Finite Group Databasebeta

← #20279022

Subgroups — #20279022

Order 6784.

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

normal subgroups only

This group has 624 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×#13133201C211yesmaximal, characteristic
2C2×#13133200C211yesmaximal, characteristic
3C2×#13133177C211yesmaximal, characteristic
4C22×#11719919C211yesmaximal, characteristic, complemented
5#16810557C212yesmaximal, complemented
6#16810552C212yesmaximal, complemented
7#16810520C212yesmaximal, complemented
8#16810484C211yesmaximal, characteristic, complemented
9#16810482C212yesmaximal, complemented
10#16810481C211yesmaximal, characteristic, complemented
11C53×#381C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133201—24no—
17#13133200—24no—
18#13133177—24no—
19C2×C4⋊Dic53C2212yes—
20C2×#11719919—24no—
21C2×#11719919C2212yes—
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×#11719919C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2212yes—
29C2×#11719921C2212yes—
30C2×C4×Dic53C2211yescharacteristic
31#13133124C2212yes—
32#13133124C2212yes—
33#13133124C2212yes—
34C23×C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2211yescharacteristic
36C2×C22⋊C4×C53C2211yescharacteristic
37#95×C53C2212yes—
38#95×C53C2212yes—
39C24×C53C2311yescharacteristic
40C23×D53C2311yescharacteristic
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C22×C4×C53C2312yes—
47C22×C4×C53C2312yes—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50C2×C22⋊D53D414yes—
51C22×Dic53—24no—
52C22×Dic53—22no—
53C22×Dic53—22no—
54C22×Dic53—22no—
55C22×Dic53D414yes—
56C22×Dic53C2312yes—
57C22×Dic53C2312yes—
58C22×Dic53C2311yescharacteristic
59C22×Dic53C2311yescharacteristic
60C22×Dic53C2311yescharacteristic
61C2×C4×D53—22no—
62C2×C4×D53—24no—
63C22⋊C4×C53—22no—
64C22⋊C4×C53—22no—
65C22⋊Dic53—22no—
66#11719921—24no—
67#11719921—24no—
68#11719921—22no—
69C4⋊Dic53—24no—
70#11719919—24no—
71#11719919—24no—
72#11719919—22no—
73#11719919—24no—
74#11719919—24no—
75C4×Dic53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53C2×D412yes—
79C23×C53C2×D412yes—
80C23×C53#4112yes—
81C23×C53C2411yescharacteristic
82C22×D53—21no—
83C22×D53—22no—
84C22×D53—21no—
85C22×D53—22no—
86C22×D53C2×D412yes—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—44no—
98C22⋊D53—28no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—28no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—44no—
110C2×Dic53C2×D412yes—
111C2×Dic53C2×D412yes—
112C2×Dic53C2×D412yes—
113C2×Dic53#4114yes—
114C4×D53—44no—
115C22×C53—22no—
116C22×C53—22no—
117C22×C53—24no—
118C22×C53—24no—
119C22×C53#12114yes—
120C22×C53C22×D411yescharacteristic
121C22×C53C22×D411yescharacteristic
122C22×C53C2×#4112yes—
123C22×C53C2×#4111yescharacteristic
124C22×C532−1+412yes—
125C2×D53—22no—
126C2×D53—24no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—42no—
130C4×C53—42no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—42no—
134Dic53—22no—
135Dic53—22no—
136Dic53—42no—
137Dic53—42no—
138Dic53#12314yescomplemented
139#2486—531nomaximal, Sylow
140C2×C53—24no—
141C2×C53C2×#12111yescharacteristic
142C2×C53C2×#12311yescharacteristic
143C2×C53#53611yescharacteristic
144C2×C53#54712yes—
145C2×C53#55212yes—
146D53—42no—
147C2×#123—531no—
148C2×#123—531no—
149C2×C4×D4—531no—
150C22×C4⋊C4—531no—
151#398—532no—
152#397—532no—
153#389—532no—
154#381—531no—
155#381D5311yescharacteristic, complemented
156#380—531no—
157#380—532no—
158C53#248611yescharacteristic, complemented, Sylow
159C22×D4—531no—
160C23×C4—531no—
161C23×C4—531no—
162C23×C4—531no—
163C23×C4C2×D5311yescharacteristic
164#123—1064no—
165#123—1064no—
166C4×D4—1064no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—531no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—532no—
173C2×C4⋊C4—1064no—
174C2×C22⋊C4—531no—
175C2×C22⋊C4—531no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4C2×D5311yescharacteristic
179C2×C22⋊C4C2×D5311yescharacteristic
180C2×C42—531no—
181#95—532no—
182#95—532no—
183#95—532no—
184#95C2×D5312yes—
185#95C2×D5312yes—
186C24—531no—
187C24C22×D5311yescharacteristic
188C2×D4—534no—
189C2×D4—1062no—
190C2×D4—1062no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—531no—
195C22×C4—534no—
196C22×C4—531no—
197C22×C4—531no—
198C22×C4—532no—
199C22×C4—532no—
200C22×C4—1062no—
201C22×C4—1064no—
202C22×C4—1064no—
203C22×C4—1062no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4C22×D5311yescharacteristic
207C22×C4C22×D5311yescharacteristic
208C22×C4C22×D5312yes—
209C22×C4C22×D5312yes—
210C4⋊C4—1064no—
211C4⋊C4—1064no—
212C4⋊C4—1062no—
213C4⋊C4—1064no—
214C4⋊C4—1064no—
215C4⋊C4—1064no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—1062no—
219C22⋊C4—1064no—
220C22⋊C4—1064no—
221C22⋊C4—1062no—
222C42—1062no—
223C23—22no—
224C23—22no—
225C23—532no—
226C23—1061no—
227C23—1062no—
228C23—1061no—
229C23—1062no—
230C23#1171994512yes—
231C23D4×D5312yes—
232C23D4×D5312yes—
233C23C23×D5311yescharacteristic
234D4—1068no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—532no—
247C2×C4—534no—
248C2×C4—532no—
249C2×C4—532no—
250C2×C4—1068no—
251C2×C4—1061no—
252C2×C4—1062no—
253C2×C4—1061no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—2124no—
261C2×C4—2124no—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—1064no—
267C22—1062no—
268C22—2122no—
269C22—2121no—
270C22—2121no—
271C22#1313319014yes—
272C22C2×D4×D5311yescharacteristic
273C22C2×D4×D5311yescharacteristic
274C22#41×D5312yes—
275C22C2×#1171994511yescharacteristic
276C22#1313326412yes—
277C4—42no—
278C4—42no—
279C4—41no—
280C4—41no—
281C4—534no—
282C4—1062no—
283C4—1062no—
284C4—2122no—
285C4—2122no—
286C2—24no—
287C2—2122no—
288C2C2×#1313319011yescharacteristic
289C2#123×D5311yescharacteristic
290C2#1681119511yescharacteristic
291C2#1681127112yes—
292C2#1681130612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#381C2
hypercenter#381D53
hypocenterC53#2486
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
4. C22×#11719919C2semidirect
5. #16810557C2semidirect
6. #16810552C2semidirect
7. #16810520C2semidirect
8. #16810484C2semidirect
9. #16810482C2semidirect
10. #16810481C2semidirect
11. C53×#381C2semidirect
138. Dic53#123semidirect
155. #381D53semidirect
158. C53#2486semidirect