Finite Group Databasebeta

← #20275162

Subgroups — #20275162

Order 6784.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#16811135C211yesmaximal, characteristic
2C2×#13133174C211yesmaximal, characteristic, complemented
3C2×#13133145C211yesmaximal, characteristic, complemented
4C2×#13133143C211yesmaximal, characteristic, complemented
5#16810586C211yesmaximal, characteristic, complemented
6#16810382C211yesmaximal, characteristic, complemented
7C53×#339C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×C4×D53C2211yescharacteristic
10C2×#11719917C2211yescharacteristic
11#13133201—21no—
12#13133197C412yescomplemented
13#13133192C412yescomplemented
14#13133187—21no—
15C4×D212—22no—
16#13133174—22no—
17#13133174—22no—
18C2×#11719919—21no—
19C2×#11719921—21no—
20C2×C4×Dic53C2211yescharacteristic
21#13133145—22no—
22#13133143—22no—
23#13133129—22no—
24C2×M16×C53C2211yescharacteristic
25#117×C53C2211yescharacteristic
26C2×C42×C53C2211yescharacteristic
27C23×D53D411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53C2311yescharacteristic
30C2×C22⋊D53—21no—
31C22×Dic53—21no—
32C22×Dic53D411yescharacteristic
33#11719945—21no—
34#11719945—21no—
35#11719945—21no—
36#11719945—21no—
37C2×D212—22no—
38C2×D212C2×C411yescharacteristic
39C2×C4×D53—21no—
40C2×C4×D53—21no—
41C2×C4×D53—21no—
42C2×C4×D53—21no—
43C2×C4×D53—21no—
44C2×C4×D53C2×C412yes—
45C2×C4×D53D412yes—
46C2×Dic106C2×C411yescharacteristic
47M16×C53—21no—
48M16×C53—21no—
49C2×C8×C53—21no—
50C4⋊C4×C53C2×C412yes—
51C22⋊C4×C53—21no—
52C42×C53—21no—
53C42×C53—21no—
54C42×C53D412yescomplemented
55#11719921—24no—
56#11719921—22no—
57#11719921—22no—
58C4⋊Dic53—22no—
59#11719919—24no—
60C4×Dic53—21no—
61C4×Dic53D412yescomplemented
62#11719917—21no—
63#11719917—21no—
64C2×#88322—21no—
65C23×C53C2×D411yescharacteristic
66C22×D53—21no—
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—42no—
70C22×D53C22⋊C412yes—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—42no—
74C2×C4×C53C22×C411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53#4111yescharacteristic
79C2×C4×C53#4111yescharacteristic
80C22⋊D53—41no—
81C22⋊D53—41no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—42no—
87C2×Dic53C22⋊C412yes—
88D212—21no—
89D212—21no—
90D212—44no—
91C4×D53—22no—
92C4×D53—22no—
93C4×D53—22no—
94C4×D53—22no—
95C4×D53—41no—
96C4×D53—41no—
97Dic106—21no—
98Dic106—21no—
99C8×C53—41no—
100#88322—41no—
101C22×C53—22no—
102C22×C53C2×C22⋊C411yescharacteristic
103C22×C53#12011yescharacteristic
104C22×C53#12311yescharacteristic
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—22no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—41no—
111C2×D53—42no—
112C4×C53—42no—
113C4×C53—42no—
114C4×C53C4×D411yescharacteristic
115C4×C53C4×D411yescharacteristic
116C4×C53#12111yescharacteristic
117C4×C53#12111yescharacteristic
118Dic53—22no—
119Dic53—42no—
120Dic53—41no—
121#1897—531nomaximal, Sylow
122C2×C53—21no—
123C2×C53—21no—
124C2×C53#38611yescharacteristic
