Finite Group Databasebeta

← #20278354

Subgroups — #20278354

Order 6784.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×Q8×D53C211yesmaximal, characteristic
2C2×#13133216C212yesmaximal, complemented
3C2×#13133195C211yesmaximal, characteristic
4#16810544C211yesmaximal, characteristic, complemented
5#16810543C211yesmaximal, characteristic, complemented
6#16810542C211yesmaximal, characteristic, complemented
7#16810527C212yesmaximal, complemented
8#16810497C212yesmaximal
9#16810494C212yesmaximal
10#16810470C211yesmaximal, characteristic
11C53×#395C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×Dic106C2211yescharacteristic
14C22×Dic106C2212yes—
15C2×Q8×D53—24no—
16C2×Q8×D53—28no—
17C22×C4×D53C2212yes—
18C22×C4×D53C2211yescharacteristic
19#13133216—28no—
20#13133195—24no—
21C2×C4⋊Dic53C2212yes—
22C2×#11719919C2212yes—
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2212yes—
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2212yes—
29C2×C4×Dic53C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2212yes—
33#13133124C2211yescharacteristic
34C22×Q8×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic
36C2×C4⋊C4×C53C2212yes—
37#95×C53C2212yes—
38#95×C53C2211yescharacteristic
39C23×D53C2311yescharacteristic
40C2×Q8×C53—22no—
41C2×Q8×C53—24no—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2312yes—
46C22×C4×C53C2312yes—
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2312yes—
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52Q8×D53—232no—
53C2×C4×D53—24no—
54C2×C4×D53—24no—
55C2×C4×D53—24no—
56C2×C4×D53—24no—
57C2×C4×D53D414yes—
58C2×Dic106—22no—
59C2×Dic106—24no—
60C2×Dic106—24no—
61C2×Dic106—24no—
62C2×Dic106—22no—
63C2×Dic106—24no—
64C2×Dic106D418yes—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—24no—
67#11719921—22no—
68#11719921—22no—
69#11719921Q818yescomplemented
70C4⋊Dic53—24no—
71#11719919—22no—
72#11719919—24no—
73#11719919—22no—
74#11719919—24no—
75C4×Dic53—24no—
76C23×C53C2411yescharacteristic
77C22×D53—22no—
78C22×D53—21no—
79C22×D53—21no—
80C22×D53C2×D412yes—
81C22×D53C2×Q814yes—
82Q8×C53—28no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53C2×D412yes—
94C2×C4×C53C2×D414yes—
95C2×C4×C53C2×Q814yescomplemented
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53C2×D414yes—
106C2×Dic53C2×D412yes—
107C2×Dic53C2×D414yes—
108C2×Dic53C2×Q814yescomplemented
109C4×D53—216no—
110C4×D53—28no—
111Dic106—28no—
112Dic106—216no—
113Dic106—44no—
114C22×C53C22×D411yescharacteristic
115C22×C53C22×D412yes—
116C22×C53C22×Q812yes—
117C22×C532−1+411yescharacteristic
118C22×C532−1+411yescharacteristic
119C2×D53—22no—
120C2×D53—22no—
121C2×D53—24no—
122C2×D53—24no—
123C2×D53C4⋊Q814yescomplemented
124C4×C53—22no—
125C4×C53—24no—
126C4×C53—24no—
127C4×C53—41no—
128C4×C53—41no—
129Dic53—24no—
130Dic53—24no—
131Dic53—41no—
132Dic53—41no—
133Dic53#12014yes—
134#2434—531nomaximal, Sylow
135C2×C53C2×#12011yescharacteristic
136C2×C53C2×C4⋊Q811yescharacteristic
137C2×C53#53611yescharacteristic
138C2×C53D4×Q812yes—
139C2×C53D4×Q812yes—
