Finite Group Databasebeta

← #20273895

Subgroups — #20273895

Order 6784.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#16811258C211yesmaximal, characteristic
2#16810969C211yesmaximal, characteristic, complemented
3#16810904C211yesmaximal, characteristic, complemented
4#16810829C211yesmaximal, characteristic, complemented
5#16810350C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#331C211yesmaximal, characteristic, complemented
8C2×D4×D53—21no—
9C2×D4×D53—21no—
10C2×#11719913—21no—
11#13133217C2211yescharacteristic, complemented
12#13133188—21no—
13#13133188—21no—
14#13133185—22no—
15C4×D212C2211yescharacteristic
16C4×D212C2211yescharacteristic
17#13133151—21no—
18#13133142—21no—
19#13133139—21no—
20#13133138—22no—
21C4×#88322C2211yescharacteristic, complemented
22#127×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic, complemented
24C4⋊C8×C53C2211yescharacteristic, complemented
25C23×D53—22no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53D412yes—
29C2×C22⋊D53—22no—
30C2×C22⋊D53—22no—
31D4×D53—42no—
32D4×D53—42no—
33D4×D53—42no—
34D4×D53—42no—
35C2×D212D411yescharacteristic
36C2×D212D411yescharacteristic
37C2×C4×D53—21no—
38C2×C4×D53—21no—
39C2×Dic106—21no—
40C2×C8×C53—21no—
41C2×C8×C53—21no—
42C42×C53C2311yescharacteristic
43C22⋊Dic53—22no—
44#11719921—21no—
45#11719921—21no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53D411yescharacteristic
49C4⋊Dic53D411yescharacteristic
50C2×#88322—21no—
51#11719913—42no—
52C23×C53—22no—
53C22×D53—21no—
54C22×D53—21no—
55C22×D53—44no—
56C22×D53—44no—
57C22×D53—82no—
58D4×C53—42no—
59D4×C53—41no—
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—41no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C22⋊D53—44no—
67C22⋊D53—44no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—41no—
71D212—21no—
72D212—21no—
73D212D812yes—
74D212SD1612yescomplemented
75C4×D53—41no—
76C4×D53—41no—
77C4×D53—41no—
78C4×D53—41no—
79Dic106—41no—
80Dic106—41no—
81C8×C53—41no—
82C8×C53—41no—
83#88322—41no—
84C22×C53—42no—
85C22×C53—42no—
86C22×C53—42no—
87C22×C53#12011yescharacteristic
88C2×D53—22no—
89C2×D53—22no—
90C2×D53—41no—
91C2×D53—41no—
92C2×D53—41no—
93C2×D53—41no—
94C2×D53—88no—
95C4×C53—21no—
96C4×C53C2×D811yescharacteristic
97C4×C53C2×SD1611yescharacteristic
98C4×C53AGL(1,ℤ/8)11yescharacteristic
99C4×C53AGL(1,ℤ/8)11yescharacteristic
100Dic53—41no—
101Dic53—41no—
102Dic53—81no—
103#1639—531nomaximal, Sylow
104C2×C53—82no—
105C2×C53#44711yescharacteristic
106C2×C53#45011yescharacteristic
107C2×C53#45511yescharacteristic
108D53—42no—
109D53—42no—
110D42—531no—
111#486—531no—
112#474—531no—
113#461—531no—
114#331D5311yescharacteristic, complemented
115D4⋊C8—531no—
116D4⋊C8—531no—
117C53#163911yescharacteristic, complemented, Sylow
118C22×D4—1061no—
119C22×D4—1061no—
120C2×SD16—1061no—
121C4⋊Q8—531no—
122#127C2×D5311yescharacteristic
123#121—1061no—
124#121—1061no—
125#120—1062no—
126C4×D4—531no—
127C4×D4—531no—
128#107—1061no—
129C4⋊C8C2×D5311yescharacteristic, complemented
130C4⋊C8C2×D5311yescharacteristic, complemented
131#103—1061no—
132#102—1061no—
133#102—1062no—
134C4×C8—531no—
135C24—1062no—
136C2×Q8—1061no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—531no—
140C2×D4—531no—
141C2×D4—1062no—
142C2×D4—1062no—
143C2×D4—2122no—
144C2×D4—2122no—
145C2×D4—2122no—
146C2×D4—2122no—
147C2×D4D21212yes—
148C22×C4—1061no—
149C22×C4—1061no—
150SD16—2122no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—1061no—
154C4⋊C4—531no—
155C4⋊C4—531no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C22⋊C4—1062no—
159C22⋊C4—1061no—
160C22⋊C4—1061no—
161C42C22×D5311yescharacteristic
162C23—22no—
163C23—1061no—
164C23—1061no—
165C23—4242no—
166C23—2124no—
167C23—2124no—
168Q8—2121no—
169Q8—2121no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—532no—
176D4—532no—
177D4—1061no—
178D4—1061no—
179D4—2124no—
180D4—2124no—
181C2×C4—1061no—
182C2×C4—1061no—
183C2×C4—2121no—
184C2×C4—2121no—
185C2×C4—2121no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4C2×D21211yescharacteristic
189C2×C4D4×D5311yescharacteristic
190C2×C4D4×D5311yescharacteristic
191C8—41no—
192C8—41no—
193C8—2121no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—1062no—
198C22—1062no—
199C22—4248no—
200C22—2121no—
201C22—2121no—
202C22—2121no—
203C22—2121no—
204C22#1313318711yescharacteristic
205C4—21no—
206C4—4241no—
207C4—2121no—
208C4—2121no—
209C4D8×D5311yescharacteristic
210C4SD16×D5311yescharacteristic
211C4#1313324111yescharacteristic
212C4#1313324411yescharacteristic
213C2—82no—
214C2—2122no—
215C2—2122no—
216C2#1681075811yescharacteristic
217C2#1681077511yescharacteristic
218C2#1681080311yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#331C2
socleC22×C53#120
hypercenter#331D53
hypocenterC53#1639
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #16810969C2semidirect
3. #16810904C2semidirect
4. #16810829C2semidirect
5. #16810350C2semidirect
6. #16810350C2semidirect
7. C53×#331C2semidirect
11. #13133217C22semidirect
21. C4×#88322C22semidirect
23. C4⋊C8×C53C22semidirect
24. C4⋊C8×C53C22semidirect
74. D212SD16semidirect
114. #331D53semidirect
117. C53#1639semidirect
129. C4⋊C8C2×D53semidirect
130. C4⋊C8C2×D53semidirect