Finite Group Databasebeta

← #20275160

Subgroups — #20275160

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#16811130C211yesmaximal, characteristic
2C2×#13133174C211yesmaximal, characteristic, complemented
3C2×#13133144C211yesmaximal, characteristic, complemented
4#16810586C211yesmaximal, characteristic, complemented
5#16810384C211yesmaximal, characteristic, complemented
6#16810383C211yesmaximal, characteristic, complemented
7C2×#104×C53C211yesmaximal, characteristic, complemented
8C22×C4×D53C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10#13133198—21no—
11#13133193C412yescomplemented
12#13133188C412yescomplemented
13#13133185—21no—
14C4×C22⋊D53—22no—
15#13133176C2211yescharacteristic
16#13133174—22no—
17#13133174—22no—
18C2×#11719919—21no—
19C2×#11719921—21no—
20C2×C4×Dic53C2211yescharacteristic
21#13133144—22no—
22#13133129—22no—
23C2×#41×C53C2211yescharacteristic
24C2×M16×C53C2211yescharacteristic
25C2×C42×C53C2211yescharacteristic
26#104×C53—22no—
27C23×D53D411yescharacteristic
28#41×C53—21no—
29#41×C53—21no—
30#41×C53—21no—
31#41×C53—21no—
32C2×Q8×C53C2×C411yescharacteristic
33C2×D4×C53—21no—
34C2×D4×C53C2×C411yescharacteristic
35C22×C4×C53—21no—
36C22×C4×C53—21no—
37C22×C4×C53C2311yescharacteristic
38C2×C22⋊D53—22no—
39C22×Dic53—21no—
40C22×Dic53D411yescharacteristic
41C2×C4×D53—21no—
42C2×C4×D53—21no—
43C2×C4×D53—21no—
44C2×C4×D53—21no—
45C2×C4×D53C2×C412yes—
46C2×C4×D53D412yes—
47M16×C53—21no—
48M16×C53—21no—
49C2×C8×C53—21no—
50C42×C53—21no—
51C42×C53D412yescomplemented
52C22⋊Dic53—21no—
53C22⋊Dic53—22no—
54#11719921—24no—
55#11719921—22no—
56C4⋊Dic53C2×C412yes—
57#11719919—24no—
58#11719919—22no—
59C4×Dic53—21no—
60C4×Dic53—21no—
61C4×Dic53D412yescomplemented
62#11719917—21no—
63#11719917—21no—
64C2×#88322—21no—
65C23×C53—21no—
66C23×C53C2×D411yescharacteristic
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—42no—
70C22×D53C22⋊C412yes—
71Q8×C53—21no—
72Q8×C53—21no—
73D4×C53—21no—
74D4×C53—21no—
75D4×C53—41no—
76D4×C53—41no—
77C2×C4×C53—21no—
78C2×C4×C53—22no—
79C2×C4×C53—42no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53C22×C411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2×D411yescharacteristic
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53#4111yescharacteristic
87C2×C4×C53#4111yescharacteristic
88C22⋊D53—44no—
89C2×Dic53—21no—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—42no—
94C2×Dic53C22⋊C412yes—
95C4×D53—22no—
96C4×D53—22no—
97C4×D53—22no—
98C4×D53—22no—
99C8×C53—41no—
100#88322—41no—
101C22×C53—22no—
102C22×C53—41no—
103C22×C53—42no—
104C22×C53C2×C22⋊C411yescharacteristic
105C22×C53#12011yescharacteristic
106C22×C53#12311yescharacteristic
107C2×D53—22no—
108C2×D53—22no—
109C2×D53—22no—
110C2×D53—42no—
111C2×D53—42no—
112C4×C53—42no—
113C4×C53—41no—
114C4×C53C4×D411yescharacteristic
115C4×C53C4×D411yescharacteristic
116C4×C53#12111yescharacteristic
117C4×C53#12111yescharacteristic
118Dic53—22no—
119Dic53—42no—
120Dic53—42no—
121#1897—531nomaximal, Sylow
122C2×C53—21no—
123C2×C53—21no—
124C2×C53—81no—
125C2×C53#38611yescharacteristic
126C2×C53#42112yes—
127D53—42no—
128#525—531no—
129C2×#117—531no—
130C2×#104—531no—
131C2×#104D5311yescharacteristic, complemented
132#407—531no—
133#339—531no—
134#339—531no—
135C53#189711yescharacteristic, complemented, Sylow
136C2×#41C2×D5311yescharacteristic
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—531no—
146#117—1062no—
147#117—1062no—
148C2×C4⋊C4—1061no—
149C2×C22⋊C4—1061no—
150C2×C42—531no—
151C2×C42C2×D5311yescharacteristic
152#104—22no—
153#104—1062no—
154C22⋊C8—1062no—
155C24—531no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160C2×Q8C4×D5311yescharacteristic
161C2×D4—21no—
162C2×D4—1062no—
163C2×D4C4×D5311yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—531no—
167C22×C4—532no—
168C22×C4—532no—
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—532no—
182C4⋊C4—1062no—
183C4⋊C4—1064no—
184C22⋊C4—1062no—
185C22⋊C4—1064no—
186C22⋊C4—1062no—
187C22⋊C4—1061no—
188C42—21no—
189C42—532no—
190C42—1061no—
191C42—1061no—
192C42C22⋊D5312yescomplemented
193C23—21no—
194C23—532no—
195C23—1061no—
196C23—1061no—
197C23—2122no—
198C23D4×D5311yescharacteristic
199Q8—21no—
200Q8—21no—
201D4—21no—
202D4—21no—
203D4—41no—
204D4—41no—
205D4—2124no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—532no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—1062no—
218C2×C4—1062no—
219C2×C4—1062no—
220C2×C4—2122no—
221C2×C4C2×C4×D5311yescharacteristic
222C2×C4#1171994511yescharacteristic
223C2×C4D4×D5311yescharacteristic
224C2×C4D4×D5311yescharacteristic
225C2×C4#1171994711yescharacteristic
226C2×C4C2×C22⋊D5311yescharacteristic
227C8—41no—
228C8—2121no—
229C22—22no—
230C22—41no—
231C22—42no—
232C22—1062no—
233C22—1062no—
234C22—1062no—
235C22—2122no—
236C22—2122no—
237C22C22⋊C4×D5311yescharacteristic
238C22#1313318511yescharacteristic
239C22#1313319911yescharacteristic
240C4—42no—
241C4—41no—
242C4—1062no—
243C4—2122no—
244C4—2122no—
245C4#1313317711yescharacteristic
246C4C4×C22⋊D5311yescharacteristic
247C4#1313318811yescharacteristic
248C4#1313319011yescharacteristic
249C2—21no—
250C2—21no—
251C2—81no—
252C2—2122no—
253C2#1681049911yescharacteristic
254C2#1681063812yes—
subgroupG/H
centerC2×C4D4×D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×#104×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#104D53
hypocenterC53#1897
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#13133174C2semidirect
3. C2×#13133144C2semidirect
4. #16810586C2semidirect
5. #16810384C2semidirect
6. #16810383C2semidirect
7. C2×#104×C53C2semidirect
11. #13133193C4semidirect
12. #13133188C4semidirect
51. C42×C53D4semidirect
61. C4×Dic53D4semidirect
131. C2×#104D53semidirect
135. C53#1897semidirect
192. C42C22⋊D53semidirect