Finite Group Databasebeta

← #20273381

Subgroups — #20273381

Order 6784.

344 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 75 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#127×D53C211yesmaximal, characteristic
2#16810675C211yesmaximal, characteristic, complemented
3#16810645C211yesmaximal, characteristic, complemented
4#16810367C211yesmaximal, characteristic, complemented
5#16810366C211yesmaximal, characteristic, complemented
6#16810366C211yesmaximal, characteristic, complemented
7C53×#331C211yesmaximal, characteristic, complemented
8C2×D4×D53—21no—
9C2×D4×D53—21no—
10C2×D4×D53C412yes—
11C2×#11719912—21no—
12#13133214C2211yescharacteristic
13#13133213—21no—
14#13133213—21no—
15#13133213C412yes—
16C42×D53C2211yescharacteristic
17C4⋊#88322C2211yescharacteristic, complemented
18C4⋊#88322C2211yescharacteristic, complemented
19#13133146—21no—
20#13133129—21no—
21C8×Dic53—21no—
22#127×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic, complemented
24C4⋊C8×C53C2211yescharacteristic, complemented
25C23×D53—22no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53C2×C412yes—
29C2×C22⋊D53—22no—
30C2×C22⋊D53—22no—
31C2×C22⋊D53—22no—
32D4×D53—42no—
33D4×D53—42no—
34D4×D53—44no—
35D4×D53—42no—
36D4×D53—42no—
37C2×D212—21no—
38C2×D212—21no—
39C2×D212C2×C412yes—
40C2×C4×D53C2×C411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C4×D53D411yescharacteristic
43C2×C8×C53—21no—
44C2×C8×C53—21no—
45C42×C53C2311yescharacteristic
46C4×Dic53C2×C411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53D411yescharacteristic
49C2×#88322—21no—
50C2×#88322—21no—
51#11719912—42no—
52C23×C53—22no—
53C22×D53—22no—
54C22×D53—44no—
55C22×D53—44no—
56C22×D53—44no—
57C22×D53C22⋊C411yescharacteristic
58D4×C53—42no—
59D4×C53—41no—
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—41no—
63C2×C4×C53C22×C411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C22⋊D53—44no—
67C22⋊D53—44no—
68C22⋊D53—44no—
69C2×Dic53C22⋊C411yescharacteristic
70C2×Dic53C22⋊C411yescharacteristic
71C2×Dic53C22⋊C411yescharacteristic
72D212—42no—
73D212—41no—
74D212—41no—
75D212—41no—
76D212—41no—
77C4×D53—21no—
78C4×D53—21no—
79C4×D53—22no—
80C4×D53D812yes—
81C4×D53SD1612yes—
82C8×C53—41no—
83C8×C53—41no—
84#88322—41no—
85#88322—41no—
86C22×C53—42no—
87C22×C53—42no—
88C22×C53—42no—
89C22×C53C2×C22⋊C411yescharacteristic
90C2×D53—21no—
91C2×D53—21no—
92C2×D53—42no—
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—88no—
96C2×D53#10212yes—
97C4×C53—21no—
98C4×C53C2×D811yescharacteristic
99C4×C53C2×SD1611yescharacteristic
100C4×C53AGL(1,ℤ/8)11yescharacteristic
101C4×C53AGL(1,ℤ/8)11yescharacteristic
102Dic53—21no—
103Dic53#10012yes—
104Dic53#10212yes—
105#1545—531nomaximal, Sylow
106C2×C53—82no—
107C2×C53C2×#10011yescharacteristic
108C2×C53C2×#10211yescharacteristic
109C2×C53#41811yescharacteristic
110D53—22no—
111D53—82no—
112C2×#127—531no—
113#431—531no—
114#423—531no—
115#331—531no—
116#331—531no—
117#331—531no—
118#331D5311yescharacteristic, complemented
119C53#154511yescharacteristic, complemented, Sylow
120C22×D4—532no—
121C22×D4—1061no—
122C22×D4—1061no—
123C2×M16—1061no—
124#127—532no—
125#127—531no—
126#127—1061no—
127#127—1061no—
128#127C2×D5311yescharacteristic
129C2×C42—531no—
130C4⋊C8—531no—
131C4⋊C8—531no—
132C4⋊C8—1061no—
133C4⋊C8C2×D5311yescharacteristic, complemented
134C4⋊C8C2×D5311yescharacteristic, complemented
135C22⋊C8—1061no—
136C4×C8—1061no—
137C24—1062no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—532no—
141C2×D4—1062no—
142C2×D4—1062no—
143C2×D4—1062no—
144C2×D4—1061no—
145C2×D4—1061no—
146C2×D4—2122no—
147C2×D4—2122no—
148C2×D4—2122no—
149C2×D4—2122no—
150C2×D4—2124no—
151C2×D4C4×D5312yes—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—531no—
155M16—2122no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—1061no—
159C2×C8—1061no—
160C42—531no—
161C42—531no—
162C42—531no—
163C42C22×D5311yescharacteristic
164C23—22no—
165C23—531no—
166C23—1062no—
167C23—2124no—
168C23—2124no—
169C23—2124no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—2122no—
176D4—2124no—
177D4—2124no—
178D4—2124no—
179D4—2121no—
180D4—2121no—
181D4—2121no—
182D4—2121no—
183C2×C4—531no—
184C2×C4—531no—
185C2×C4—532no—
186C2×C4—532no—
187C2×C4—531no—
188C2×C4—1062no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4C2×C4×D5311yescharacteristic
192C2×C4D4×D5311yescharacteristic
193C2×C4D4×D5311yescharacteristic
194C8—41no—
195C8—41no—
196C8—2121no—
197C8—2121no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—532no—
202C22—1061no—
203C22—1061no—
204C22—4248no—
205C22—2122no—
206C22—2122no—
207C22—2122no—
208C22C22⋊C4×D5311yescharacteristic
209C4—21no—
210C4—532no—
211C4—532no—
212C4—1061no—
213C4D8×D5311yescharacteristic
214C4SD16×D5311yescharacteristic
215C4#1313324111yescharacteristic
216C4#1313324411yescharacteristic
217C2—82no—
218C2—1062no—
219C2—4242no—
220C2#100×D5311yescharacteristic
221C2#102×D5311yescharacteristic
222C2#1681062611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC42C22×D53
FittingC53×#331C2
socleC22×C53C2×C22⋊C4
hypercenter#331D53
hypocenterC53#1545
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #16810675C2semidirect
3. #16810645C2semidirect
4. #16810367C2semidirect
5. #16810366C2semidirect
6. #16810366C2semidirect
7. C53×#331C2semidirect
17. C4⋊#88322C22semidirect
18. C4⋊#88322C22semidirect
23. C4⋊C8×C53C22semidirect
24. C4⋊C8×C53C22semidirect
118. #331D53semidirect
119. C53#1545semidirect
133. C4⋊C8C2×D53semidirect
134. C4⋊C8C2×D53semidirect