Finite Group Databasebeta

← #20273893

Subgroups — #20273893

Order 6784.

308 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D212C211yesmaximal, characteristic
2#16810971C211yesmaximal, characteristic, complemented
3#16810825C211yesmaximal, characteristic, complemented
4#16810366C211yesmaximal, characteristic, complemented
5D4⋊#88322C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#474C211yesmaximal, characteristic, complemented
8C22×D212—21no—
9C2×D4×D53—21no—
10C2×#11719924—21no—
11#13133214C2211yescharacteristic
12#13133192—21no—
13#13133191—21no—
14#13133187—22no—
15C4×D212C2211yescharacteristic
16C4⋊#88322C2211yescharacteristic, complemented
17#13133141—21no—
18#13133137—22no—
19C4×#88322C2211yescharacteristic, complemented
20C4⋊Q8×C53C2211yescharacteristic, complemented
21C4×D4×C53C2211yescharacteristic
22#107×C53—21no—
23C4⋊C8×C53C2211yescharacteristic, complemented
24#102×C53—21no—
25C23×D53—22no—
26C2×Q8×C53—21no—
27C2×D4×C53D411yescharacteristic
28C22×C4×C53—21no—
29C2×C22⋊D53—22no—
30D4×D53—42no—
31D4×D53—42no—
32C2×D212—22no—
33C2×D212—21no—
34C2×D212—21no—
35C2×D212—42no—
36C2×D212—42no—
37C2×D212D411yescharacteristic
38C2×D212D412yes—
39C2×C4×D53—21no—
40C2×C8×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53D411yescharacteristic
44C22⋊C4×C53—21no—
45C42×C53C2311yescharacteristic
46#11719924—42no—
47#11719921—22no—
48#11719921—21no—
49C4⋊Dic53D411yescharacteristic
50C2×#88322—21no—
51C2×#88322—21no—
52C23×C53—21no—
53C22×D53—21no—
54C22×D53—22no—
55C22×D53—44no—
56C22×D53—44no—
57C22×D53—82no—
58Q8×C53—41no—
59Q8×C53—41no—
60D4×C53—21no—
61D4×C53SD1612yescomplemented
62C2×C4×C53—21no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C22⋊D53—44no—
70C2×Dic53—21no—
71D212—21no—
72D212—42no—
73D212—44no—
74D212—41no—
75D212—41no—
76D212—41no—
77D212—41no—
78D212D812yes—
79C4×D53—41no—
80C4×D53—41no—
81C8×C53—41no—
82#88322—41no—
83#88322—41no—
84C22×C53—22no—
85C22×C53—41no—
86C22×C53—41no—
87C22×C53#12011yescharacteristic
88C2×D53—22no—
89C2×D53—41no—
90C2×D53—41no—
91C2×D53—42no—
92C2×D53—42no—
93C2×D53—42no—
94C2×D53—88no—
95C4×C53—21no—
96C4×C53—41no—
97C4×C53—81no—
98C4×C53C2×D811yescharacteristic
99C4×C53C2×SD1611yescharacteristic
100C4×C53AGL(1,ℤ/8)11yescharacteristic
101C4×C53AGL(1,ℤ/8)11yescharacteristic
102Dic53—41no—
103#1639—531nomaximal, Sylow
104C2×C53—42no—
105C2×C53#44711yescharacteristic
106C2×C53#45011yescharacteristic
107C2×C53#45511yescharacteristic
108D53—42no—
109D53—82no—
110D42—531no—
111#486—531no—
112#474D5311yescharacteristic, complemented
113#461—531no—
114#331—531no—
115D4⋊C8—531no—
116D4⋊C8—531no—
117C53#163911yescharacteristic, complemented, Sylow
118C22×D4—1061no—
119C22×D4—1061no—
120C2×SD16—1061no—
121C4⋊Q8C2×D5311yescharacteristic, complemented
122#127—531no—
123#121—1061no—
124#121—1061no—
125#120—1062no—
126C4×D4—531no—
127C4×D4C2×D5311yescharacteristic
128#107—21no—
129C4⋊C8—531no—
130C4⋊C8C2×D5311yescharacteristic, complemented
131#103—1061no—
132#102—21no—
133#102—1062no—
134C4×C8—531no—
135C24—1062no—
136C2×Q8—21no—
137C2×D4—531no—
138C2×D4—532no—
139C2×D4—1062no—
140C2×D4—1062no—
141C2×D4—1061no—
142C2×D4—1061no—
143C2×D4—2122no—
144C2×D4—2122no—
145C2×D4—2122no—
146C2×D4—2122no—
147C2×D4C22⋊D5311yescharacteristic
148C22×C4—21no—
149C22×C4—1061no—
150SD16—2122no—
151C2×C8—21no—
152C2×C8—1061no—
153C2×C8—1061no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—531no—
157C4⋊C4C22⋊D5311yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—1062no—
160C22⋊C4—1061no—
161C42C22×D5311yescharacteristic
162C23—21no—
163C23—1061no—
164C23—1062no—
165C23—4242no—
166C23—2124no—
167C23—2124no—
168Q8—41no—
169Q8—41no—
170D4—21no—
171D4—532no—
172D4—1061no—
173D4—2122no—
174D4—2124no—
175D4—2124no—
176D4—2121no—
177D4—2121no—
178D4—2121no—
179D4—2121no—
180D4#1171992412yescomplemented
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—1061no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4D4×D5311yescharacteristic
189C2×C4D4×D5311yescharacteristic
190C2×C4C2×C22⋊D5311yescharacteristic
191C8—41no—
192C8—2121no—
193C8—2121no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—1062no—
198C22—4248no—
199C22—2122no—
200C22—2122no—
201C22—2122no—
202C22—2121no—
203C22—2121no—
204C22#1313318511yescharacteristic
205C4—21no—
206C4—41no—
207C4—81no—
208C4—2121no—
209C4D8×D5311yescharacteristic
210C4C2×#1171992411yescharacteristic
211C4#1313324111yescharacteristic
212C4#1313324211yescharacteristic
213C2—42no—
214C2—4242no—
215C2—2122no—
216C2#1681075511yescharacteristic
217C2#1681077311yescharacteristic
218C2#1681079911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#474C2
socleC22×C53#120
hypercenter#474D53
hypocenterC53#1639
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #16810971C2semidirect
3. #16810825C2semidirect
4. #16810366C2semidirect
5. D4⋊#88322C2semidirect
6. #16810350C2semidirect
7. C53×#474C2semidirect
16. C4⋊#88322C22semidirect
19. C4×#88322C22semidirect
20. C4⋊Q8×C53C22semidirect
23. C4⋊C8×C53C22semidirect
61. D4×C53SD16semidirect
112. #474D53semidirect
117. C53#1639semidirect
121. C4⋊Q8C2×D53semidirect
130. C4⋊C8C2×D53semidirect
180. D4#11719924semidirect