Finite Group Databasebeta

← #20280463

Subgroups — #20280463

Order 6784.

632 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133201C211yesmaximal, characteristic, complemented
2C2×#13133198C211yesmaximal, characteristic, complemented
3C2×#13133189C211yesmaximal, characteristic
4C22×#11719919C211yesmaximal, characteristic, complemented
5#16810557C212yesmaximal, complemented
6#16810547C212yesmaximal, complemented
7#16810546C212yesmaximal, complemented
8#16810538C212yesmaximal, complemented
9#16810502C211yesmaximal, characteristic, complemented
10#16810481C211yesmaximal, characteristic, complemented
11C53×#386C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15C23×Dic53C2211yescharacteristic
16#13133201—24no—
17#13133198—24no—
18#13133189—24no—
19C2×C4⋊Dic53C2212yes—
20C2×#11719919—24no—
21C2×#11719919C2212yes—
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2212yes—
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2212yes—
28C2×C22⋊Dic53C2211yescharacteristic
29C2×#11719921C2212yes—
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2212yes—
32#13133124C2212yes—
33#13133124C2212yes—
34C22×D4×C53C2211yescharacteristic
35C23×C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yes—
38#95×C53C2212yes—
39C24×C53C2311yescharacteristic
40C24×C53C2311yescharacteristic
41C23×D53C2311yescharacteristic
42C2×D4×C53—22no—
43C2×D4×C53—22no—
44C2×D4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—42no—
47C22×C4×C53C2311yescharacteristic
48C22×C4×C53C2312yes—
49C22×C4×C53C2312yes—
50C2×C22⋊D53—22no—
51C2×C22⋊D53—22no—
52C2×C22⋊D53—22no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—42no—
55C22×Dic53—22no—
56C22×Dic53—22no—
57C22×Dic53—42no—
58C22×Dic53D414yes—
59C22×Dic53D414yescomplemented
60C22×Dic53C2312yes—
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2312yes—
65C22⋊C4×C53—22no—
66C22⋊C4×C53—42no—
67C22⋊Dic53—22no—
68C22⋊Dic53—22no—
69C22⋊Dic53—24no—
70#11719921—22no—
71#11719921—24no—
72#11719921—42no—
73C4⋊Dic53—24no—
74#11719919—24no—
75#11719919—24no—
76#11719919—22no—
77#11719919—24no—
78#11719919—42no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—21no—
82C23×C53—21no—
83C23×C53—21no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53C2×D412yes—
87C23×C53C2×D412yes—
88C23×C53#4112yes—
89C23×C53C2411yescharacteristic
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—22no—
94C22×D53—22no—
95D4×C53—44no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—44no—
106C22⋊D53—44no—
107C22⋊D53—44no—
108C2×Dic53—28no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—22no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—22no—
119C2×Dic53—44no—
120C2×Dic53C2×D412yes—
121C2×Dic53C2×D412yes—
122C2×Dic53C2×D414yes—
123C22×C53—22no—
124C22×C53—22no—
125C22×C53—24no—
126C22×C53—24no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—42no—
131C22×C53—42no—
132C22×C53#12114yes—
133C22×C53C22×D411yescharacteristic
134C22×C53C22×D411yescharacteristic
135C22×C53C2×#4111yescharacteristic
136C22×C53H(5,𝔽2)12yes—
137C22×C532−1+412yes—
138C2×D53—41no—
139C2×D53—41no—
140C2×D53—41no—
141C2×D53—42no—
142C2×D53—42no—
143C4×C53—41no—
144C4×C53—42no—
145C4×C53—42no—
146Dic53—24no—
147Dic53—22no—
148Dic53—22no—
149Dic53—41no—
150Dic53—42no—
151#2618—531nomaximal, Sylow
152C2×C53—24no—
153C2×C53—81no—
154C2×C53C2×#12111yescharacteristic
