Finite Group Databasebeta

← #20279663

Subgroups — #20279663

Order 6784.

676 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 171 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×#13133216C211yesmaximal, characteristic, complemented
2C2×#13133195C212yesmaximal, complemented
3C2×#13133194C212yesmaximal, complemented
4C2×#13133177C211yesmaximal, characteristic
5#16810560C211yesmaximal, characteristic, complemented
6#16810559C211yesmaximal, characteristic, complemented
7#16810558C211yesmaximal, characteristic, complemented
8#16810523C212yesmaximal
9#16810492C212yesmaximal, complemented
10#16810474C211yesmaximal, characteristic, complemented
11C53×#399C211yesmaximal, characteristic, complemented
12C22×Dic106C2212yes—
13C22×C22⋊D53C2211yescharacteristic
14C23×Dic53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133216—24no—
17#13133195—28no—
18#13133194—28no—
19#13133177—24no—
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2212yes—
22C2×#11719919C2212yes—
23C2×#11719919C2212yes—
24C2×#11719919C2212yes—
25C2×#11719919C2211yescharacteristic
26C2×#11719919C2212yes—
27C2×C22⋊Dic53C2212yes—
28C2×#11719921C2212yes—
29C2×#11719921C2212yes—
30C2×#11719921C2211yescharacteristic
31C2×C4×Dic53C2212yes—
32C2×C4×Dic53C2211yescharacteristic
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
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2312yes—
45C2×C22⋊D53—22no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53Q814yescomplemented
48C22×Dic53—22no—
49C22×Dic53—24no—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C2×C4×D53—22no—
56C2×C4×D53—24no—
57C2×Dic106—24no—
58C2×Dic106—24no—
59C2×Dic106—24no—
60C4⋊C4×C53—24no—
61C4⋊C4×C53—22no—
62C22⋊C4×C53—22no—
63C22⋊C4×C53—24no—
64C22⋊Dic53—24no—
65#11719921—22no—
66#11719921—24no—
67#11719921—24no—
68C4⋊Dic53—24no—
69C4⋊Dic53—22no—
70#11719919—22no—
71#11719919—24no—
72#11719919—24no—
73#11719919—24no—
74#11719919D418yescomplemented
75C4×Dic53—24no—
76C4×Dic53D414yes—
77C23×C53—21no—
78C23×C53—22no—
79C23×C53—21no—
80C23×C53—22no—
81C23×C53C2×Q812yes—
82C23×C53C2411yescharacteristic
83C22×D53—21no—
84C22×D53—22no—
85C22×D53—21no—
86C22×D53—22no—
87C22×D53C2×Q812yes—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53C2×D412yes—
97C2×C4×C53C2×D414yes—
98C22⋊D53—28no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—44no—
107C2×Dic53C2×D414yes—
108C2×Dic53C2×D412yes—
109C2×Dic53C2×D412yes—
110C2×Dic53C2×D414yes—
111C2×Dic53C2×Q812yes—
112C2×Dic53#4112yes—
113C4×D53—44no—
114Dic106—48no—
115C22×C53—22no—
116C22×C53—24no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—42no—
120C22×C53C22×D411yescharacteristic
121C22×C53C22×D412yes—
122C22×C53C22×Q811yescharacteristic
123C22×C53C2×#4111yescharacteristic
124C22×C532−1+412yes—
125C2×D53—22no—
126C2×D53—24no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—42no—
130C4×C53—22no—
131C4×C53—24no—
132C4×C53—42no—
133C4×C53—41no—
134Dic53—24no—
135Dic53—42no—
136Dic53—41no—
137Dic53C22⋊Q814yescomplemented
138Dic53#12714yes—
139#2542—531nomaximal, Sylow
140C2×C53—42no—
141C2×C53C2×C22⋊Q811yescharacteristic
142C2×C53C2×#12711yescharacteristic
143C2×C53#53611yescharacteristic
