C2×C22⋊Q8D53semidirect
155. C53#3007semidirect
162. C22×Q8C2×D53semidirect
174. C22⋊Q8C2×D53semidirect
176. C2×C4⋊C4C2×D53semidirect
277. C22#13133248semidirect