Finite Group Databasebeta

← #20279181

Subgroups — #20279181

Order 6784.

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

normal subgroups only

This group has 708 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
1C2×#13133202C212yesmaximal, complemented
2C2×#13133199C212yesmaximal, complemented
3C2×#13133190C211yesmaximal, characteristic
4C2×C4⋊C4×D53C211yesmaximal, characteristic, complemented
5#16810560C211yesmaximal, characteristic, complemented
6#16810559C211yesmaximal, characteristic, complemented
7#16810558C211yesmaximal, characteristic, complemented
8#16810523C212yesmaximal, complemented
9#16810492C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13C22×C22⋊D53C2211yescharacteristic
14C22×C22⋊D53C2211yescharacteristic
15C22×C4×D53C2212yes—
16C22×C4×D53C2211yescharacteristic
17#13133202—28no—
18#13133199—28no—
19#13133190—24no—
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2212yes—
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×#11719919C2211yescharacteristic
26C4⋊C4×D53—24no—
27C2×C22⋊Dic53C2212yes—
28C2×#11719921C2212yes—
29C2×#11719921C2212yes—
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2212yes—
32C2×C4×Dic53C2212yes—
33#13133124C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2212yes—
37C2×C22⋊C4×C53C2211yescharacteristic
38#95×C53C2211yescharacteristic
39C24×C53C2311yescharacteristic
40C23×D53C2311yescharacteristic
41C23×D53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2312yes—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—24no—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C2×D212—22no—
55C2×D212—24no—
56C2×C4×D53—24no—
57C2×C4×D53—24no—
58C2×C4×D53—24no—
59C2×C4×D53—24no—
60C2×C4×D53D414yes—
61C4⋊C4×C53—22no—
62C4⋊C4×C53—24no—
63C22⋊C4×C53—22no—
64C22⋊C4×C53—24no—
65C22⋊Dic53—24no—
66#11719921—22no—
67#11719921—24no—
68#11719921—24no—
69#11719921D418yescomplemented
70C4⋊Dic53—22no—
71C4⋊Dic53—24no—
72#11719919—24no—
73#11719919—22no—
74#11719919—24no—
75#11719919—24no—
76C4×Dic53—24no—
77C23×C53—21no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53C2411yescharacteristic
83C22×D53—22no—
84C22×D53—21no—
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—21no—
90C22×D53—21no—
91C22×D53C2×D412yes—
92C22×D53C2×D414yes—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53C2×D412yes—
102C2×C4×C53C2×D414yes—
103C22⋊D53—44no—
104C22⋊D53—44no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53C2×D414yes—
113C2×Dic53C2×D412yes—
114C2×Dic53#4114yes—
115D212—44no—
116C4×D53—28no—
117C4×D53—48no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53C22×D411yescharacteristic
124C22×C53C22×D412yes—
125C22×C53C2×#4112yes—
126C22×C532−1+411yescharacteristic
127C22×C532−1+411yescharacteristic
128C2×D53—22no—
129C2×D53—22no—
130C2×D53—24no—
131C2×D53—24no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—41no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53#12714yes—
138C4×C53—22no—
139C4×C53—24no—
140C4×C53—42no—
141C4×C53—41no—
142Dic53—24no—
143Dic53—42no—
144Dic53—41no—
145Dic53#12314yescomplemented
146#2501—531nomaximal, Sylow
147C2×C53—81no—
148C2×C53C2×#12311yescharacteristic
149C2×C53C2×#12711yescharacteristic
150C2×C53#53611yescharacteristic
151C2×C53#54712yes—
152C2×C53#54712yes—
