Finite Group Databasebeta

← #20284664

Subgroups — #20284664

Order 6784.

410 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 137 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×#13133196C211yesmaximal, characteristic
2C2×#13133176C211yesmaximal, characteristic
3#16810965C214yesmaximal, complemented
4#16810959C212yesmaximal, complemented
5#16810954C212yesmaximal, complemented
6#16810665C212yesmaximal, complemented
7#16810633C212yesmaximal, complemented
8C53×#407C211yesmaximal, characteristic
9C22×Dic106C2211yescharacteristic
10#13133196—21no—
11#13133196—21no—
12#13133196C2212yes—
13#13133196C2212yes—
14#13133176—21no—
15#13133176—21no—
16#13133176C2212yes—
17#13133176C2212yes—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×#11719919—21no—
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53—21no—
24C2×C4×Dic53—21no—
25#13133151C2214yescomplemented
26#13133149C2214yescomplemented
27#13133142C2214yescomplemented
28#13133142C2214yescomplemented
29C23×C4×C53C2211yescharacteristic
30C2×M16×C53C2212yes—
31C22⋊C8×C53C2212yes—
32C22⋊C8×C53C2212yes—
33C24×C53D411yescharacteristic
34C22×C4×C53—24no—
35C22×C4×C53D411yescharacteristic
36C22×C4×C53D411yescharacteristic
37C22×C4×C53D411yescharacteristic
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2311yescharacteristic
40C22×C4×C53C2311yescharacteristic
41C22×Dic53—21no—
42C22×Dic53—21no—
43C22×Dic53—21no—
44C22×Dic53—21no—
45C2×Dic106—24no—
46C2×Dic106—21no—
47C2×Dic106C2312yes—
48M16×C53—24no—
49C2×C8×C53C2314yes—
50C22⋊Dic53—22no—
51C22⋊Dic53—22no—
52C22⋊Dic53—22no—
53C22⋊Dic53—22no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53C2312yes—
58C4⋊Dic53C2312yes—
59C4⋊Dic53C2312yes—
60#11719919—22no—
61#11719919—22no—
62C4×Dic53—22no—
63C4×Dic53—22no—
64C23×C53—22no—
65C23×C53—22no—
66C23×C53—41no—
67C23×C53C2×D411yescharacteristic
68C23×C53C2×D411yescharacteristic
69C23×C53C2×D411yescharacteristic
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53#4114yes—
79C2×C4×C53C2411yescharacteristic
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88Dic106—24no—
89Dic106—42no—
90C8×C53—24no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—44no—
99C22×C53#12312yes—
100C22×C53#13712yes—
101C22×C53C22×D411yescharacteristic
102C4×C53—21no—
103C4×C53—21no—
104C4×C53#12312yes—
105C4×C53C2×#4112yes—
106Dic53—42no—
107Dic53—42no—
108Dic53—42no—
109Dic53—42no—
110#3095—531nomaximal, Sylow
111C2×C53—22no—
112C2×C53—22no—
113C2×C53—41no—
114C2×C53C2×#12311yescharacteristic
115C2×C53C2×#13711yescharacteristic
116C2×C53#57511yescharacteristic
117C2×C22⋊Q8—531no—
118C2×#117—531no—
119#485—534no—
120#484—532no—
121#483—532no—
122#428—532no—
123#419—532no—
124#407D5311yescharacteristic
125C53#309511yescharacteristic, complemented, Sylow
126C22×Q8—531no—
127C23×C4C2×D5311yescharacteristic
128C2×M16C2×D5312yes—
129C22⋊Q8—532no—
130C22⋊Q8—532no—
131C22⋊Q8—1061no—
132C22⋊Q8—1061no—
133#117—532no—
134#117—532no—
135#117—1061no—
136#117—1061no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4—531no—
139C2×C4⋊C4—531no—
140C2×C4⋊C4—1061no—
141C2×C22⋊C4—1061no—
142C2×C22⋊C4—1061no—
143C2×C42—1061no—
144#107—534no—
145#106—534no—
146#103—534no—
147#103—534no—
148C22⋊C8C2×D5312yes—
149C22⋊C8C2×D5312yes—
150C24D21211yescharacteristic
151C2×Q8—532no—
152C2×Q8—1061no—
153C2×Q8—1064no—
154C22×C4—24no—
155C22×C4—1061no—
156C22×C4—1061no—
157C22×C4—1061no—
158C22×C4—1061no—
159C22×C4D21211yescharacteristic
160C22×C4D21211yescharacteristic
161C22×C4D21211yescharacteristic
162C22×C4C22×D5311yescharacteristic
163C22×C4C22×D5311yescharacteristic
164C22×C4C22×D5311yescharacteristic
165M16—24no—
166C2×C8C22×D5314yes—
167C4⋊C4—532no—
168C4⋊C4—532no—
169C4⋊C4—532no—
170C4⋊C4—1061no—
171C4⋊C4—1061no—
172C4⋊C4—1061no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C22⋊C4—1062no—
176C22⋊C4—1062no—
177C22⋊C4—1062no—
178C22⋊C4—1062no—
179C42—1062no—
180C42—1062no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23C2×D21211yescharacteristic
185C23C2×D21211yescharacteristic
186C23C2×D21211yescharacteristic
187Q8—1064no—
188Q8—2122no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—1062no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4—2122no—
202C2×C4C2×D21211yescharacteristic
203C2×C4C2×D21211yescharacteristic
204C2×C4C2×D21211yescharacteristic
205C2×C4#1171994714yes—
206C2×C4C23×D5311yescharacteristic
207C8—24no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22#1313320112yes—
217C22#1313325012yes—
218C22C22×D21211yescharacteristic
219C4—21no—
220C4—21no—
221C4—2122no—
222C4—2122no—
223C4—2122no—
224C4—2122no—
225C4#1313320112yes—
226C4C2×#1171994712yes—
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#1313320111yescharacteristic
231C2C2×#1313325011yescharacteristic
232C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3095
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #16810965C2semidirect
4. #16810959C2semidirect
5. #16810954C2semidirect
6. #16810665C2semidirect
7. #16810633C2semidirect
25. #13133151C22semidirect
26. #13133149C22semidirect
27. #13133142C22semidirect
28. #13133142C22semidirect
125. C53#3095semidirect