Finite Group Databasebeta

← #20278368

Subgroups — #20278368

Order 6784.

852 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 161 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×Q8×D53C211yesmaximal, characteristic, complemented
2C2×#13133203C212yesmaximal, complemented
3C2×#13133190C211yesmaximal, characteristic
4#16810536C211yesmaximal, characteristic, complemented
5#16810535C211yesmaximal, characteristic, complemented
6#16810534C211yesmaximal, characteristic, complemented
7#16810527C212yesmaximal, complemented
8#16810501C212yesmaximal, complemented
9#16810499C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#393C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2212yes—
14C2×Q8×D53—24no—
15C2×Q8×D53—28no—
16C22×D212C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C4×D53C2212yes—
20C22×C4×D53C2211yescharacteristic
21#13133203—28no—
22#13133190—24no—
23C2×#11719919C2211yescharacteristic
24C2×C22⋊Dic53C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×#11719921C2211yescharacteristic
29C2×C4×Dic53C2212yes—
30#13133124C2212yes—
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×D53C2311yescharacteristic
41C23×D53C2311yescharacteristic
42C2×Q8×C53—22no—
43C2×Q8×C53—24no—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2312yes—
46C22×C4×C53C2311yescharacteristic
47C22×C4×C53C2312yes—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—24no—
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2312yes—
54C22×Dic53C2312yes—
55C22×Dic53C2311yescharacteristic
56Q8×D53—232no—
57C2×D212—22no—
58C2×D212—24no—
59C2×C4×D53—24no—
60C2×C4×D53—24no—
61C2×C4×D53—24no—
62C2×C4×D53—24no—
63C2×C4×D53D414yes—
64C2×Dic106—22no—
65C2×Dic106—24no—
66C2×Dic106—24no—
67C2×Dic106—24no—
68C2×Dic106D418yescomplemented
69C22⋊C4×C53—22no—
70C22⋊C4×C53—24no—
71C22⋊Dic53—24no—
72#11719921—24no—
73#11719921—22no—
74#11719921—24no—
75#11719921—24no—
76#11719919—22no—
77C4×Dic53—24no—
78C23×C53—21no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53C2411yescharacteristic
84C22×D53—22no—
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—21no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—21no—
91C22×D53—21no—
92C22×D53C2×D412yes—
93C22×D53#4114yes—
94Q8×C53—28no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53C2×D412yes—
104C2×C4×C53C2×D414yes—
105C22⋊D53—44no—
106C22⋊D53—44no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—21no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53C2×D414yes—
115C2×Dic53C2×D412yes—
116C2×Dic53C2×D414yes—
117D212—44no—
118C4×D53—216no—
119C4×D53—28no—
120Dic106—28no—
121Dic106—216no—
122C22×C53—41no—
123C22×C53—41no—
124C22×C53—41no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53C22×D411yescharacteristic
128C22×C53C22×D412yes—
129C22×C53C2×#4112yes—
130C22×C532−1+411yescharacteristic
131C22×C532−1+411yescharacteristic
132C2×D53—22no—
133C2×D53—22no—
134C2×D53—24no—
135C2×D53—24no—
136C2×D53—41no—
137C2×D53—41no—
138C2×D53—41no—
139C2×D53—42no—
140C2×D53—42no—
141C2×D53#12414yescomplemented
142C4×C53—22no—
143C4×C53—24no—
144C4×C53—41no—
145C4×C53—42no—
146Dic53—24no—
147Dic53—41no—
148Dic53—42no—
149Dic53#12014yes—
150#2436—531nomaximal, Sylow
151C2×C53—81no—
152C2×C53C2×#12011yescharacteristic
153C2×C53C2×#12411yescharacteristic
154C2×C53#53611yescharacteristic
155C2×C53#54812yes—