153D53—24no—
154D53—81no—
155C2×#123—532no—
156C2×#123—532no—
157C2×#121—531no—
158C22×C4⋊C4—531no—
159#399—531no—
160#399—531no—
161#399—531no—
162#399D5311yescharacteristic, complemented
163#390—532no—
164#384—532no—
165C4×C22⋊C4—531no—
166C53#250111yescharacteristic, complemented, Sylow
167C22×D4—531no—
168C22×D4—531no—
169C22×D4—531no—
170C23×C4—531no—
171C23×C4—532no—
172#123—1068no—
173#123—1068no—
174#121—1064no—
175C2×C4⋊C4—532no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—531no—
178C2×C4⋊C4—532no—
179C2×C4⋊C4—532no—
180C2×C4⋊C4—532no—
181C2×C4⋊C4—1064no—
182C2×C4⋊C4C2×D5311yescharacteristic
183C2×C4⋊C4C2×D5312yes—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—531no—
187C2×C22⋊C4—532no—
188C2×C22⋊C4—532no—
189C2×C22⋊C4C2×D5311yescharacteristic
190C2×C22⋊C4C2×D5312yes—
191C2×C42—532no—
192#95—531no—
193#95C2×D5311yescharacteristic
194C24—531no—
195C24—531no—
196C24C22×D5311yescharacteristic
197C2×D4—1062no—
198C2×D4—1064no—
199C2×D4—1062no—
200C2×D4—1064no—
201C2×D4—1062no—
202C2×D4—1064no—
203C22×C4—531no—
204C22×C4—532no—
205C22×C4—534no—
206C22×C4—532no—
207C22×C4—531no—
208C22×C4—1064no—
209C22×C4—1064no—
210C22×C4—1064no—
211C22×C4—1064no—
212C22×C4C22×D5312yes—
213C22×C4C22×D5311yescharacteristic
214C22×C4C22×D5311yescharacteristic
215C22×C4C22×D5312yes—
216C4⋊C4—22no—
217C4⋊C4—24no—
218C4⋊C4—1064no—
219C4⋊C4—1062no—
220C4⋊C4—1064no—
221C4⋊C4—1064no—
222C4⋊C4—1062no—
223C4⋊C4—1064no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—538no—
227C22⋊C4—1064no—
228C22⋊C4—1064no—
229C22⋊C4—1062no—
230C22⋊C4—1064no—
231C42—1064no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—534no—
238C23—532no—
239C23—1062no—
240C23—1061no—
241C23—1061no—
242C23—1062no—
243C23—1061no—
244C23—1062no—
245C23—1061no—
246C23—1061no—
247C23C23×D5311yescharacteristic
248D4—2124no—
249D4—2124no—
250D4—2124no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—532no—
260C2×C4—534no—
261C2×C4—534no—
262C2×C4—1068no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—1062no—
267C2×C4—1061no—
268C2×C4—1062no—
269C2×C4—1062no—
270C2×C4—2128no—
271C2×C4D4×D5314yes—
272C2×C4D4×D5312yes—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—534no—
279C22—1064no—
280C22—1064no—
281C22—1062no—
282C22—1062no—
283C22—2122no—
284C22—2122no—
285C22—2121no—
286C22—2121no—
287C22—2121no—
288C22C2×D4×D5312yes—
289C22C2×D4×D5311yescharacteristic
290C22#41×D5312yes—
291C22#1313326411yescharacteristic
292C22#1313326511yescharacteristic
293C4—24no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—534no—
298C4—1064no—
299C4—2122no—
300C4—2121no—
301C2—81no—
302C2—1064no—
303C2—4241no—
304C2#123×D5311yescharacteristic
305C2#127×D5311yescharacteristic
306C2#1681119611yescharacteristic
307C2#1681127112yes—
308C2#1681127512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2501
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133202C2semidirect
2. C2×#13133199C2semidirect
4. C2×C4⋊C4×D53C2semidirect
5. #16810560C2semidirect
6. #16810559C2semidirect
7. #16810558C2semidirect
8. #16810523C2semidirect
9. #16810492C2semidirect
10. #16810470C2semidirect
11. C53×#399C2semidirect
69. #11719921D4semidirect
145. Dic53#123semidirect
162. #399D53semidirect
166. C53#2501semidirect