Finite Group Databasebeta

← #20281006

Subgroups — #20281006

Order 6784.

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

normal subgroups only

This group has 804 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
1C2×#13133192C212yesmaximal, complemented
2C2×#13133191C212yesmaximal, complemented
3C2×#13133190C212yesmaximal, complemented
4#16810545C212yesmaximal, complemented
5#16810533C212yesmaximal, complemented
6#16810532C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810497C212yesmaximal, complemented
9C53×#385C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C22×D212C2212yes—
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16C22×C4×D53C2212yes—
17#13133192—28no—
18#13133191—28no—
19#13133190—28no—
20C2×C4⋊Dic53C2212yes—
21C2×#11719919C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27#13133124C2211yescharacteristic
28#13133124C2211yescharacteristic
29C23×C4×C53C2211yescharacteristic
30C2×C4⋊C4×C53C2212yes—
31C2×C22⋊C4×C53C2211yescharacteristic
32C2×C22⋊C4×C53C2211yescharacteristic
33#95×C53C2212yes—
34C24×C53C2311yescharacteristic
35C23×D53C2312yes—
36C23×D53C2312yes—
37C22×C4×C53—24no—
38C22×C4×C53—41no—
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2312yes—
43C2×C22⋊D53—24no—
44C2×C22⋊D53—24no—
45C2×C22⋊D53—24no—
46C2×C22⋊D53—24no—
47C2×C22⋊D53—42no—
48C2×C22⋊D53—42no—
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C2×D212—24no—
52C2×D212—24no—
53C2×D212—24no—
54C2×D212—24no—
55C2×D212—24no—
56C2×D212—42no—
57C2×D212—42no—
58C2×D212—41no—
59C2×D212—42no—
60C2×D212—41no—
61C2×C4×D53—24no—
62C2×C4×D53—24no—
63C2×C4×D53—42no—
64C4⋊C4×C53—24no—
65C22⋊C4×C53—41no—
66C22⋊C4×C53D414yescomplemented
67#11719921—24no—
68#11719921—24no—
69#11719921—24no—
70#11719921—42no—
71#11719921D418yescomplemented
72C4⋊Dic53—24no—
73#11719919—24no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53C2×D412yes—
79C23×C53C2411yescharacteristic
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93C22×D53C2×D414yes—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—44no—
103C2×C4×C53C2×D414yes—
104C2×C4×C53C2×D414yes—
105C22⋊D53—48no—
106C22⋊D53—48no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53C2×D414yes—
113D212—48no—
114D212—48no—
115D212—44no—
116D212—44no—
117C4×D53—48no—
118C22×C53—22no—
119C22×C53—24no—
120C22×C53—42no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53C22×D412yes—
124C22×C53C22×D411yescharacteristic
125C22×C53H(5,𝔽2)11yescharacteristic
126C22×C53H(5,𝔽2)12yes—
127C22×C532−1+411yescharacteristic
128C2×D53—24no—
129C2×D53—24no—
130C2×D53—24no—
131C2×D53—42no—
132C2×D53—42no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—42no—
141C2×D53—42no—
142C4×C53—24no—
143C4×C53—24no—
144C4×C53—41no—
145C4×C53—41no—
146Dic53—24no—
147Dic53—42no—
148#2675—531nomaximal, Sylow
149C2×C53—42no—
150C2×C53#53712yes—
151C2×C53#53711yescharacteristic
152C2×C53D4211yescharacteristic
153C2×C53D4212yes—
154C2×C53#56011yescharacteristic
155D53—44no—
156D53—82no—
157C2×#121—532no—
158C2×#121—532no—
159C2×#121—532no—
160#396—532no—
161#392—531no—
162#392—531no—
163#392—532no—
164#385—532no—
165#385D5311yescharacteristic, complemented
166C53#267511yescharacteristic, complemented, Sylow
167C22×D4—532no—
168C22×D4—532no—
169C22×D4—531no—
170C22×D4—532no—
171C22×D4—532no—
172C22×D4—531no—
173C23×C4—532no—
174C23×C4C2×D5311yescharacteristic
175#121—1068no—
176#121—1068no—
177#121—1068no—
178C2×C4⋊C4—532no—
179C2×C4⋊C4—532no—
180C2×C4⋊C4C2×D5312yes—
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4C2×D5311yescharacteristic
187C2×C22⋊C4C2×D5311yescharacteristic
188#95—531no—
189#95—531no—
190#95C2×D5312yes—
191C24—532no—
192C24—532no—
193C24C22×D5311yescharacteristic
194C2×D4—1064no—
195C2×D4—1064no—
196C2×D4—1064no—
197C2×D4—1064no—
198C2×D4—1064no—
199C2×D4—1064no—
200C2×D4—1064no—
201C2×D4—1064no—
202C2×D4—1064no—
203C2×D4—2122no—
204C2×D4—2122no—
205C2×D4—2121no—
206C2×D4—2121no—
207C2×D4—2122no—
208C2×D4—2122no—
209C2×D4—2122no—
210C22×C4—24no—
211C22×C4—41no—
212C22×C4—532no—
213C22×C4—532no—
214C22×C4—1064no—
215C22×C4—1064no—
216C22×C4—2122no—
217C22×C4C22×D5311yescharacteristic
218C22×C4C22×D5311yescharacteristic
219C22×C4C22×D5312yes—
220C22×C4C22×D5312yes—
221C4⋊C4—24no—
222C4⋊C4—1064no—
223C4⋊C4—1064no—
224C22⋊C4—41no—
225C22⋊C4—538no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—1064no—
229C22⋊C4—2122no—
230C22⋊C4D21214yescomplemented
231C23—22no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—534no—
236C23—1062no—
237C23—1062no—
238C23—1062no—
239C23—1062no—
240C23—1062no—
241C23—1062no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23—1062no—
248C23—1062no—
249C23C2×D21212yes—
250C23C23×D5311yescharacteristic
251D4—2128no—
252D4—2128no—
253D4—2124no—
254D4—2124no—
255D4—2128no—
256D4—2128no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—44no—
266C2×C4—534no—
267C2×C4—1062no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—1062no—
271C2×C4—1062no—
272C2×C4—2128no—
273C2×C4C2×D21214yes—
274C2×C4D4×D5314yes—
275C22—22no—
276C22—24no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—1064no—
281C22—1064no—
282C22—1064no—
283C22—2122no—
284C22—2122no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22—2122no—
289C22—2122no—
290C22—2122no—
291C22—2122no—
292C22—2122no—
293C22—2122no—
294C22C2×D4×D5312yes—
295C22C22×D21211yescharacteristic
296C22#1313326211yescharacteristic
297C22#1313326312yes—
298C22#1313326511yescharacteristic
299C4—24no—
300C4—24no—
301C4—41no—
302C4—41no—
303C4—1064no—
304C4—2122no—
305C2—42no—
306C2—4242no—
307C2—2124no—
308C2#1681120312yes—
309C2#1681120411yescharacteristic
310C2#1681125511yescharacteristic
311C2D4×D21212yes—
312C2#1681136311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#385C2
hypercenter#385D53
hypocenterC53#2675
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#13133192C2semidirect
2. C2×#13133191C2semidirect
3. C2×#13133190C2semidirect
4. #16810545C2semidirect
5. #16810533C2semidirect
6. #16810532C2semidirect
7. #16810532C2semidirect
8. #16810497C2semidirect
9. C53×#385C2semidirect
66. C22⋊C4×C53D4semidirect
71. #11719921D4semidirect
165. #385D53semidirect
166. C53#2675semidirect
230. C22⋊C4D212semidirect