Finite Group Databasebeta

← #20278320

Subgroups — #20278320

Order 6784.

914 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 157 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, complemented
2C2×#13133202C211yesmaximal, characteristic, complemented
3C2×#13133190C212yesmaximal, complemented
4#16810543C211yesmaximal, characteristic, complemented
5#16810536C211yesmaximal, characteristic, complemented
6#16810531C211yesmaximal, characteristic, complemented
7#16810500C212yesmaximal, complemented
8#16810499C212yesmaximal, complemented
9#16810495C212yesmaximal, complemented
10#95×D53C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C2×#11719947—28no—
13C2×#11719947—24no—
14C22×Dic106C2211yescharacteristic
15C22×D212C2211yescharacteristic
16C22×C22⋊D53C2212yes—
17C22×C22⋊D53C2212yes—
18C23×Dic53C2212yes—
19C22×C4×D53C2212yes—
20C22×C4×D53C2211yescharacteristic
21#13133202—24no—
22#13133190—28no—
23C2×#11719919C2212yes—
24C2×#11719919C2211yescharacteristic
25C2×C22⋊Dic53C2212yes—
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2212yes—
29C2×#11719921C2211yescharacteristic
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
41C23×D53C2311yescharacteristic
42C2×D4×C53—24no—
43C2×D4×C53—22no—
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—24no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53D418yescomplemented
54C22×Dic53—24no—
55C22×Dic53—24no—
56C22×Dic53—24no—
57C22×Dic53C2311yescharacteristic
58C22×Dic53C2311yescharacteristic
59C22×Dic53C2312yes—
60C22×Dic53C2312yes—
61#11719947—232no—
62C2×D212—24no—
63C2×D212—22no—
64C2×C4×D53—24no—
65C2×C4×D53—24no—
66C2×C4×D53—24no—
67C2×C4×D53—42no—
68C2×C4×D53D414yescomplemented
69C2×Dic106—22no—
70C2×Dic106—24no—
71C4⋊C4×C53—22no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—42no—
74C22⋊Dic53—24no—
75#11719921—22no—
76#11719921—22no—
77#11719921—24no—
78#11719921—22no—
79#11719919—24no—
80#11719919—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53C2×D414yes—
88C23×C53C2411yescharacteristic
89C22×D53—22no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—22no—
93C22×D53—21no—
94C22×D53—22no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53C2×D414yes—
98C22×D53C2×D412yes—
99D4×C53—28no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53C2×D412yes—
110C22⋊D53—216no—
111C22⋊D53—48no—
112C2×Dic53—216no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—21no—
119C2×Dic53—22no—
120C2×Dic53C2×D414yes—
121C2×Dic53C2×D412yes—
122C2×Dic53#4114yescomplemented
123D212—44no—
124C4×D53—28no—
125C4×D53—48no—
126Dic106—28no—
127C22×C53—24no—
128C22×C53—24no—
129C22×C53—24no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53C22×D411yescharacteristic
135C22×C53C22×D412yes—
136C22×C53C2×#4112yes—
137C22×C53H(5,𝔽2)11yescharacteristic
138C22×C532−1+411yescharacteristic
139C2×D53—22no—
140C2×D53—22no—
141C2×D53—24no—
142C2×D53—24no—
143C2×D53—41no—
144C2×D53—41no—
145C2×D53—41no—
146C2×D53—42no—
147C2×D53—42no—
148C2×D53#12014yes—
149C4×C53—22no—
150C4×C53—41no—
151C4×C53—41no—
152C4×C53—42no—
153Dic53—24no—
154Dic53—22no—
155Dic53—24no—
156Dic53—41no—
157#2430—531nomaximal, Sylow
158C2×C53—44no—
159C2×C53C2×#12011yescharacteristic
160C2×C53#53211yescharacteristic
161C2×C53#53711yescharacteristic
162C2×C53#54612yes—
163C2×C53#54712yes—
164D53—24no—
165D53—81no—
166C22×#41—531no—
167C2×#123—531no—
