Finite Group Databasebeta

← #20279621

Subgroups — #20279621

Order 6784.

604 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 147 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×#13133195C212yesmaximal, complemented
2C2×#13133182C211yesmaximal, characteristic, complemented
3C2×#13133179C211yesmaximal, characteristic
4#16810554C211yesmaximal, characteristic, complemented
5#16810541C211yesmaximal, characteristic
6#16810540C211yesmaximal, characteristic
7#16810513C212yesmaximal, complemented
8#16810512C212yesmaximal, complemented
9#16810487C211yesmaximal, characteristic, complemented
10#16810482C212yesmaximal, complemented
11C53×#398C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×D212C2211yescharacteristic
14C22×C4×D53C2212yes—
15#13133195—28no—
16#13133182—24no—
17#13133179—24no—
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28C2×C4×Dic53C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33C2×C4⋊C4×C53C2212yes—
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36#95×C53C2212yes—
37#95×C53C2211yescharacteristic
38C23×D53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C2×D212—24no—
50C2×D212—22no—
51C2×C4×D53—24no—
52C2×C4×D53—24no—
53C2×C4×D53—24no—
54C2×Dic106—22no—
55C2×Dic106—22no—
56C2×Dic106D414yescomplemented
57C4⋊C4×C53—22no—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—24no—
60#11719921—24no—
61#11719921—24no—
62#11719921—24no—
63#11719921—24no—
64#11719921—24no—
65#11719919—22no—
66#11719919—22no—
67#11719919—24no—
68#11719919—24no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53D414yes—
72C23×C53C2411yescharacteristic
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—22no—
76C22×D53—22no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53#4114yes—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53C2×D412yes—
92C2×C4×C53C2×D412yes—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53C2×D412yes—
102C2×Dic53C2×D412yes—
103C2×Dic53C2×D412yes—
104C2×Dic53C2×D412yes—
105C2×Dic53#4112yes—
106D212—28no—
107C4×D53—48no—
108Dic106—28no—
109C22×C53C22×D411yescharacteristic
110C22×C53C22×D411yescharacteristic
111C22×C53C2×#4111yescharacteristic
112C22×C53C2×#4112yes—
113C22×C532−1+412yes—
114C2×D53—24no—
115C2×D53—24no—
116C2×D53—24no—
117C2×D53—42no—
118C2×D53—42no—
119C2×D53—42no—
120C2×D53—42no—
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—42no—
124C4×C53—41no—
125C4×C53—42no—
126Dic53—22no—
127Dic53—22no—
128Dic53—41no—
129Dic53—42no—
130Dic53#12114yes—
131#2536—531nomaximal, Sylow
132C2×C53C2×#12111yescharacteristic
133C2×C53#52511yescharacteristic
134C2×C53#53611yescharacteristic
135C2×C53#54812yes—
136C2×C53#55212yes—
137D53—44no—
138C2×C22⋊Q8—532no—
139C2×C4×Q8—531no—
140C2×C4×D4—531no—
141#398—531no—
142#398D5311yescharacteristic, complemented
143#394—531no—
144#394—531no—
145#388—532no—
146#388—532no—
147#382—531no—
148#380—532no—
149C53#253611yescharacteristic, complemented, Sylow
150C22×Q8—531no—
151C22×D4—531no—
152C23×C4—532no—
153C22⋊Q8—1068no—
154C4×Q8—1064no—
155C4×D4—1064no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—531no—
158C2×C4⋊C4—532no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4C2×D5311yescharacteristic
161C2×C4⋊C4C2×D5311yescharacteristic
162C2×C4⋊C4C2×D5312yes—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—532no—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42—531no—
171#95—532no—
172#95—532no—
173#95—531no—
174#95C2×D5312yes—
175#95C2×D5311yescharacteristic
176C24—532no—
177C2×Q8—534no—
178C2×Q8—1062no—
179C2×Q8—1062no—
180C2×D4—1064no—
181C2×D4—1062no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—532no—
186C22×C4—531no—
187C22×C4—1064no—
188C22×C4—1064no—
189C22×C4—1064no—
190C22×C4C22×D5312yes—
191C22×C4C22×D5312yes—
192C22×C4C22×D5311yescharacteristic
193C22×C4C22×D5311yescharacteristic
194C22×C4C22×D5311yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—1062no—
199C4⋊C4—1064no—
200C4⋊C4—1064no—
201C4⋊C4—1062no—
202C22⋊C4—1064no—
203C22⋊C4—1064no—
204C22⋊C4—1064no—
205C22⋊C4—1064no—
206C22⋊C4—1064no—
207C42—534no—
208C42—1062no—
209C42—1062no—
210C23—534no—
211C23—1062no—
212C23—1062no—
213C23—1062no—
214C23—1062no—
215C23—1062no—
216C23—1062no—
217C23C23×D5311yescharacteristic
218Q8—1068no—
219D4—1068no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—532no—
232C2×C4—532no—
233C2×C4—532no—
234C2×C4—532no—
235C2×C4—532no—
236C2×C4—1061no—
237C2×C4—1061no—
238C2×C4—1062no—
239C2×C4—1061no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—2128no—
245C2×C4D4×D5312yes—
246C2×C4D4×D5312yes—
247C22—1064no—
248C22—1064no—
249C22—1064no—
250C22—2122no—
251C22—2122no—
252C22—2122no—
253C22—2122no—
254C22C2×D4×D5311yescharacteristic
255C22C2×D4×D5311yescharacteristic
256C22#41×D5312yes—
257C22#41×D5311yescharacteristic
258C22#1313326512yes—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—42no—
263C4—41no—
264C4—534no—
265C4—1062no—
266C4—1062no—
267C4—2122no—
268C4—2121no—
269C2—2124no—
270C2#121×D5311yescharacteristic
271C2#1681113111yescharacteristic
272C2#1681119811yescharacteristic
273C2#1681128312yes—
274C2#1681130512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#398C2
hypercenter#398D53
hypocenterC53#2536
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#13133195C2semidirect
2. C2×#13133182C2semidirect
4. #16810554C2semidirect
7. #16810513C2semidirect
8. #16810512C2semidirect
9. #16810487C2semidirect
10. #16810482C2semidirect
11. C53×#398C2semidirect
56. C2×Dic106D4semidirect
142. #398D53semidirect
149. C53#2536semidirect