Finite Group Databasebeta

← #20284114

Subgroups — #20284114

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#16811185C212yesmaximal, complemented
3#16811182C211yesmaximal, characteristic, complemented
4#16810814C214yesmaximal, complemented
5#16810811C214yesmaximal, complemented
6C2×#13133131C212yesmaximal, complemented
7C2×#99×C53C211yesmaximal, characteristic, complemented
8#13133262—22no—
9#13133262—24no—
10#13133262C2214yes—
11C2×#11719945—21no—
12C2×#11719947—22no—
13C22×C22⋊D53—21no—
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2211yescharacteristic, complemented
16C2×D4×D53—22no—
17#13133201—22no—
18#13133201C2214yescomplemented
19#13133200—21no—
20#13133198C2212yescomplemented
21#13133191—24no—
22#13133189—22no—
23#13133187—22no—
24#13133187C2214yescomplemented
25#13133186—21no—
26#13133185C2212yescomplemented
27C2×C22⋊Dic53C2211yescharacteristic, complemented
28#13133131C2218yescomplemented
29C22×D4×C53C2211yescharacteristic
30C2×C22⋊C4×C53C2212yescomplemented
31#99×C53C2214yescomplemented
32C24×C53D411yescharacteristic, complemented
33C24×C53C2311yescharacteristic
34C23×D53—21no—
35C23×D53D412yescomplemented
36C23×D53C2311yescharacteristic, complemented
37C2×D4×C53—22no—
38C2×D4×C53—22no—
39C2×D4×C53D412yescomplemented
40C2×D4×C53C2312yes—
41C22×C4×C53—22no—
42C22×C4×C53D411yescharacteristic, complemented
43C2×C22⋊D53—22no—
44C2×C22⋊D53—24no—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—21no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—21no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53D414yescomplemented
56C2×C22⋊D53C2314yes—
57C22×Dic53—21no—
58C22×Dic53—21no—
59C22×Dic53D412yescomplemented
60C22×Dic53C2311yescharacteristic, complemented
61#11719947—28no—
62#11719947—28no—
63D4×D53—28no—
64D4×D53—28no—
65#11719945—24no—
66#11719945—24no—
67C2×D212—21no—
68C2×D212—24no—
69C2×C4×D53—22no—
70C2×Dic106—21no—
71C22⋊C4×C53—22no—
72C22⋊C4×C53C2314yescomplemented
73C22⋊Dic53—21no—
74C22⋊Dic53—22no—
75C22⋊Dic53C2312yescomplemented
76#11719921—24no—
77#11719921—22no—
78#11719921—24no—
79C4⋊Dic53—24no—
80#11719919—22no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—81no—
84C23×C53—81no—
85C23×C53C2×D412yes—
86C23×C53C2×D411yescharacteristic
87C23×C53C2×D411yescharacteristic
88C23×C53C2×D412yes—
89C23×C53C2×D412yes—
90C23×C53C2411yescharacteristic
91C22×D53—22no—
92C22×D53—24no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—21no—
96C22×D53—81no—
97C22×D53—82no—
98C22×D53—82no—
99C22×D53C2×D414yes—
100D4×C53—24no—
101D4×C53—24no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—24no—
105C2×C4×C53—82no—
106C2×C4×C53C2×D412yes—
107C22⋊D53—28no—
108C22⋊D53—24no—
109C22⋊D53—24no—
110C22⋊D53—24no—
111C22⋊D53—24no—
112C22⋊D53—41no—
113C22⋊D53—42no—
114C22⋊D53—41no—
115C22⋊D53—82no—
116C2×Dic53—24no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—21no—
121C2×Dic53—22no—
122C2×Dic53—81no—
123C2×Dic53C2×D414yes—
124D212—24no—
125D212—84no—
126C4×D53—28no—
127Dic106—24no—
128C22×C53—22no—
129C22×C53—41no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—81no—
137C22×C53#12011yescharacteristic
138C22×C53#12012yes—
139C22×C53#12011yescharacteristic
140C22×C53C22×D412yes—
141C22×C53C22×D411yescharacteristic
142C2×D53—24no—
143C2×D53—44no—
144C2×D53—44no—
145C2×D53—44no—
146C2×D53—42no—
147C2×D53—41no—
148C2×D53—82no—
149C2×D53—84no—
150C4×C53—22no—
151C4×C53—44no—
152Dic53—24no—
153Dic53—42no—
154Dic53—41no—
155#3026—531nomaximal, Sylow
156C2×C53—21no—
157C2×C53—22no—
158C2×C53—21no—
159C2×C53—22no—
160C2×C53—21no—
161C2×C53—42no—
162C2×C53C2×#12011yescharacteristic
163D53—44no—
164D53—81no—
165C2×H(5,𝔽2)—531no—
166#534—531no—
167#534—532no—
168#458—534no—
169B(4,𝔽2)—534no—
170C2×#99—532no—
