Finite Group Databasebeta

← #20273833

Subgroups — #20273833

Order 6784.

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

normal subgroups only

This group has 314 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#16810996C211yesmaximal, characteristic, complemented
3#16810914C211yesmaximal, characteristic, complemented
4#16810913C211yesmaximal, characteristic, complemented
5#16810366C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C22×D212—21no—
9C2×D4×D53—21no—
10C2×#11719913—22no—
11#13133217C2211yescharacteristic, complemented
12#13133214C2211yescharacteristic
13#13133192—21no—
14#13133191—21no—
15#13133187—22no—
16C4×D212C2211yescharacteristic
17#13133149—21no—
18#13133148—21no—
19C4⋊#88322C2211yescharacteristic, complemented
20#13133139—21no—
21#13133138—21no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic, complemented
24C4×C8×C53C2211yescharacteristic, complemented
25C23×D53—22no—
26C2×D4×C53D411yescharacteristic
27C22×C4×C53—21no—
28C2×C22⋊D53—22no—
29D4×D53—42no—
30D4×D53—42no—
31C2×D212—22no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—42no—
35C2×D212—42no—
36C2×D212D411yescharacteristic
37C2×D212D412yes—
38C2×C4×D53—21no—
39C2×Dic106—21no—
40C2×C8×C53—21no—
41C2×C8×C53—21no—
42C4⋊C4×C53D411yescharacteristic
43C22⋊C4×C53—21no—
44C42×C53C2311yescharacteristic
45#11719921—22no—
46#11719921—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53—21no—
49C4⋊Dic53D411yescharacteristic
50C2×#88322—21no—
51#11719913—44no—
52C23×C53—21no—
53C22×D53—21no—
54C22×D53—22no—
55C22×D53—44no—
56C22×D53—44no—
57C22×D53—82no—
58D4×C53—21no—
59D4×C53SD1612yescomplemented
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C22⋊D53—44no—
67C2×Dic53—21no—
68C2×Dic53—41no—
69D212—21no—
70D212—42no—
71D212—44no—
72D212—41no—
73D212—41no—
74D212—41no—
75D212—41no—
76D212SD1612yescomplemented
77C4×D53—41no—
78C4×D53—41no—
79Dic106—42no—
80C8×C53—22no—
81C8×C53—41no—
82#88322—41no—
83C22×C53—22no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53#12011yescharacteristic
87C2×D53—22no—
88C2×D53—41no—
89C2×D53—41no—
90C2×D53—42no—
91C2×D53—42no—
92C2×D53—42no—
93C2×D53—88no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53C2×SD1611yescharacteristic
97C4×C53C2×SD1611yescharacteristic
98C4×C53AGL(1,ℤ/8)11yescharacteristic
99C4×C53AGL(1,ℤ/8)11yescharacteristic
100Dic53—41no—
101Dic53—81no—
102#1629—531nomaximal, Sylow
103C2×C53—42no—
104C2×C53#45011yescharacteristic
105C2×C53#45011yescharacteristic
106C2×C53#45311yescharacteristic
107D53—42no—
108D53—82no—
109D42—531no—
110#492—531no—
111#476—531no—
112#476—531no—
113#331—531no—
114D4⋊C8—531no—
115D4⋊C8D5311yescharacteristic, complemented
116C53#162911yescharacteristic, complemented, Sylow
117C22×D4—1061no—
118C22×D4—1061no—
119C2×SD16—1062no—
120C4⋊Q8—531no—
121#127—531no—
122#121—1061no—
123#121—1061no—
124#120—1062no—
125C4×D4—531no—
126C4×D4C2×D5311yescharacteristic
127#106—1061no—
128#106—1061no—
129C4⋊C8—531no—
130C4⋊C8C2×D5311yescharacteristic, complemented
131#102—1061no—
132#102—1061no—
133C4×C8C2×D5311yescharacteristic, complemented
134C24—1062no—
135C2×Q8—1061no—
136C2×D4—531no—
137C2×D4—532no—
138C2×D4—1062no—
139C2×D4—1062no—
140C2×D4—1061no—
141C2×D4—1061no—
142C2×D4—2122no—
143C2×D4—2122no—
144C2×D4—2122no—
145C2×D4—2122no—
146C2×D4D21211yescharacteristic
147C22×C4—21no—
148C22×C4—1061no—
149SD16—2124no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—1061no—
153C4⋊C4—531no—
154C4⋊C4—1061no—
155C4⋊C4—1061no—
156C4⋊C4D21211yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—1062no—
159C22⋊C4—1061no—
160C42C22×D5311yescharacteristic
161C23—21no—
162C23—1061no—
163C23—1062no—
164C23—4242no—
165C23—2124no—
166C23—2124no—
167Q8—2122no—
168D4—21no—
169D4—532no—
170D4—1061no—
171D4—2122no—
172D4—2124no—
173D4—2124no—
174D4—2121no—
175D4—2121no—
176D4—2121no—
177D4—2121no—
178D4#1171991312yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—1061no—
183C2×C4—2121no—
184C2×C4—2121no—
185C2×C4—2121no—
186C2×C4C2×D21211yescharacteristic
187C2×C4D4×D5311yescharacteristic
188C2×C4D4×D5311yescharacteristic
189C8—22no—
190C8—41no—
191C8—2121no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—1062no—
196C22—4248no—
197C22—2122no—
198C22—2122no—
199C22—2122no—
200C22—2121no—
201C22—2121no—
202C22#1313318711yescharacteristic
203C4—21no—
204C4—41no—
205C4—4241no—
206C4—2121no—
207C4SD16×D5311yescharacteristic
208C4C2×#1171991311yescharacteristic
209C4#1313324111yescharacteristic
210C4#1313324511yescharacteristic
211C2—42no—
212C2—4242no—
213C2—2122no—
214C2#1681077511yescharacteristic
215C2#1681077611yescharacteristic
216C2#1681079211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1629
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #16810996C2semidirect
3. #16810914C2semidirect
4. #16810913C2semidirect
5. #16810366C2semidirect
6. #16810352C2semidirect
7. C53×D4⋊C8C2semidirect
11. #13133217C22semidirect
19. C4⋊#88322C22semidirect
23. C4⋊C8×C53C22semidirect
24. C4×C8×C53C22semidirect
59. D4×C53SD16semidirect
76. D212SD16semidirect
115. D4⋊C8D53semidirect
116. C53#1629semidirect
130. C4⋊C8C2×D53semidirect
133. C4×C8C2×D53semidirect
178. D4#11719913semidirect