Finite Group Databasebeta

← #20284111

Subgroups — #20284111

Order 6784.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13133262C211yesmaximal, characteristic, complemented
2#16811182C212yesmaximal, complemented
3#16810812C214yesmaximal, complemented
4#16810809C214yesmaximal, complemented
5C2×#13133132C211yesmaximal, characteristic, complemented
6C2×#13133131C212yesmaximal, complemented
7C53×#534C211yesmaximal, characteristic, complemented
8#13133262—22no—
9#13133262—24no—
10#13133262C2214yes—
11C2×#11719945—21no—
12C2×#11719947—22no—
13C22×C22⋊D53—21no—
14C22×C22⋊D53C2212yes—
15C2×D4×D53—22no—
16#13133200C2214yescomplemented
17#13133198—22no—
18#13133189—24no—
19#13133186C2214yescomplemented
20#13133185—22no—
21C2×C22⋊Dic53C2212yescomplemented
22#13133132C2214yescomplemented
23#13133131C2218yescomplemented
24C22×D4×C53C2211yescharacteristic
25C22×D4×C53C2211yescharacteristic, complemented
26#123×C53—21no—
27#123×C53C2212yescomplemented
28#121×C53—22no—
29#120×C53—21no—
30#120×C53C2212yescomplemented
31C2×C22⋊C4×C53C2211yescharacteristic, complemented
32C24×C53D411yescharacteristic, complemented
33C24×C53C2311yescharacteristic
34C24×C53C2311yescharacteristic, complemented
35C23×D53—21no—
36C23×D53D412yescomplemented
37C2×D4×C53—22no—
38C2×D4×C53—22no—
39C2×D4×C53—21no—
40C2×D4×C53—21no—
41C2×D4×C53—22no—
42C2×D4×C53—22no—
43C2×D4×C53—22no—
44C2×D4×C53D412yescomplemented
45C2×D4×C53C2312yes—
46C22×C4×C53—21no—
47C22×C4×C53D411yescharacteristic, complemented
48C22×C4×C53C2311yescharacteristic, complemented
49C2×C22⋊D53—22no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—22no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53D414yescomplemented
58C2×C22⋊D53C2314yes—
59C22×Dic53—21no—
60C22×Dic53—22no—
61C22×Dic53D412yescomplemented
62#11719947—28no—
63#11719947—28no—
64D4×D53—28no—
65D4×D53—28no—
66#11719945—24no—
67#11719945—24no—
68C2×D212—21no—
69C2×C4×D53—22no—
70C2×Dic106—21no—
71C4⋊C4×C53—22no—
72C22⋊C4×C53—21no—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—22no—
75C22⋊C4×C53C2312yescomplemented
76C22⋊Dic53—22no—
77C22⋊Dic53—24no—
78C22⋊Dic53C2314yescomplemented
79#11719921—24no—
80#11719919—24no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—22no—
84C23×C53—21no—
85C23×C53—81no—
86C23×C53—81no—
87C23×C53—81no—
88C23×C53C2×D412yes—
89C23×C53C2×D412yes—
90C23×C53C2×D412yes—
91C23×C53C2×D411yescharacteristic
92C23×C53C2×D411yescharacteristic
93C23×C53C2411yescharacteristic
94C22×D53—22no—
95C22×D53—24no—
96C22×D53—22no—
97C22×D53—82no—
98C22×D53—82no—
99C22×D53C2×D414yes—
100D4×C53—24no—
101D4×C53—24no—
102D4×C53—41no—
103D4×C53—42no—
104D4×C53—41no—
105D4×C53—82no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—81no—
112C2×C4×C53C2×D412yes—
113C22⋊D53—28no—
114C22⋊D53—24no—
115C22⋊D53—24no—
116C22⋊D53—24no—
117C22⋊D53—24no—
118C22⋊D53—84no—
119C2×Dic53—24no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—24no—
123C2×Dic53—82no—
124C2×Dic53C2×D414yes—
125D212—24no—
126C4×D53—28no—
127Dic106—24no—
128C22×C53—22no—
129C22×C53—42no—
130C22×C53—41no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—44no—
137C22×C53—42no—
138C22×C53—41no—
139C22×C53—81no—
140C22×C53#12011yescharacteristic
141C22×C53#12012yes—
142C22×C53#12011yescharacteristic
143C22×C53C22×D412yes—
144C22×C53C22×D411yescharacteristic
145C2×D53—24no—
146C2×D53—44no—
147C2×D53—44no—
148C2×D53—82no—
149C2×D53—84no—
150C4×C53—22no—
151C4×C53—42no—
152C4×C53—41no—
153Dic53—24no—
154Dic53—44no—
155#3026—531nomaximal, Sylow
156C2×C53—22no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—22no—
160C2×C53—21no—
161C2×C53—42no—
162C2×C53—81no—
163C2×C53C2×#12011yescharacteristic
164D53—44no—
165C2×H(5,𝔽2)—531no—
166#534—532no—
167#534D5311yescharacteristic, complemented
168#458—534no—
169B(4,𝔽2)—534no—
170C2×#99—531no—
171C2×#99—532no—
172C53#302611yescharacteristic, complemented, Sylow
