Finite Group Databasebeta

← #20273782

Subgroups — #20273782

Order 6784.

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

normal subgroups only

This group has 278 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#16811295C211yesmaximal, characteristic, complemented
2#16811000C211yesmaximal, characteristic, complemented
3#16810823C211yesmaximal, characteristic, complemented
4#16810819C211yesmaximal, characteristic, complemented
5#16810366C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719949—21no—
9C2×#11719924—21no—
10C2×D424—21no—
11C2×D424—22no—
12#13133214—21no—
13#13133214C2211yescharacteristic
14#13133214C2211yescharacteristic
15#13133192—21no—
16#13133192—22no—
17C4×D212—21no—
18C4×D212C2211yescharacteristic
19C4⋊#88322C2211yescharacteristic
20#13133139—21no—
21#13133137—21no—
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53D411yescharacteristic
26#11719949—42no—
27#11719949—42no—
28C2×D212—22no—
29C2×D212—21no—
30C2×D212—21no—
31C2×D212—21no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—22no—
35C2×D212D411yescharacteristic
36C2×D212D412yes—
37C2×C4×D53—22no—
38C2×C4×D53—21no—
39C2×C8×C53—21no—
40C2×C8×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53D411yescharacteristic
43C42×C53—21no—
44C42×C53C2311yescharacteristic
45#11719924—42no—
46#11719921—22no—
47#11719921—21no—
48C4⋊Dic53D411yescharacteristic
49C2×#88322—21no—
50D424—44no—
51D424—42no—
52C22×D53—21no—
53C22×D53—22no—
54C22×D53—41no—
55Q8×C53—21no—
56Q8×C53D812yescomplemented
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C2×Dic53—21no—
63D212—21no—
64D212—21no—
65D212—42no—
66D212—44no—
67D212—42no—
68D212—41no—
69D212—41no—
70D212—41no—
71D212—41no—
72D212—41no—
73D212—41no—
74D212—41no—
75D212—41no—
76D212—82no—
77C4×D53—44no—
78C4×D53—41no—
79C4×D53—41no—
80C8×C53—22no—
81C8×C53—41no—
82#88322—41no—
83C22×C53#12011yescharacteristic
84C2×D53—42no—
85C2×D53—41no—
86C2×D53—41no—
87C2×D53—42no—
88C2×D53—42no—
89C2×D53—41no—
90C2×D53—81no—
91C2×D53—81no—
92C2×D53—81no—
93C4×C53—22no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53C2×D811yescharacteristic
97C4×C53#13511yescharacteristic
98C4×C53AGL(1,ℤ/8)11yescharacteristic
99C4×C53AGL(1,ℤ/8)11yescharacteristic
100Dic53—41no—
101#1621—531nomaximal, Sylow
102C2×C53#44711yescharacteristic
103C2×C53#44911yescharacteristic
104C2×C53#45311yescharacteristic
105D53—81no—
106D53—82no—
107D53—161no—
108#550—531no—
109#493—531no—
110#460—531no—
111#459—531no—
112#331—531no—
113Q8⋊C8D5311yescharacteristic, complemented
114D4⋊C8—531no—
115C53#162111yescharacteristic, complemented, Sylow
116C2×#41—1061no—
117C2×SD16—1061no—
118C2×D8—1061no—
119C2×D8—1062no—
120#127—531no—
121#127—531no—
122#127—1061no—
123#121—1062no—
124#121—1061no—
125C4×Q8C2×D5311yescharacteristic
126C4×D4—531no—
127C4×D4—1061no—
128C4⋊C8—531no—
129C4⋊C8C2×D5311yescharacteristic
130#102—1061no—
131#102—1061no—
132C4×C8C2×D5311yescharacteristic
133#41—2122no—
134#41—2122no—
135C2×Q8D21211yescharacteristic
136C2×D4—531no—
137C2×D4—532no—
138C2×D4—1062no—
139C2×D4—1062no—
140C2×D4—1061no—
141C2×D4—1061no—
142C2×D4—1061no—
143C2×D4—1061no—
144C2×D4—1061no—
145C22×C4—1062no—
146C22×C4—1061no—
147SD16—2122no—
148D8—2124no—
149D8—2122no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—1061no—
153C4⋊C4—21no—
154C4⋊C4—531no—
155C4⋊C4D21211yescharacteristic
156C22⋊C4—1062no—
157C22⋊C4—1061no—
158C42—21no—
159C42C22×D5311yescharacteristic
160C23—1061no—
161C23—1062no—
162C23—2121no—
163Q8—21no—
164Q8D42412yescomplemented
165D4—1061no—
166D4—1061no—
167D4—4242no—
168D4—2122no—
169D4—2122no—
170D4—2124no—
171D4—2121no—
172D4—2121no—
173D4—2121no—
174D4—2121no—
175D4—2121no—
176D4—2121no—
177D4—2121no—
178D4—2121no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—1061no—
182C2×C4—2124no—
183C2×C4—2121no—
184C2×C4—2121no—
185C2×C4C2×D21211yescharacteristic
186C2×C4D4×D5311yescharacteristic
187C2×C4D4×D5311yescharacteristic
188C8—22no—
189C8—41no—
190C8—2121no—
191C22—4241no—
192C22—4241no—
193C22—4241no—
194C22—2122no—
195C22—2121no—
196C22—2121no—
197C22—2122no—
198C22—2121no—
199C22—2122no—
200C22#1313318711yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—2121no—
205C4C2×D42411yescharacteristic
206C4#1313324011yescharacteristic
207C4#1313324411yescharacteristic
208C4#1313324511yescharacteristic
209C2—8481no—
210C2—4241no—
211C2—4242no—
212C2#1681075911yescharacteristic
213C2#1681077111yescharacteristic
214C2#1681079211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×Q8⋊C8C2
socleC22×C53#120
hypercenterQ8⋊C8D53
hypocenterC53#1621
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #16811295C2semidirect
2. #16811000C2semidirect
3. #16810823C2semidirect
4. #16810819C2semidirect
5. #16810366C2semidirect
6. #16810352C2semidirect
7. C53×Q8⋊C8C2semidirect
56. Q8×C53D8semidirect
113. Q8⋊C8D53semidirect
115. C53#1621semidirect
164. Q8D424semidirect