171C2×#99D5311yescharacteristic, complemented
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—531no—
180C22×D4—1062no—
181C22×D4—1061no—
182C22×D4C2×D5311yescharacteristic
183#123—532no—
184#123—534no—
185#123—1062no—
186#123—1061no—
187#121—1064no—
188#121—1062no—
189#120—532no—
190#120—534no—
191#120—1062no—
192#120—1061no—
193C2×C22⋊C4—531no—
194C2×C22⋊C4C2×D5312yescomplemented
195#99—538no—
196#99C2×D5314yescomplemented
197C24—532no—
198C24—531no—
199C24—1061no—
200C24D21211yescharacteristic, complemented
201C24C22×D5311yescharacteristic
202#41—1064no—
203#41—1068no—
204#41—1068no—
205#41—1064no—
206C2×Q8—1061no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—534no—
210C2×D4—534no—
211C2×D4—1061no—
212C2×D4—1064no—
213C2×D4—1068no—
214C2×D4—1068no—
215C2×D4—1062no—
216C2×D4—1064no—
217C2×D4—1062no—
218C2×D4—1062no—
219C2×D4—1064no—
220C2×D4—1062no—
221C2×D4—1062no—
222C2×D4—1062no—
223C2×D4—1061no—
224C2×D4—1061no—
225C2×D4—1062no—
226C2×D4—1062no—
227C2×D4D21212yescomplemented
228C2×D4C22×D5312yes—
229C22×C4—22no—
230C22×C4—532no—
231C22×C4—531no—
232C22×C4—1062no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4D21211yescharacteristic, complemented
236C4⋊C4—1064no—
237C4⋊C4—1062no—
238C22⋊C4—22no—
239C22⋊C4—532no—
240C22⋊C4—1064no—
241C22⋊C4—1064no—
242C22⋊C4—1061no—
243C22⋊C4—1062no—
244C22⋊C4—1062no—
245C22⋊C4C22×D5314yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—534no—
251C23—1064no—
252C23—1062no—
253C23—1062no—
254C23—1062no—
255C23—1061no—
256C23—4242no—
257C23—4242no—
258C23—4241no—
259C23C2×D21212yes—
260C23C2×D21211yescharacteristic
261C23C2×D21211yescharacteristic
262C23D4×D5312yes—
263C23D4×D5312yes—
264C23C23×D5311yescharacteristic
265Q8—1064no—
266D4—24no—
267D4—24no—
268D4—1064no—
269D4—1068no—
270D4—1064no—
271D4—1064no—
272D4—1064no—
273D4—1064no—
274D4—4244no—
275D4—4242no—
276D4—2121no—
277D4—2121no—
278D4—2122no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—534no—
284C2×C4—1068no—
285C2×C4—1064no—
286C2×C4—1062no—
287C2×C4—1062no—
288C2×C4—1062no—
289C2×C4—1062no—
290C2×C4—1061no—
291C2×C4—4241no—
292C2×C4C2×D21212yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—1064no—
303C22—4242no—
304C22—4244no—
305C22—2124no—
306C22—2122no—
307C22—2124no—
308C22—2124no—
309C22—2121no—
310C22#1313318711yescharacteristic
311C22#1313318711yescharacteristic
312C22#1313318712yes—
313C22C2×D4×D5312yes—
314C22C22×D21211yescharacteristic
315C4—22no—
316C4—44no—
317C4—1064no—
318C4—2122no—
319C4—2121no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—4241no—
327C2—2124no—
328C2C2×#1313318711yescharacteristic
subgroupG/H
centerC2C2×#13133187
commutatorC23×C53C24
FrattiniC23C23×D53
FittingC2×#99×C53C2
socleC2×C53C2×#120
hypercenterC2×#99D53
hypocenterC53#3026
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#13133262C2semidirect
2. #16811185C2semidirect
3. #16811182C2semidirect
4. #16810814C2semidirect
5. #16810811C2semidirect
6. C2×#13133131C2semidirect
7. C2×#99×C53C2semidirect
15. C22×C22⋊D53C22semidirect
18. #13133201C22semidirect
20. #13133198C22semidirect
24. #13133187C22semidirect
26. #13133185C22semidirect
27. C2×C22⋊Dic53C22semidirect
28. #13133131C22semidirect
30. C2×C22⋊C4×C53C22semidirect
31. #99×C53C22semidirect
32. C24×C53D4semidirect
35. C23×D53D4semidirect
36. C23×D53C23semidirect
39. C2×D4×C53D4semidirect
42. C22×C4×C53D4semidirect
55. C2×C22⋊D53D4semidirect
59. C22×Dic53D4semidirect
60. C22×Dic53C23semidirect
72. C22⋊C4×C53C23semidirect
75. C22⋊Dic53C23semidirect
171. C2×#99D53semidirect
172. C53#3026semidirect
194. C2×C22⋊C4C2×D53semidirect
196. #99C2×D53semidirect
200. C24D212semidirect
227. C2×D4D212semidirect
235. C22×C4D212semidirect
245. C22⋊C4C22×D53semidirect