173H(5,𝔽2)—534no—
174H(5,𝔽2)—1064no—
175H(5,𝔽2)—1062no—
176C2×#41—1062no—
177C2×#41—1061no—
178C22×D4—532no—
179C22×D4—1062no—
180C22×D4—1061no—
181C22×D4C2×D5311yescharacteristic
182C22×D4C2×D5311yescharacteristic, complemented
183#123—21no—
184#123—534no—
185#123—1062no—
186#123C2×D5312yescomplemented
187#121—22no—
188#121—1064no—
189#120—21no—
190#120—534no—
191#120—1062no—
192#120C2×D5312yescomplemented
193C2×C22⋊C4—532no—
194C2×C22⋊C4C2×D5311yescharacteristic, complemented
195#99—534no—
196#99—538no—
197C24—532no—
198C24—1061no—
199C24C22⋊D5311yescharacteristic, complemented
200C24C22×D5311yescharacteristic
201C24C22×D5311yescharacteristic, complemented
202#41—1064no—
203#41—1068no—
204#41—1068no—
205#41—1064no—
206C2×Q8—1061no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—534no—
215C2×D4—534no—
216C2×D4—1061no—
217C2×D4—1068no—
218C2×D4—1068no—
219C2×D4—1062no—
220C2×D4—1064no—
221C2×D4—1062no—
222C2×D4—1062no—
223C2×D4—1062no—
224C2×D4—1062no—
225C2×D4—1064no—
226C2×D4—1064no—
227C2×D4C22⋊D5312yescomplemented
228C2×D4C22×D5312yes—
229C22×C4—21no—
230C22×C4—532no—
231C22×C4—1062no—
232C22×C4—1061no—
233C22×C4—1062no—
234C22×C4C22⋊D5311yescharacteristic, complemented
235C22×C4C22×D5311yescharacteristic, complemented
236C4⋊C4—22no—
237C4⋊C4—1064no—
238C22⋊C4—21no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—534no—
242C22⋊C4—1064no—
243C22⋊C4—1064no—
244C22⋊C4—1062no—
245C22⋊C4C22×D5312yescomplemented
246C23—21no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—534no—
254C23—1064no—
255C23—1062no—
256C23—1062no—
257C23—4242no—
258C23—4242no—
259C23D4×D5312yes—
260C23D4×D5312yes—
261C23C2×C22⋊D5312yes—
262C23C2×C22⋊D5311yescharacteristic
263C23C2×C22⋊D5311yescharacteristic
264C23C23×D5311yescharacteristic
265Q8—1064no—
266D4—24no—
267D4—24no—
268D4—41no—
269D4—41no—
270D4—42no—
271D4—82no—
272D4—1064no—
273D4—1068no—
274D4—1064no—
275D4—1064no—
276D4—1064no—
277D4—1064no—
278D4—4244no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—534no—
286C2×C4—1068no—
287C2×C4—1064no—
288C2×C4—1062no—
289C2×C4—1062no—
290C2×C4—1064no—
291C2×C4—4242no—
292C2×C4C2×C22⋊D5312yes—
293C22—22no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—44no—
303C22—41no—
304C22—81no—
305C22—1064no—
306C22—4242no—
307C22—4244no—
308C22—2124no—
309C22—2124no—
310C22#1313318511yescharacteristic
311C22#1313318511yescharacteristic
312C22#1313318512yes—
313C22C2×D4×D5312yes—
314C22C22×C22⋊D5311yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—1064no—
319C4—2124no—
320C2—21no—
321C2—22no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—81no—
327C2—2124no—
328C2C2×#1313318511yescharacteristic
subgroupG/H
centerC2C2×#13133185
commutatorC23×C53C24
FrattiniC23C23×D53
FittingC53×#534C2
socleC2×C53C2×#120
hypercenter#534D53
hypocenterC53#3026
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#13133262C2semidirect
2. #16811182C2semidirect
3. #16810812C2semidirect
4. #16810809C2semidirect
5. C2×#13133132C2semidirect
6. C2×#13133131C2semidirect
7. C53×#534C2semidirect
16. #13133200C22semidirect
19. #13133186C22semidirect
21. C2×C22⋊Dic53C22semidirect
22. #13133132C22semidirect
23. #13133131C22semidirect
25. C22×D4×C53C22semidirect
27. #123×C53C22semidirect
30. #120×C53C22semidirect
31. C2×C22⋊C4×C53C22semidirect
32. C24×C53D4semidirect
34. C24×C53C23semidirect
36. C23×D53D4semidirect
44. C2×D4×C53D4semidirect
47. C22×C4×C53D4semidirect
48. C22×C4×C53C23semidirect
57. C2×C22⋊D53D4semidirect
61. C22×Dic53D4semidirect
75. C22⋊C4×C53C23semidirect
78. C22⋊Dic53C23semidirect
167. #534D53semidirect
172. C53#3026semidirect
182. C22×D4C2×D53semidirect
186. #123C2×D53semidirect
192. #120C2×D53semidirect
194. C2×C22⋊C4C2×D53semidirect
199. C24C22⋊D53semidirect
201. C24C22×D53semidirect
227. C2×D4C22⋊D53semidirect
234. C22×C4C22⋊D53semidirect
235. C22×C4C22×D53semidirect
245. C22⋊C4C22×D53semidirect