155C2×C53#53411yescharacteristic
156C2×C53#53611yescharacteristic
157C2×C53#53712yes—
158C2×C53#53912yes—
159D53—81no—
160C2×#123—531no—
161C2×#123—531no—
162C2×#121—531no—
163C22×C4⋊C4—531no—
164#398—532no—
165#396—532no—
166#396—532no—
167#394—532no—
168#386—531no—
169#386D5311yescharacteristic, complemented
170#380—531no—
171C53#261811yescharacteristic, complemented, Sylow
172C22×D4—531no—
173C22×D4—531no—
174C22×D4C2×D5311yescharacteristic
175C23×C4—531no—
176C23×C4—531no—
177C23×C4C2×D5311yescharacteristic
178#123—1064no—
179#123—1064no—
180#121—1064no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—532no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—532no—
185C2×C4⋊C4—532no—
186C2×C4⋊C4—532no—
187C2×C4⋊C4—1064no—
188C2×C22⋊C4—531no—
189C2×C22⋊C4—531no—
190C2×C22⋊C4—532no—
191C2×C22⋊C4—531no—
192C2×C22⋊C4—532no—
193C2×C22⋊C4—532no—
194C2×C22⋊C4C2×D5311yescharacteristic
195C2×C22⋊C4C2×D5312yes—
196#95—532no—
197#95—532no—
198#95C2×D5312yes—
199C24—531no—
200C24C22×D5311yescharacteristic
201C24C22×D5311yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—1062no—
206C2×D4—1062no—
207C2×D4—1062no—
208C2×D4—1062no—
209C2×D4—2122no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—534no—
213C22×C4—531no—
214C22×C4—532no—
215C22×C4—531no—
216C22×C4—534no—
217C22×C4—532no—
218C22×C4—531no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—2122no—
222C22×C4C22×D5312yes—
223C22×C4C22×D5312yes—
224C22×C4C22×D5311yescharacteristic
225C4⋊C4—1064no—
226C4⋊C4—1064no—
227C4⋊C4—1062no—
228C4⋊C4—1064no—
229C4⋊C4—1064no—
230C4⋊C4—2122no—
231C22⋊C4—22no—
232C22⋊C4—42no—
233C22⋊C4—1062no—
234C22⋊C4—1064no—
235C22⋊C4—1062no—
236C22⋊C4—1064no—
237C22⋊C4—1062no—
238C22⋊C4—2122no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—1061no—
247C23—1061no—
248C23—1061no—
249C23—1062no—
250C23—1062no—
251C23#1171994512yes—
252C23D4×D5312yes—
253C23D4×D5312yes—
254C23C23×D5311yescharacteristic
255D4—44no—
256D4—2124no—
257D4—2124no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—44no—
268C2×C4—534no—
269C2×C4—532no—
270C2×C4—532no—
271C2×C4—1068no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—1062no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1062no—
281C2×C4—1062no—
282C2×C4—2124no—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—2121no—
293C22—2121no—
294C22—2121no—
295C22—2122no—
296C22—2122no—
297C22#1313319014yes—
298C22C2×D4×D5311yescharacteristic
299C22C2×D4×D5311yescharacteristic
300C22C2×#1171994511yescharacteristic
301C22#1313326212yes—
302C22#1313326412yes—
303C4—41no—
304C4—42no—
305C4—42no—
306C4—1064no—
307C4—1062no—
308C4—1062no—
309C4—2121no—
310C4—2122no—
311C2—24no—
312C2—81no—
313C2—4241no—
314C2C2×#1313319011yescharacteristic
315C2#1681118311yescharacteristic
316C2#1681119511yescharacteristic
317C2#1681120012yes—
318C2#1681121512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#386C2
hypercenter#386D53
hypocenterC53#2618
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#13133201C2semidirect
2. C2×#13133198C2semidirect
4. C22×#11719919C2semidirect
5. #16810557C2semidirect
6. #16810547C2semidirect
7. #16810546C2semidirect
8. #16810538C2semidirect
9. #16810502C2semidirect
10. #16810481C2semidirect
11. C53×#386C2semidirect
59. C22×Dic53D4semidirect
169. #386D53semidirect
171. C53#2618semidirect