Finite Group Databasebeta

← #20273259

Subgroups — #20273259

Order 6784.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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#16810483C211yesmaximal, characteristic, complemented
3#16810361C212yesmaximal, complemented
4#16810358C212yesmaximal, complemented
5C53×#407C211yesmaximal, characteristic, complemented
6C2×#11719945C411yescharacteristic, complemented
7#13133200—21no—
8#13133196C411yescharacteristic, complemented
9#13133196C2211yescharacteristic
10#13133191C411yescharacteristic, complemented
11#13133191C2211yescharacteristic
12#13133186—21no—
13C4×C22⋊D53—21no—
14C4×C22⋊D53—21no—
15#13133176C411yescharacteristic, complemented
16C2×C4⋊Dic53—21no—
17C2×C22⋊Dic53—21no—
18#13133124C2212yes—
19C23×C4×C53C2211yescharacteristic
20C2×M16×C53—21no—
21C22⋊C8×C53—21no—
22C22⋊C8×C53C2212yes—
23C24×C53D411yescharacteristic
24C22×C4×C53—42no—
25C22×C4×C53C2×C411yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53D411yescharacteristic
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C22×Dic53—22no—
35#11719945—42no—
36C2×D212C2×C411yescharacteristic
37C2×C4×D53—21no—
38C2×Dic106C2×C411yescharacteristic
39M16×C53—42no—
40C2×C8×C53—22no—
41C22⋊Dic53—21no—
42C22⋊Dic53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—42no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53—42no—
48C4⋊Dic53C2×C411yescharacteristic
49C4⋊Dic53C2×C411yescharacteristic
50#11719919—21no—
51#11719919—21no—
52C4×Dic53—21no—
53C23×C53—21no—
54C23×C53—21no—
55C23×C53—41no—
56C23×C53—41no—
57C23×C53C22⋊C411yescharacteristic
58C23×C53C22⋊C411yescharacteristic
59C23×C53C2×D411yescharacteristic
60C22×D53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53C22⋊C411yescharacteristic
68C2×C4×C53C22⋊C411yescharacteristic
69C2×C4×C53C22×C411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C22⋊D53—42no—
72C22⋊D53—42no—
73C22⋊D53—41no—
74C22⋊D53—41no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—42no—
79C2×Dic53—42no—
80C2×Dic53—42no—
81D212—41no—
82C4×D53—42no—
83Dic106—41no—
84C8×C53—42no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53C2×C22⋊C411yescharacteristic
96C22×C53AGL(1,ℤ/8)11yescharacteristic
97C22×C53#13711yescharacteristic
98C2×D53—41no—
99C2×D53—41no—
100C2×D53—41no—
101C4×C53—21no—
102C4×C53—21no—
103C4×C53—21no—
104C4×C53#9912yes—
105Dic53—41no—
106Dic53—41no—
107Dic53—41no—
108Dic53—82no—
109#1513—531nomaximal, Sylow
110C2×C53—21no—
111C2×C53—21no—
112C2×C53—41no—
113C2×C53—41no—
114C2×C53C2×#9911yescharacteristic
115C2×C53#41711yescharacteristic
116C2×C53#42111yescharacteristic
117D53—81no—
118#525—531no—
119#407D5311yescharacteristic, complemented
120#380—531no—
121#328—532no—
122#327—532no—
123C53#151311yescharacteristic, complemented, Sylow
124C2×#41—531no—
125C23×C4C2×D5311yescharacteristic
126C2×M16—21no—
127#123—1061no—
128C22⋊Q8—531no—
129C22⋊Q8—531no—
130#121—531no—
131#121—531no—
132#120—1061no—
133C4×D4—1061no—
134C4×D4—1061no—
135#117—531no—
136C2×C4⋊C4—1061no—
137C2×C22⋊C4—1061no—
138C22⋊C8—21no—
139C22⋊C8C2×D5312yes—
140#95—532no—
141C24D21211yescharacteristic
142#41—2122no—
143C2×Q8—531no—
144C2×D4—531no—
145C2×D4—1061no—
146C2×D4—1061no—
147C2×D4—1061no—
148C22×C4—42no—
149C22×C4—1061no—
150C22×C4—1062no—
151C22×C4C4×D5311yescharacteristic
152C22×C4C4×D5311yescharacteristic
153C22×C4D21211yescharacteristic
154C22×C4C22⋊D5311yescharacteristic
155C22×C4C22⋊D5311yescharacteristic
156C22×C4C22×D5311yescharacteristic
157M16—42no—
158C2×C8—22no—
159C4⋊C4—531no—
160C4⋊C4—531no—
161C4⋊C4—1061no—
162C4⋊C4—1061no—
163C4⋊C4—2122no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4—1061no—
169C22⋊C4—2122no—
170C42—1061no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—1061no—
176C23#1171992111yescharacteristic
177C23#1171992111yescharacteristic
178C23C2×D21211yescharacteristic
179Q8—2121no—
180D4—2121no—
181D4—2122no—
182D4—2122no—
183D4—2121no—
184D4—2121no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4#1171992111yescharacteristic
199C2×C4#1171992111yescharacteristic
200C2×C4C2×C4×D5311yescharacteristic
201C2×C4C2×C22⋊D5311yescharacteristic
202C8—42no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—2121no—
214C22—2121no—
215C22—2121no—
216C22C2×#1171992111yescharacteristic
217C22#1313324511yescharacteristic
218C22#1313325011yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—4242no—
223C4—2121no—
224C4—2121no—
225C4—2121no—
226C4#1313313112yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—4241no—
232C2C2×#1313313111yescharacteristic
233C2#1681062511yescharacteristic
234C2#1681063911yescharacteristic
subgroupG/H
centerC22C2×#11719921
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#407C2
socleC22×C53C2×C22⋊C4
hypercenter#407D53
hypocenterC53#1513
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #16810483C2semidirect
3. #16810361C2semidirect
4. #16810358C2semidirect
5. C53×#407C2semidirect
6. C2×#11719945C4semidirect
8. #13133196C4semidirect
10. #13133191C4semidirect
15. #13133176C4semidirect
119. #407D53semidirect
123. C53#1513semidirect