Finite Group Databasebeta

← #20278328

Subgroups — #20278328

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×#11719949C211yesmaximal, characteristic, complemented
2C2×#13133199C211yesmaximal, characteristic, complemented
3C2×#13133190C212yesmaximal, complemented
4#16810542C211yesmaximal, characteristic, complemented
5#16810541C211yesmaximal, characteristic, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810501C212yesmaximal, complemented
8#16810499C212yesmaximal, complemented
9#16810497C212yesmaximal, complemented
10#95×D53C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic, complemented
12C2×#11719949—28no—
13C2×#11719949—24no—
14C22×D212C2211yescharacteristic
15C22×D212C2212yes—
16C22×C22⋊D53C2211yescharacteristic
17C22×C22⋊D53C2212yes—
18C22×C4×D53C2212yes—
19C22×C4×D53C2212yes—
20C22×C4×D53C2211yescharacteristic
21#13133199—24no—
22#13133190—28no—
23C2×C4⋊Dic53C2211yescharacteristic
24C2×#11719919C2212yes—
25C2×#11719919C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2212yes—
31#13133124C2212yes—
32#13133124C2212yes—
33#13133124C2211yescharacteristic
34C22×Q8×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2211yescharacteristic
37#95×C53C2212yes—
38#95×C53C2211yescharacteristic
39C24×C53C2311yescharacteristic
40C23×D53C2312yes—
41C23×D53C2311yescharacteristic
42C2×Q8×C53—24no—
43C2×Q8×C53—22no—
44C22×C4×C53C2312yes—
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2312yes—
47C22×C4×C53C2311yescharacteristic
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—24no—
52C2×C22⋊D53—42no—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2312yes—
57#11719949—232no—
58C2×D212—24no—
59C2×D212—24no—
60C2×D212—24no—
61C2×D212—22no—
62C2×D212D418yescomplemented
63C2×C4×D53—24no—
64C2×C4×D53—24no—
65C2×C4×D53—24no—
66C2×C4×D53—24no—
67C2×C4×D53—24no—
68C2×C4×D53—24no—
69C2×C4×D53—42no—
70C2×C4×D53D414yescomplemented
71C22⋊C4×C53—24no—
72C22⋊C4×C53—22no—
73C22⋊Dic53—22no—
74#11719921—24no—
75#11719921—22no—
76#11719921—24no—
77#11719921—42no—
78C4⋊Dic53—22no—
79#11719919—24no—
80#11719919—22no—
81C23×C53—22no—
82C23×C53—21no—
83C23×C53—22no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53C2411yescharacteristic
87C22×D53—22no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53—22no—
94C22×D53—22no—
95C22×D53—22no—
96C22×D53C2×D414yes—
97C22×D53C2×D414yes—
98C22×D53C2×D412yes—
99Q8×C53—28no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53C2×D414yes—
109C2×C4×C53C2×D412yes—
110C22⋊D53—48no—
111C22⋊D53—44no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—21no—
116C2×Dic53—22no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53C2×D412yes—
121C2×Dic53#4114yescomplemented
122D212—28no—
123D212—216no—
124C4×D53—216no—
125C4×D53—28no—
126C4×D53—48no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53C22×D411yescharacteristic
133C22×C53C22×D412yes—
134C22×C53C2×#4112yes—
135C22×C53H(5,𝔽2)11yescharacteristic
136C22×C532−1+411yescharacteristic
137C2×D53—22no—
138C2×D53—22no—
139C2×D53—24no—
140C2×D53—24no—
141C2×D53—24no—
142C2×D53—24no—
143C2×D53—24no—
144C2×D53—42no—
145C2×D53—42no—
146C2×D53—42no—
147C2×D53—42no—
148C2×D53#12014yes—
149C4×C53—24no—
150C4×C53—22no—
151C4×C53—41no—
152C4×C53—42no—
153Dic53—22no—
154Dic53—24no—
