Finite Group Databasebeta

← #20277956

Subgroups — #20277956

Order 6784.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#13133179C212yesmaximal, complemented
2C2×#13133179C211yesmaximal, characteristic, complemented
3C2×#13133175C211yesmaximal, characteristic, complemented
4#16810549C211yesmaximal, characteristic, complemented
5#16810545C212yesmaximal, complemented
6#16810541C212yesmaximal, complemented
7#16810540C211yesmaximal, characteristic, complemented
8#16810533C211yesmaximal, characteristic
9#16810474C211yesmaximal, characteristic, complemented
10#16810470C212yesmaximal, complemented
11C53×#398C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×D212C2212yes—
14C22×C4×D53C2212yes—
15C22×C4×D53C2211yescharacteristic
16#13133179—24no—
17#13133179—28no—
18#13133175—24no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719921C2212yes—
22C2×#11719921C2211yescharacteristic
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29C2×C4×Dic53C2212yes—
30C2×C4×Dic53C2211yescharacteristic
31#13133124C2211yescharacteristic
32#13133124C2212yes—
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35C2×C4⋊C4×C53C2211yescharacteristic
36#95×C53C2211yescharacteristic
37#95×C53C2212yes—
38C23×D53C2311yescharacteristic
39C23×D53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C2×D212—24no—
50C2×D212—24no—
51C2×D212—24no—
52C2×D212—22no—
53C2×D212—24no—
54C2×C4×D53—22no—
55C2×C4×D53—22no—
56C2×C4×D53—24no—
57C2×C4×D53—24no—
58C2×C4×D53—24no—
59C2×C4×D53D414yescomplemented
60C4⋊C4×C53—22no—
61C4⋊C4×C53—24no—
62C4⋊C4×C53—22no—
63#11719921—24no—
64#11719921—24no—
65#11719921—22no—
66#11719921—24no—
67#11719921—24no—
68#11719921—22no—
69#11719921—24no—
70C4⋊Dic53—22no—
71#11719919—24no—
72C4×Dic53—22no—
73C4×Dic53—24no—
74C4×Dic53D414yescomplemented
75C23×C53C2411yescharacteristic
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—21no—
79C22×D53—21no—
80C22×D53—22no—
81C22×D53—22no—
82C22×D53—22no—
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53C2×D412yes—
87C22×D53#4114yes—
88C2×C4×C53—21no—
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—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53C2×D412yes—
99C2×C4×C53C2×D412yes—
100C2×C4×C53#4114yes—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53C2×D412yes—
108C2×Dic53C2×D412yes—
109C2×Dic53C2×D412yes—
110C2×Dic53#4112yes—
111D212—28no—
112D212—216no—
113C4×D53—28no—
114C4×D53—48no—
115C22×C53C22×D411yescharacteristic
116C22×C53C22×D411yescharacteristic
117C22×C53C2×#4111yescharacteristic
118C22×C53C2×#4112yes—
119C22×C53C2×#4112yes—
120C2×D53—22no—
121C2×D53—24no—
122C2×D53—24no—
123C2×D53—24no—
124C2×D53—24no—
125C2×D53—42no—
126C2×D53—41no—
127C2×D53—42no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—41no—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—24no—
135C4×C53—42no—
136C4×C53—41no—
137Dic53—22no—
138Dic53—42no—
139Dic53—41no—
140Dic53#12114yes—
141#2392—531nomaximal, Sylow
142C2×C53C2×#12111yescharacteristic
143C2×C53#52511yescharacteristic
144C2×C53#52512yes—
145C2×C53#52912yes—
146C2×C53#53211yescharacteristic
147D53—42no—
148D53—44no—
149C2×C4×D4—531no—
150C2×C4×D4—532no—
151C2×#117—531no—
152#398D5311yescharacteristic, complemented
153#397—531no—
154#396—532no—
155#394—531no—
156#394—532no—
157#392—531no—
158C4×C4⋊C4—531no—
159C4×C22⋊C4—532no—
160C53#239211yescharacteristic, complemented, Sylow
161C22×D4—531no—
162C22×D4—532no—
163C23×C4—531no—
164C23×C4—532no—
165C4×D4—1064no—
166C4×D4—1068no—
167#117—1064no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4C2×D5311yescharacteristic
171C2×C4⋊C4C2×D5312yes—
172C2×C4⋊C4C2×D5311yescharacteristic
173C2×C22⋊C4—531no—
174C2×C22⋊C4—531no—
175C2×C22⋊C4—532no—
176C2×C22⋊C4—532no—
177C2×C22⋊C4—532no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C42—531no—
181C2×C42—531no—
182C2×C42—532no—
183#95—532no—
184#95—531no—
185#95C2×D5312yes—
186#95C2×D5311yescharacteristic
187C24—531no—
188C24—532no—
189C2×D4—1064no—
190C2×D4—1064no—
191C2×D4—1064no—
192C2×D4—1062no—
193C2×D4—1064no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—534no—
197C22×C4—532no—
198C22×C4—531no—
199C22×C4—1062no—
200C22×C4—1062no—
201C22×C4—1064no—
202C22×C4—1064no—
203C22×C4—1064no—
204C22×C4C22×D5312yes—
205C22×C4C22×D5311yescharacteristic
206C22×C4C22×D5311yescharacteristic
207C22×C4C22×D5311yescharacteristic
208C22×C4C22×D5312yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—1062no—
213C4⋊C4—1064no—
214C22⋊C4—1062no—
215C22⋊C4—1064no—
216C22⋊C4—1064no—
217C22⋊C4—1064no—
218C22⋊C4—1062no—
219C22⋊C4—1064no—
220C22⋊C4—1064no—
221C42—534no—
222C42—1062no—
223C42—1064no—
224C23—532no—
225C23—534no—
226C23—1061no—
227C23—1062no—
228C23—1062no—
229C23—1061no—
230C23—1062no—
231C23—1062no—
232C23—1062no—
233C23—1062no—
234C23—1062no—
235C23—1062no—
236C23C23×D5311yescharacteristic
237D4—1068no—
238D4—10616no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—532no—
250C2×C4—532no—
251C2×C4—532no—
252C2×C4—532no—
253C2×C4—1068no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1061no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—2128no—
261C2×C4D4×D5312yes—
262C2×C4D4×D5312yes—
263C2×C4#1171994914yes—
264C22—1062no—
265C22—1064no—
266C22—1064no—
267C22—1064no—
268C22—1064no—
269C22—2122no—
270C22—2121no—
271C22—2122no—
272C22—2122no—
273C22—2122no—
274C22—2121no—
275C22—2122no—
276C22C2×D4×D5311yescharacteristic
277C22C2×D4×D5311yescharacteristic
278C22#41×D5311yescharacteristic
279C22#41×D5312yes—
280C22C2×#1171994912yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—534no—
287C4—1062no—
288C4—2122no—
289C4—2121no—
290C2—2122no—
291C2—2124no—
292C2#121×D5311yescharacteristic
293C2#1681113111yescharacteristic
294C2#1681113612yes—
295C2#1681115812yes—
296C2#1681117411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2392
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13133179C2semidirect
2. C2×#13133179C2semidirect
3. C2×#13133175C2semidirect
4. #16810549C2semidirect
5. #16810545C2semidirect
6. #16810541C2semidirect
7. #16810540C2semidirect
9. #16810474C2semidirect
10. #16810470C2semidirect
11. C53×#398C2semidirect
59. C2×C4×D53D4semidirect
74. C4×Dic53D4semidirect
152. #398D53semidirect
160. C53#2392semidirect