Finite Group Databasebeta

← #20278045

Subgroups — #20278045

Order 6784.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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×#11719947C211yesmaximal, characteristic
2C2×#13133199C212yesmaximal, complemented
3C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810544C211yesmaximal, characteristic, complemented
5#16810543C211yesmaximal, characteristic, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810499C212yesmaximal, complemented
8#16810498C212yesmaximal, complemented
9#16810495C212yesmaximal, complemented
10#16810485C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C2×#11719947—28no—
13C2×#11719947—24no—
14C22×Dic106C2211yescharacteristic
15C22×C22⋊D53C2212yes—
16C23×Dic53C2212yes—
17C22×C4×D53C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133199—28no—
21C2×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊Dic53C2212yes—
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2212yes—
25C2×#11719919C2211yescharacteristic
26C4⋊C4×D53—24no—
27C2×C22⋊Dic53C2212yes—
28C2×C22⋊Dic53C2212yes—
29C2×#11719921C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2212yes—
33#13133124C2211yescharacteristic
34C22×D4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2212yes—
37C2×C22⋊C4×C53C2212yes—
38#95×C53C2211yescharacteristic
39C24×C53C2312yes—
40C23×D53C2311yescharacteristic
41C2×D4×C53—24no—
42C2×D4×C53—22no—
43C22×C4×C53C2312yes—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53D418yes—
50C22×Dic53—24no—
51C22×Dic53—24no—
52C22×Dic53—24no—
53C22×Dic53C2312yes—
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2312yes—
57C22×Dic53C2311yescharacteristic
58#11719947—232no—
59C2×C4×D53—22no—
60C2×C4×D53—24no—
61C2×C4×D53—22no—
62C2×C4×D53—22no—
63C2×C4×D53—22no—
64C2×C4×D53—24no—
65C2×C4×D53D414yes—
66C2×Dic106—22no—
67C2×Dic106—24no—
68C4⋊C4×C53—22no—
69C22⋊C4×C53—24no—
70C22⋊C4×C53—24no—
71C22⋊Dic53—24no—
72C22⋊Dic53—24no—
73#11719921—24no—
74C4⋊Dic53—22no—
75C4⋊Dic53—24no—
76#11719919—22no—
77#11719919—22no—
78#11719919—24no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53C2×D414yes—
86C23×C53C2411yescharacteristic
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—22no—
90C22×D53C2×D412yes—
91C22×D53C2×D414yes—
92D4×C53—28no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53C2×D412yes—
102C2×C4×C53#4112yescomplemented
103C22⋊D53—216no—
104C2×Dic53—216no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53C2×D414yes—
116C2×Dic53C2×D412yes—
117C2×Dic53#4112yescomplemented
118C4×D53—28no—
119C4×D53—28no—
120C4×D53—44no—
121Dic106—28no—
122C22×C53—24no—
123C22×C53—24no—
124C22×C53—24no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53C22×D411yescharacteristic
130C22×C53C22×D412yes—
131C22×C53C2×#4111yescharacteristic
132C22×C53C2×#4111yescharacteristic
133C22×C532−1+412yes—
134C2×D53—22no—
135C2×D53—22no—
136C2×D53—24no—
137C2×D53—24no—
138C2×D53#12014yes—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—41no—
142C4×C53—42no—
143Dic53—24no—
144Dic53—22no—
145Dic53—22no—
146Dic53—41no—
147Dic53—42no—
148#2406—531nomaximal, Sylow
149C2×C53—44no—
150C2×C53C2×#12011yescharacteristic
151C2×C53#52511yescharacteristic
152C2×C53#53611yescharacteristic
153C2×C53#54712yes—
154C2×C53#54712yes—
155D53—24no—
156C22×#41—531no—
157C2×#123—532no—
158C22×C4⋊C4—531no—
159#395—531no—
160#395—531no—
161#394—531no—
162#394D5311yescharacteristic, complemented
163#386—532no—
164#385—532no—
165#385—532no—
166#381—531no—
167C53#240611yescharacteristic, complemented, Sylow
168C2×#41—1064no—
169C2×#41—1068no—
170C22×Q8—531no—
171C22×D4—532no—
172C22×D4C2×D5311yescharacteristic
173C23×C4—531no—
174C23×C4—532no—
175C23×C4—531no—
176C23×C4—531no—
177#123—1068no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—532no—
183C2×C4⋊C4—1064no—
184C2×C4⋊C4C2×D5311yescharacteristic
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4—532no—
188C2×C22⋊C4C2×D5312yes—
189C2×C22⋊C4C2×D5312yes—
190#95—532no—
191#95—531no—
192#95—532no—
193#95—532no—
194#95C2×D5311yescharacteristic
195C24—531no—
196C24C22×D5312yes—
197#41—10632no—
198C2×Q8—1064no—
199C2×Q8—1062no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—538no—
203C2×D4—1064no—
204C2×D4—1064no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—532no—
208C22×C4—534no—
209C22×C4—532no—
210C22×C4—531no—
211C22×C4—1062no—
212C22×C4—1062no—
213C22×C4—1064no—
214C22×C4—1062no—
215C22×C4—1064no—
216C22×C4—1064no—
217C22×C4—1064no—
218C22×C4—1064no—
219C22×C4—1062no—
220C22×C4C22×D5312yes—
221C22×C4C22×D5311yescharacteristic
222C22×C4C22×D5311yescharacteristic
223C22×C4C22×D5311yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—1062no—
226C4⋊C4—1062no—
227C4⋊C4—1064no—
228C4⋊C4—1064no—
229C4⋊C4—1062no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—1064no—
233C22⋊C4—1064no—
234C22⋊C4—1064no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—534no—
242C23—532no—
243C23—1061no—
244C23—1061no—
245C23—1062no—
246C23D4×D5314yes—
247C23C23×D5311yescharacteristic
248Q8—1068no—
249D4—28no—
250D4—10616no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—532no—
260C2×C4—534no—
261C2×C4—532no—
262C2×C4—1068no—
263C2×C4—1068no—
264C2×C4—10616no—
265C2×C4—1061no—
266C2×C4—1062no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1061no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—1061no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—2124no—
276C2×C4D4×D5312yes—
277C2×C4#1171994712yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—534no—
286C22—1064no—
287C22—1064no—
288C22—1062no—
289C22—1062no—
290C22C2×D4×D5312yes—
291C22C2×D4×D5311yescharacteristic
292C22#41×D5311yescharacteristic
293C22C2×#1171994711yescharacteristic
294C22#1313326412yes—
295C4—22no—
296C4—22no—
297C4—42no—
298C4—41no—
299C4—1062no—
300C4—1064no—
301C4—1062no—
302C4—2122no—
303C4—2121no—
304C2—44no—
305C2—1064no—
306C2#120×D5311yescharacteristic
307C2#1681113311yescharacteristic
308C2#1681119311yescharacteristic
309C2#1681127112yes—
310C2#1681127312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2406
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133199C2semidirect
3. C2×C4⋊C4×D53C2semidirect
4. #16810544C2semidirect
5. #16810543C2semidirect
6. #16810537C2semidirect
7. #16810499C2semidirect
8. #16810498C2semidirect
9. #16810495C2semidirect
10. #16810485C2semidirect
11. C53×#394C2semidirect
102. C2×C4×C53#41semidirect
117. C2×Dic53#41semidirect
162. #394D53semidirect
167. C53#2406semidirect
277. C2×C4#11719947semidirect