125C2×C53#42112yes—
126D53—42no—
127D53—81no—
128#525—531no—
129C2×#117—531no—
130C2×#104—531no—
131C2×#104—531no—
132#407—531no—
133#339—531no—
134#339D5311yescharacteristic, complemented
135C53#189711yescharacteristic, complemented, Sylow
136C2×#41—531no—
137C23×C4—531no—
138C2×M16—531no—
139C2×M16C2×D5311yescharacteristic
140#123—1061no—
141C22⋊Q8—532no—
142#121—532no—
143#120—1061no—
144C4×D4—1062no—
145#117—1062no—
146#117—1062no—
147#117C2×D5311yescharacteristic
148C2×C4⋊C4—1061no—
149C2×C22⋊C4—1061no—
150C2×C42—531no—
151C2×C42C2×D5311yescharacteristic
152#104—1062no—
153#104—1062no—
154C22⋊C8—1062no—
155C24—531no—
156#41—1061no—
157#41—1061no—
158#41—1061no—
159#41—1061no—
160C2×Q8—531no—
161C2×D4—531no—
162C2×D4—1062no—
163C2×D4—1061no—
164C22×C4—21no—
165C22×C4—531no—
166C22×C4—532no—
167C22×C4—532no—
168C22×C4—1061no—
169C22×C4—1061no—
170C22×C4—1061no—
171C22×C4—1061no—
172C22×C4—1061no—
173C22×C4—1061no—
174C22×C4C22×D5311yescharacteristic
175M16—21no—
176M16—21no—
177M16—1061no—
178M16—1061no—
179C2×C8—21no—
180C2×C8—1061no—
181C4⋊C4—1062no—
182C4⋊C4—1064no—
183C4⋊C4C4×D5312yes—
184C22⋊C4—21no—
185C22⋊C4—1062no—
186C22⋊C4—1062no—
187C22⋊C4—1064no—
188C42—21no—
189C42—21no—
190C42—532no—
191C42—1061no—
192C42D21212yescomplemented
193C23—532no—
194C23—1061no—
195C23—1061no—
196C23—1061no—
197C23—2122no—
198C23D4×D5311yescharacteristic
199Q8—1061no—
200Q8—1061no—
201D4—1061no—
202D4—1061no—
203D4—2124no—
204D4—2121no—
205D4—2121no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—532no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1061no—
217C2×C4—1062no—
218C2×C4—2122no—
219C2×C4—2121no—
220C2×C4—2121no—
221C2×C4C2×C4×D5311yescharacteristic
222C2×C4C2×D21211yescharacteristic
223C2×C4#1171994511yescharacteristic
224C2×C4D4×D5311yescharacteristic
225C2×C4D4×D5311yescharacteristic
226C2×C4#1171994911yescharacteristic
227C8—41no—
228C8—2121no—
229C22—22no—
230C22—1062no—
231C22—1062no—
232C22—1062no—
233C22—2122no—
234C22—2122no—
235C22—2121no—
236C22—2122no—
237C22C22⋊C4×D5311yescharacteristic
238C22#1313318711yescharacteristic
239C22#1313320211yescharacteristic
240C4—42no—
241C4—42no—
242C4—1062no—
243C4—2122no—
244C4—2121no—
245C4#1313317911yescharacteristic
246C4C4×D21211yescharacteristic
247C4#1313319011yescharacteristic
248C4#1313319211yescharacteristic
249C2—21no—
250C2—21no—
251C2—4241no—
252C2—2122no—
253C2#1681050111yescharacteristic
254C2#1681063812yes—
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#339C2
socleC22×C53C2×C22⋊C4
hypercenter#339D53
hypocenterC53#1897
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#13133174C2semidirect
3. C2×#13133145C2semidirect
4. C2×#13133143C2semidirect
5. #16810586C2semidirect
6. #16810382C2semidirect
7. C53×#339C2semidirect
12. #13133197C4semidirect
13. #13133192C4semidirect
54. C42×C53D4semidirect
61. C4×Dic53D4semidirect
134. #339D53semidirect
135. C53#1897semidirect
192. C42D212semidirect