155Dic53—41no—
156Dic53—41no—
157#2430—531nomaximal, Sylow
158C2×C53—81no—
159C2×C53C2×#12011yescharacteristic
160C2×C53#53211yescharacteristic
161C2×C53#53711yescharacteristic
162C2×C53#54612yes—
163C2×C53#54712yes—
164D53—24no—
165D53—44no—
166C22×#41—531no—
167C2×#123—531no—
168C2×#121—532no—
169#395—531no—
170#394—531no—
171#393D5311yescharacteristic, complemented
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×Q8C2×D5311yescharacteristic
181C22×D4—531no—
182C22×D4—532no—
183C22×D4—532no—
184C22×D4—531no—
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⋊C4—531no—
193C2×C22⋊C4—531no—
194C2×C22⋊C4—532no—
195C2×C22⋊C4—531no—
196C2×C22⋊C4—532no—
197C2×C22⋊C4—532no—
198C2×C22⋊C4C2×D5312yes—
199C2×C22⋊C4C2×D5311yescharacteristic
200#95—532no—
201#95—531no—
202#95—532no—
203#95C2×D5311yescharacteristic
204#95C2×D5312yes—
205C24—531no—
206C24—532no—
207C24C22×D5311yescharacteristic
208#41—10632no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×D4—538no—
212C2×D4—1064no—
213C2×D4—1064no—
214C2×D4—1064no—
215C2×D4—1064no—
216C2×D4—1064no—
217C2×D4—1064no—
218C2×D4—1062no—
219C2×D4—1062no—
220C2×D4—2122no—
221C22×C4—532no—
222C22×C4—531no—
223C22×C4—534no—
224C22×C4—531no—
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×D5312yes—
237C4⋊C4—1062no—
238C4⋊C4—1064no—
239C4⋊C4—1062no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—1064no—
243C22⋊C4—1064no—
244C22⋊C4—1062no—
245C22⋊C4—1062no—
246C22⋊C4—2122no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—532no—
253C23—534no—
254C23—534no—
255C23—1062no—
256C23—1061no—
257C23—1061no—
258C23—1062no—
259C23—1062no—
260C23—1062no—
261C23—1062no—
262C23—1062no—
263C23—1062no—
264C23C23×D5311yescharacteristic
265Q8—28no—
266D4—10616no—
267D4—1068no—
268D4—2128no—
269D4—2124no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—532no—
279C2×C4—534no—
280C2×C4—1068no—
281C2×C4—10616no—
282C2×C4—1062no—
283C2×C4—1062no—
284C2×C4—1062no—
285C2×C4—1061no—
286C2×C4—1062no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—2128no—
291C2×C4D4×D5312yes—
292C2×C4D4×D5314yes—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—534no—
299C22—1064no—
300C22—1064no—
301C22—1062no—
302C22—1062no—
303C22—1064no—
304C22—1064no—
305C22—1064no—
306C22—2122no—
307C22—2122no—
308C22—2122no—
309C22—2122no—
310C22C2×D4×D5312yes—
311C22C2×D4×D5311yescharacteristic
312C22#41×D5312yes—
313C22#1313326311yescharacteristic
314C22#1313326511yescharacteristic
315C4—22no—
316C4—24no—
317C4—42no—
318C4—41no—
319C4—1064no—
320C4—1062no—
321C4—2121no—
322C4—2121no—
323C2—81no—
324C2—1064no—
325C2—2124no—
326C2#120×D5311yescharacteristic
327C2#1681117211yescharacteristic
328C2#1681120311yescharacteristic
329C2#1681126612yes—
330C2#1681127512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2430
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#11719949C2semidirect
2. C2×#13133199C2semidirect
3. C2×#13133190C2semidirect
4. #16810542C2semidirect
5. #16810541C2semidirect
6. #16810532C2semidirect
7. #16810501C2semidirect
8. #16810499C2semidirect
9. #16810497C2semidirect
10. #95×D53C2semidirect
11. C53×#393C2semidirect
62. C2×D212D4semidirect
70. C2×C4×D53D4semidirect
121. C2×Dic53#41semidirect
171. #393D53semidirect
177. C53#2430semidirect