Finite Group Databasebeta

← #20273077

Subgroups — #20273077

Order 6784.

270 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 65 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×#13133195C211yesmaximal, characteristic
2#16810585C211yesmaximal, characteristic
3C22⋊C8×D53C211yesmaximal, characteristic
4#16810349C211yesmaximal, characteristic, complemented
5#16810348C211yesmaximal, characteristic, complemented
6#16810348C211yesmaximal, characteristic, complemented
7C53×#324C211yesmaximal, characteristic, complemented
8C22×Dic106C411yescharacteristic
9C22×C4×D53C2211yescharacteristic
10C2×#11719912—21no—
11C2×C8×D53—21no—
12#13133195—21no—
13#13133195—21no—
14#13133195—21no—
15#13133195—21no—
16C2×#11719919C411yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719921C411yescharacteristic
19C2×#11719921C411yescharacteristic
20C22⋊#88322C2211yescharacteristic
21C22⋊#88322C2211yescharacteristic
22#13133129—21no—
23#13133129—21no—
24C2×C4⋊C4×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26C22⋊C8×C53C2211yescharacteristic
27C23×D53C2×C411yescharacteristic
28C22×C4×C53C2×C411yescharacteristic
29C22×C4×C53C2×C411yescharacteristic
30C22×C4×C53C2311yescharacteristic
31C22×Dic53C2×C411yescharacteristic
32C22×Dic53C2×C411yescharacteristic
33C22×Dic53C2×C411yescharacteristic
34C2×C4×D53—42no—
35C2×C4×D53D411yescharacteristic
36C2×C4×D53D411yescharacteristic
37C2×C4×D53D411yescharacteristic
38C2×C4×D53D411yescharacteristic
39C2×Dic106—21no—
40C2×Dic106—21no—
41C2×Dic106—42no—
42C2×C8×C53—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53—21no—
46#11719921—21no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53#11719919—21no—
54C2×#88322—21no—
55C2×#88322—21no—
56#11719912—42no—
57C8×D53—42no—
58C23×C53C22×C411yescharacteristic
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—22no—
62C22×D53—41no—
63C22×D53C22⋊C411yescharacteristic
64C22×D53C22⋊C411yescharacteristic
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×Dic53—22no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—41no—
79C2×Dic53C22⋊C411yescharacteristic
80C2×Dic53C22⋊C411yescharacteristic
81C4×D53—22no—
82C4×D53—22no—
83C4×D53—41no—
84C4×D53—41no—
85Dic106—42no—
86Dic106—42no—
87C8×C53—41no—
88C8×C53—41no—
89#88322—41no—
90#88322—41no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53C2×C22⋊C411yescharacteristic
94C22×C53#13111yescharacteristic
95C22×C53#13111yescharacteristic
96C2×D53—21no—
97C2×D53—21no—
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—21no—
101C2×D53—42no—
102C2×D53—42no—
103C2×D53#10112yes—
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—41no—
107C4×C53—41no—
108Dic53—21no—
109Dic53—41no—
110Dic53—41no—
111Dic53#9912yes—
112#1472—531nomaximal, Sylow
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#40811yescharacteristic
116C2×C53C2×#9911yescharacteristic
117C2×C53C2×#10111yescharacteristic
118D53—22no—
119D53—41no—
120C2×C22⋊Q8—531no—
121#407—531no—
122C2×C22⋊C8—531no—
123#324—531no—
124#324—531no—
125#324—531no—
126#324D5311yescharacteristic, complemented
127C53#147211yescharacteristic, complemented, Sylow
128C22×Q8—531no—
129C23×C4—531no—
130C2×M16—1061no—
131C22×C8—1061no—
132C22⋊Q8—1061no—
133C22⋊Q8—1061no—
134C22⋊Q8—1061no—
135C22⋊Q8—1061no—
136C2×C4⋊C4—531no—
137C2×C4⋊C4—531no—
138C2×C4⋊C4C2×D5311yescharacteristic
139C2×C22⋊C4—531no—
140C2×C22⋊C4—531no—
141C22⋊C8—531no—
142C22⋊C8—531no—
143C22⋊C8—1061no—
144C22⋊C8—1061no—
145C22⋊C8C2×D5311yescharacteristic
146C22⋊C8C2×D5311yescharacteristic
147C24—531no—
148C2×Q8—1061no—
149C2×Q8—1061no—
150C2×Q8—2122no—
151C22×C4—531no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—531no—
157C22×C4—531no—
158C22×C4—2122no—
159C22×C4C4×D5311yescharacteristic
160C22×C4C4×D5311yescharacteristic
161C22×C4C22×D5311yescharacteristic
162M16—2122no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1061no—
166C2×C8—1061no—
167C2×C8—2122no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—1061no—
171C4⋊C4—1061no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C22⋊C4—1061no—
175C22⋊C4—1061no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C23—531no—
179C23—531no—
180C23—1062no—
181C23—1061no—
182C23—1061no—
183C23—2121no—
184C23C2×C4×D5311yescharacteristic
185Q8—2122no—
186Q8—2122no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—531no—
194C2×C4—531no—
195C2×C4—1062no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—2121no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4D4×D5311yescharacteristic
206C2×C4D4×D5311yescharacteristic
207C8—41no—
208C8—41no—
209C8—2121no—
210C8—2121no—
211C22—21no—
212C22—21no—
213C22—532no—
214C22—1061no—
215C22—1061no—
216C22—1061no—
217C22—1061no—
218C22—1061no—
219C22—2122no—
220C22—2122no—
221C22C22⋊C4×D5311yescharacteristic
222C22#1313322311yescharacteristic
223C22#1313322511yescharacteristic
224C4—21no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—532no—
229C4—1061no—
230C4—2121no—
231C4—2121no—
232C2—21no—
233C2—21no—
234C2—1062no—
235C2—2121no—
236C2#1681058811yescharacteristic
237C2#99×D5311yescharacteristic
238C2#101×D5311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#324C2
socleC22×C53C2×C22⋊C4
hypercenter#324D53
hypocenterC53#1472
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
4. #16810349C2semidirect
5. #16810348C2semidirect
6. #16810348C2semidirect
7. C53×#324C2semidirect
126. #324D53semidirect
127. C53#1472semidirect