168C2×#121—532no—
169#395—531no—
170#394D5311yescharacteristic, complemented
171#393—531no—
172#392—531no—
173#386—532no—
174#386—532no—
175#385—532no—
176C2×#95—531no—
177C53#243011yescharacteristic, complemented, Sylow
178C2×#41—1064no—
179C2×#41—1068no—
180C22×Q8—531no—
181C22×D4—532no—
182C22×D4—532no—
183C22×D4—531no—
184C22×D4C2×D5311yescharacteristic
185C23×C4—531no—
186C23×C4—532no—
187C23×C4—532no—
188#123—1064no—
189#121—1068no—
190C2×C4⋊C4—532no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4C2×D5311yescharacteristic
193C2×C22⋊C4—532no—
194C2×C22⋊C4—532no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4—531no—
197C2×C22⋊C4—531no—
198C2×C22⋊C4C2×D5312yes—
199C2×C22⋊C4C2×D5312yes—
200#95—532no—
201#95—531no—
202#95—532no—
203#95—532no—
204#95C2×D5311yescharacteristic
205C24—531no—
206C24—531no—
207C24C22×D5312yes—
208#41—10632no—
209C2×Q8—1064no—
210C2×Q8—1062no—
211C2×D4—24no—
212C2×D4—22no—
213C2×D4—538no—
214C2×D4—1064no—
215C2×D4—1062no—
216C2×D4—1064no—
217C2×D4—1064no—
218C2×D4—1064no—
219C2×D4—1064no—
220C2×D4—2122no—
221C22×C4—532no—
222C22×C4—531no—
223C22×C4—532no—
224C22×C4—534no—
225C22×C4—531no—
226C22×C4—1064no—
227C22×C4—1064no—
228C22×C4—1064no—
229C22×C4—1064no—
230C22×C4—1064no—
231C22×C4—1064no—
232C22×C4—2122no—
233C22×C4C22×D5312yes—
234C22×C4C22×D5311yescharacteristic
235C22×C4C22×D5311yescharacteristic
236C22×C4C22×D5311yescharacteristic
237C4⋊C4—22no—
238C4⋊C4—1064no—
239C4⋊C4—1062no—
240C22⋊C4—24no—
241C22⋊C4—42no—
242C22⋊C4—1064no—
243C22⋊C4—1062no—
244C22⋊C4—1062no—
245C22⋊C4—1064no—
246C22⋊C4—1062no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—532no—
254C23—534no—
255C23—1062no—
256C23—1061no—
257C23—1061no—
258C23—1061no—
259C23—1062no—
260C23—1062no—
261C23—1061no—
262C23—1061no—
263C23D4×D5314yes—
264C23C23×D5311yescharacteristic
265Q8—1068no—
266D4—28no—
267D4—10616no—
268D4—2124no—
269D4—2128no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—532no—
280C2×C4—534no—
281C2×C4—534no—
282C2×C4—1068no—
283C2×C4—10616no—
284C2×C4—1062no—
285C2×C4—1062no—
286C2×C4—1062no—
287C2×C4—1062no—
288C2×C4—1062no—
289C2×C4—1061no—
290C2×C4—1062no—
291C2×C4—2128no—
292C2×C4D4×D5312yes—
293C22—24no—
294C22—24no—
295C22—24no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—534no—
301C22—1064no—
302C22—1062no—
303C22—1062no—
304C22—1064no—
305C22—2122no—
306C22—2121no—
307C22—2121no—
308C22—2121no—
309C22—2122no—
310C22C2×D4×D5312yes—
311C22C2×D4×D5311yescharacteristic
312C22#41×D5312yes—
313C22#1313326211yescharacteristic
314C22#1313326411yescharacteristic
315C4—22no—
316C4—41no—
317C4—41no—
318C4—42no—
319C4—1062no—
320C4—1064no—
321C4—1064no—
322C4—2121no—
323C2—44no—
324C2—1064no—
325C2—4241no—
326C2#120×D5311yescharacteristic
327C2#1681117211yescharacteristic
328C2#1681120211yescharacteristic
329C2#1681126212yes—
330C2#1681127112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2430
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C22×#11719947C2semidirect
2. C2×#13133202C2semidirect
3. C2×#13133190C2semidirect
4. #16810543C2semidirect
5. #16810536C2semidirect
6. #16810531C2semidirect
7. #16810500C2semidirect
8. #16810499C2semidirect
9. #16810495C2semidirect
10. #95×D53C2semidirect
11. C53×#394C2semidirect
53. C2×C22⋊D53D4semidirect
68. C2×C4×D53D4semidirect
122. C2×Dic53#41semidirect
170. #394D53semidirect
177. C53#2430semidirect