Finite Group Databasebeta

← #20273217

Subgroups — #20273217

Order 6784.

334 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 77 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#16811134C211yesmaximal, characteristic
2#16810587C211yesmaximal, characteristic, complemented
3#16810359C212yesmaximal, complemented
4#16810356C212yesmaximal, complemented
5C2×#95×C53C211yesmaximal, characteristic, complemented
6C2×#11719945C411yescharacteristic, complemented
7C2×#11719917—21no—
8#13133200—21no—
9#13133196C411yescharacteristic, complemented
10#13133196C2211yescharacteristic
11#13133191C411yescharacteristic, complemented
12#13133191C2211yescharacteristic
13#13133186—21no—
14C4×C22⋊D53—21no—
15C4×C22⋊D53—21no—
16#13133176C411yescharacteristic, complemented
17C22⋊#88322—21no—
18C22⋊#88322C2212yes—
19C23×C4×C53—21no—
20C23×C4×C53C2211yescharacteristic
21#95×C53—21no—
22#95×C53C2212yes—
23C24×C53D411yescharacteristic
24C22×C4×C53—24no—
25C22×C4×C53—22no—
26C22×C4×C53—42no—
27C22×C4×C53—42no—
28C22×C4×C53—42no—
29C22×C4×C53C2×C411yescharacteristic
30C22×C4×C53C2×C411yescharacteristic
31C22×C4×C53D411yescharacteristic
32C22×C4×C53D411yescharacteristic
33C22×C4×C53D411yescharacteristic
34C22×C4×C53C2311yescharacteristic
35C2×C22⋊D53—21no—
36C2×C22⋊D53—21no—
37C2×C22⋊D53—21no—
38#11719945—42no—
39C2×D212C2×C411yescharacteristic, complemented
40C2×C4×D53—21no—
41C2×Dic106C2×C411yescharacteristic, complemented
42C22⋊Dic53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53C2×C411yescharacteristic, complemented
48C4⋊Dic53C2×C411yescharacteristic, complemented
49#11719919—21no—
50#11719919—21no—
51C4×Dic53—21no—
52#11719917—42no—
53C2×#88322—22no—
54C23×C53—22no—
55C23×C53—22no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53C22⋊C411yescharacteristic
59C23×C53C22⋊C411yescharacteristic
60C23×C53C2×D411yescharacteristic
61C22×D53—21no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—24no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53—48no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53C22⋊C411yescharacteristic
73C2×C4×C53C22⋊C411yescharacteristic
74C2×C4×C53C22×C411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C22⋊D53—42no—
77C22⋊D53—42no—
78C22⋊D53—41no—
79C22⋊D53—41no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83D212—41no—
84C4×D53—42no—
85Dic106—41no—
86#88322—42no—
87C22×C53—24no—
88C22×C53—24no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53#9912yes—
96C22×C53#10412yes—
97C22×C53C2×C22⋊C411yescharacteristic
98C22×C53AGL(1,ℤ/8)11yescharacteristic
99C22×C53#13711yescharacteristic
100C2×D53—41no—
101C2×D53—41no—
102C2×D53—41no—
103C4×C53—21no—
104C4×C53—21no—
105C4×C53—21no—
106C4×C53—21no—
107C4×C53—44no—
108Dic53—41no—
109Dic53—41no—
110Dic53—41no—
111#1504—531nomaximal, Sylow
112C2×C53—21no—
113C2×C53—22no—
114C2×C53—22no—
115C2×C53—21no—
116C2×C53C2×#9911yescharacteristic
117C2×C53#41711yescharacteristic
118C2×C53C2×#10411yescharacteristic
119D53—81no—
120#525—531no—
121#407—531no—
122C2×#95D5311yescharacteristic, complemented
123#328—532no—
124#327—532no—
125C53#150411yescharacteristic, complemented, Sylow
126C2×#41—531no—
127C23×C4—21no—
128C23×C4C2×D5311yescharacteristic
129C2×M16—1061no—
130#123—1061no—
131C22⋊Q8—531no—
132C22⋊Q8—531no—
133#121—531no—
134#121—531no—
135#120—1061no—
136C4×D4—1061no—
137C4×D4—1061no—
138#117—531no—
139C22⋊C8—532no—
140C22⋊C8—1061no—
141#95—21no—
142#95C2×D5312yes—
143C24C22⋊D5311yescharacteristic
144#41—2122no—
145C2×Q8—531no—
146C2×D4—531no—
147C2×D4—1061no—
148C2×D4—1061no—
149C2×D4—1061no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—1061no—
156C22×C4C4×D5311yescharacteristic
157C22×C4C4×D5311yescharacteristic
158C22×C4D21211yescharacteristic
159C22×C4D21211yescharacteristic
160C22×C4C22⋊D5311yescharacteristic
161C22×C4C22×D5311yescharacteristic
162M16—2122no—
163C2×C8—1062no—
164C4⋊C4—531no—
165C4⋊C4—531no—
166C4⋊C4—1061no—
167C4⋊C4—1061no—
168C22⋊C4—1061no—
169C22⋊C4—1061no—
170C22⋊C4—1061no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C42—1061no—
174C23—22no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—1061no—
179C23#1171992111yescharacteristic
180C23#1171992111yescharacteristic
181C23C2×C22⋊D5311yescharacteristic
182Q8—2121no—
183D4—2121no—
184D4—2122no—
185D4—2122no—
186D4—2121no—
187D4—2121no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—48no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—2122no—
202C2×C4#1171992111yescharacteristic
203C2×C4#1171992111yescharacteristic
204C2×C4C2×C4×D5311yescharacteristic
205C2×C4C2×D21211yescharacteristic
206C8—2122no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—2121no—
216C22—2121no—
217C22—2121no—
218C22#1313313112yes—
219C22#1313314512yes—
220C22C2×#1171992111yescharacteristic
221C22#1313324311yescharacteristic
222C22#1313324811yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—44no—
228C4—2121no—
229C4—2121no—
230C4—2121no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2—21no—
235C2—4241no—
236C2C2×#1313313111yescharacteristic
237C2#1681062211yescharacteristic
238C2C2×#1313314511yescharacteristic
subgroupG/H
centerC22C2×#11719921
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×#95×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#95D53
hypocenterC53#1504
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. #16810587C2semidirect
3. #16810359C2semidirect
4. #16810356C2semidirect
5. C2×#95×C53C2semidirect
6. C2×#11719945C4semidirect
9. #13133196C4semidirect
11. #13133191C4semidirect
16. #13133176C4semidirect
39. C2×D212C2×C4semidirect
41. C2×Dic106C2×C4semidirect
47. C4⋊Dic53C2×C4semidirect
48. C4⋊Dic53C2×C4semidirect
122. C2×#95D53semidirect
125. C53#1504semidirect