144C2×C53#54712yes—
145C2×C53D4×Q812yes—
146D53—42no—
147C2×C4⋊Q8—531no—
148C2×C22⋊Q8—532no—
149C2×C22⋊Q8—532no—
150C2×C4×D4—531no—
151#399—531no—
152#399—531no—
153#399—531no—
154#399D5311yescharacteristic, complemented
155#390—532no—
156#384—532no—
157C4×C4⋊C4—531no—
158C53#254211yescharacteristic, complemented, Sylow
159C22×Q8—532no—
160C22×D4—531no—
161C23×C4—531no—
162C23×C4—531no—
163C4⋊Q8—1064no—
164C22⋊Q8—1068no—
165C22⋊Q8—1068no—
166C4×D4—1064no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—532no—
172C2×C4⋊C4—532no—
173C2×C4⋊C4—531no—
174C2×C4⋊C4C2×D5312yes—
175C2×C4⋊C4C2×D5311yescharacteristic
176C2×C22⋊C4—532no—
177C2×C22⋊C4—531no—
178C2×C22⋊C4—532no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4C2×D5311yescharacteristic
181C2×C22⋊C4C2×D5312yes—
182C2×C42—531no—
183C2×C42—532no—
184#95—531no—
185#95C2×D5311yescharacteristic
186C24—531no—
187C24C22×D5311yescharacteristic
188C2×Q8—1064no—
189C2×Q8—1064no—
190C2×Q8—1064no—
191C2×D4—534no—
192C2×D4—1062no—
193C2×D4—1062no—
194C22×C4—531no—
195C22×C4—531no—
196C22×C4—532no—
197C22×C4—532no—
198C22×C4—531no—
199C22×C4—1062no—
200C22×C4—1064no—
201C22×C4—1062no—
202C22×C4—1064no—
203C22×C4C22×D5312yes—
204C22×C4C22×D5311yescharacteristic
205C22×C4C22×D5311yescharacteristic
206C22×C4C22×D5312yes—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—538no—
210C4⋊C4—1062no—
211C4⋊C4—1064no—
212C4⋊C4—1064no—
213C4⋊C4—1064no—
214C4⋊C4—1064no—
215C4⋊C4—1062no—
216C22⋊C4—22no—
217C22⋊C4—24no—
218C22⋊C4—1064no—
219C22⋊C4—1064no—
220C22⋊C4—1062no—
221C22⋊C4—1064no—
222C42—534no—
223C42—1064no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—532no—
229C23—1062no—
230C23—1061no—
231C23—1061no—
232C23—1062no—
233C23Q8×D5312yes—
234C23C23×D5311yescharacteristic
235Q8—2128no—
236D4—1068no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—532no—
246C2×C4—532no—
247C2×C4—534no—
248C2×C4—532no—
249C2×C4—532no—
250C2×C4—534no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1061no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—2124no—
259C2×C4—2124no—
260C2×C4D4×D5312yes—
261C2×C4D4×D5314yes—
262C22—24no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—1064no—
268C22—1062no—
269C22—2122no—
270C22—2121no—
271C22—2121no—
272C22C2×D4×D5312yes—
273C22C2×D4×D5311yescharacteristic
274C22C2×Q8×D5311yescharacteristic
275C22#41×D5311yescharacteristic
276C22#1313326412yes—
277C4—24no—
278C4—22no—
279C4—41no—
280C4—42no—
281C4—534no—
282C4—534no—
283C4—1064no—
284C4—2121no—
285C4—2122no—
286C2—42no—
287C2—2122no—
288C2C22⋊Q8×D5311yescharacteristic
289C2#127×D5311yescharacteristic
290C2#1681119511yescharacteristic
291C2#1681127112yes—
292C2#1681128612yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#399C2
hypercenter#399D53
hypocenterC53#2542
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#13133216C2semidirect
2. C2×#13133195C2semidirect
3. C2×#13133194C2semidirect
5. #16810560C2semidirect
6. #16810559C2semidirect
7. #16810558C2semidirect
9. #16810492C2semidirect
10. #16810474C2semidirect
11. C53×#399C2semidirect
47. C2×C22⋊D53Q8semidirect
74. #11719919D4semidirect
137. Dic53C22⋊Q8semidirect
154. #399D53semidirect
158. C53#2542semidirect