140D53—24no—
141C23×Q8—531no—
142C2×C4⋊Q8—532no—
143C2×C22⋊Q8—531no—
144#395—531no—
145#395—531no—
146#395—531no—
147#395D5311yescharacteristic, complemented
148#391—532no—
149#385—532no—
150#385—532no—
151C4×C22⋊C4—531no—
152C53#243411yescharacteristic, complemented, Sylow
153C22×Q8—532no—
154C22×Q8—531no—
155C22×Q8—531no—
156C22×Q8—1064no—
157C22×Q8—1068no—
158C22×Q8C2×D5311yescharacteristic
159C23×C4—531no—
160C23×C4—532no—
161C4⋊Q8—1068no—
162C22⋊Q8—1064no—
163C2×C4⋊C4—531no—
164C2×C4⋊C4—531no—
165C2×C4⋊C4—532no—
166C2×C4⋊C4—532no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4C2×D5311yescharacteristic
169C2×C4⋊C4C2×D5312yes—
170C2×C22⋊C4—532no—
171C2×C22⋊C4—531no—
172C2×C22⋊C4—531no—
173C2×C42—532no—
174#95—531no—
175#95—532no—
176#95—532no—
177#95—532no—
178#95C2×D5311yescharacteristic
179#95C2×D5312yes—
180C24—531no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—538no—
184C2×Q8—1064no—
185C2×Q8—1064no—
186C2×Q8—1064no—
187C2×Q8—1062no—
188C2×Q8—1064no—
189C2×Q8—10632no—
190C2×Q8—1062no—
191C22×C4—531no—
192C22×C4—532no—
193C22×C4—532no—
194C22×C4—534no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—1064no—
198C22×C4—1064no—
199C22×C4—1064no—
200C22×C4—1064no—
201C22×C4C22×D5311yescharacteristic
202C22×C4C22×D5311yescharacteristic
203C22×C4C22×D5311yescharacteristic
204C22×C4C22×D5312yes—
205C22×C4C22×D5312yes—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—1062no—
209C4⋊C4—1064no—
210C4⋊C4—1064no—
211C4⋊C4—1062no—
212C4⋊C4—1064no—
213C22⋊C4—538no—
214C22⋊C4—1062no—
215C22⋊C4—1062no—
216C42—1064no—
217C23—532no—
218C23—534no—
219C23—1062no—
220C23—1061no—
221C23—1061no—
222C23C23×D5311yescharacteristic
223Q8—28no—
224Q8—10616no—
225Q8—1068no—
226Q8—2124no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—534no—
238C2×C4—532no—
239C2×C4—534no—
240C2×C4—534no—
241C2×C4—1068no—
242C2×C4—10616no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1062no—
246C2×C4—1062no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—1061no—
252C2×C4D4×D5312yes—
253C2×C4D4×D5314yes—
254C2×C4Q8×D5314yescomplemented
255C22—534no—
256C22—1064no—
257C22—1064no—
258C22—1062no—
259C22—1062no—
260C22C2×D4×D5312yes—
261C22C2×D4×D5311yescharacteristic
262C22C2×Q8×D5312yes—
263C22#1313326411yescharacteristic
264C22#1313326511yescharacteristic
265C4—24no—
266C4—22no—
267C4—24no—
268C4—41no—
269C4—41no—
270C4—534no—
271C4—1064no—
272C4—1064no—
273C4—2121no—
274C4—2121no—
275C2—1064no—
276C2#120×D5311yescharacteristic
277C2C4⋊Q8×D5311yescharacteristic
278C2#1681119611yescharacteristic
279C2#1681128612yes—
280C2#1681128812yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2434
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#13133216C2semidirect
4. #16810544C2semidirect
5. #16810543C2semidirect
6. #16810542C2semidirect
7. #16810527C2semidirect
11. C53×#395C2semidirect
69. #11719921Q8semidirect
95. C2×C4×C53C2×Q8semidirect
108. C2×Dic53C2×Q8semidirect
123. C2×D53C4⋊Q8semidirect
147. #395D53semidirect
152. C53#2434semidirect
254. C2×C4Q8×D53semidirect