156C2×C53#54812yes—
157D53—24no—
158D53—81no—
159C23×Q8—531no—
160C2×#124—532no—
161C2×#121—531no—
162#393—531no—
163#393—531no—
164#393—531no—
165#393D5311yescharacteristic, complemented
166#391—532no—
167#386—532no—
168#386—532no—
169C4×C22⋊C4—531no—
170C53#243611yescharacteristic, complemented, Sylow
171C22×Q8—532no—
172C22×Q8—531no—
173C22×Q8—1064no—
174C22×Q8—1068no—
175C22×Q8C2×D5311yescharacteristic
176C22×D4—531no—
177C22×D4—531no—
178C22×D4—531no—
179C23×C4—531no—
180C23×C4—532no—
181#124—1068no—
182#121—1064no—
183C2×C4⋊C4—531no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—531no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4—532no—
188C2×C22⋊C4—532no—
189C2×C22⋊C4C2×D5311yescharacteristic
190C2×C22⋊C4C2×D5312yes—
191C2×C42—532no—
192#95—531no—
193#95—532no—
194#95—532no—
195#95—532no—
196#95C2×D5311yescharacteristic
197#95C2×D5312yes—
198C24—531no—
199C24—531no—
200C24C22×D5311yescharacteristic
201C2×Q8—22no—
202C2×Q8—24no—
203C2×Q8—538no—
204C2×Q8—1064no—
205C2×Q8—1064no—
206C2×Q8—1064no—
207C2×Q8—10632no—
208C2×Q8—1062no—
209C2×D4—1062no—
210C2×D4—1064no—
211C2×D4—1062no—
212C2×D4—1064no—
213C2×D4—1062no—
214C2×D4—1064no—
215C22×C4—531no—
216C22×C4—532no—
217C22×C4—534no—
218C22×C4—531no—
219C22×C4—532no—
220C22×C4—1064no—
221C22×C4—1064no—
222C22×C4—1064no—
223C22×C4—1064no—
224C22×C4C22×D5311yescharacteristic
225C22×C4C22×D5312yes—
226C22×C4C22×D5311yescharacteristic
227C22×C4C22×D5312yes—
228C4⋊C4—1062no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—1064no—
232C22⋊C4—1064no—
233C22⋊C4—1064no—
234C22⋊C4—1062no—
235C22⋊C4—1064no—
236C42—1064no—
237C23—22no—
238C23—21no—
239C23—22no—
240C23—21no—
241C23—21no—
242C23—532no—
243C23—534no—
244C23—1062no—
245C23—1061no—
246C23—1061no—
247C23—1062no—
248C23—1061no—
249C23—1062no—
250C23—1061no—
251C23—1061no—
252C23C23×D5311yescharacteristic
253Q8—28no—
254Q8—10616no—
255Q8—1068no—
256D4—2124no—
257D4—2124no—
258D4—2124no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—534no—
268C2×C4—532no—
269C2×C4—534no—
270C2×C4—1068no—
271C2×C4—10616no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1061no—
276C2×C4—1062no—
277C2×C4—1062no—
278C2×C4—1062no—
279C2×C4D4×D5312yes—
280C2×C4D4×D5314yes—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—534no—
287C22—1064no—
288C22—1064no—
289C22—1062no—
290C22—1062no—
291C22—2122no—
292C22—2122no—
293C22—2121no—
294C22—2121no—
295C22—2121no—
296C22C2×D4×D5312yes—
297C22C2×D4×D5311yescharacteristic
298C22#41×D5312yes—
299C22#1313326411yescharacteristic
300C22#1313326511yescharacteristic
301C4—22no—
302C4—24no—
303C4—42no—
304C4—41no—
305C4—534no—
306C4—1064no—
307C4—2122no—
308C4—2121no—
309C2—81no—
310C2—1064no—
311C2—4241no—
312C2#120×D5311yescharacteristic
313C2#124×D5311yescharacteristic
314C2#1681119611yescharacteristic
315C2#1681128112yes—
316C2#1681128312yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2436
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D53C2semidirect
2. C2×#13133203C2semidirect
4. #16810536C2semidirect
5. #16810535C2semidirect
6. #16810534C2semidirect
7. #16810527C2semidirect
8. #16810501C2semidirect
9. #16810499C2semidirect
10. #16810470C2semidirect
11. C53×#393C2semidirect
68. C2×Dic106D4semidirect
141. C2×D53#124semidirect
165. #393D53semidirect
170. C53#2436semidirect