Finite Group Databasebeta

← #20273819

Subgroups — #20273819

Order 6784.

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

normal subgroups only

This group has 250 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#16811288C211yesmaximal, characteristic, complemented
2#16810971C211yesmaximal, characteristic, complemented
3#16810824C211yesmaximal, characteristic, complemented
4#16810819C211yesmaximal, characteristic
5#16810353C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#332C211yesmaximal, characteristic, complemented
8C2×Q8×D53—21no—
9C2×#11719913—21no—
10C2×D424—21no—
11#13133216—21no—
12#13133214C2211yescharacteristic
13#13133195—22no—
14#13133193—21no—
15C4×Dic106C2211yescharacteristic
16C4×D212C2211yescharacteristic
17#13133179—21no—
18#13133139—21no—
19#13133139—21no—
20#13133137—22no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26Q8×D53—42no—
27Q8×D53—42no—
28C2×D212—21no—
29C2×D212—21no—
30C2×D212—21no—
31C2×D212D411yescharacteristic
32C2×C4×D53—22no—
33C2×C4×D53—21no—
34C2×Dic106—22no—
35C2×Dic106D411yescharacteristic
36C2×C8×C53—21no—
37C2×C8×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53D412yes—
40C42×C53C2311yescharacteristic
41#11719921—22no—
42#11719921—21no—
43C4⋊Dic53D411yescharacteristic
44C4⋊Dic53D411yescharacteristic
45#11719919—22no—
46#11719919—21no—
47C4×Dic53—21no—
48C2×#88322—21no—
49D424—42no—
50#11719913—42no—
51C22×D53—21no—
52C22×D53—41no—
53Q8×C53—41no—
54Q8×C53—41no—
55C2×C4×C53—22no—
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×Dic53—21no—
60C2×Dic53—21no—
61C2×Dic53—21no—
62D212—21no—
63D212—42no—
64D212—41no—
65D212—41no—
66D212—41no—
67D212—41no—
68D212SD1612yescomplemented
69C4×D53—44no—
70C4×D53—41no—
71C4×D53—41no—
72Dic106—21no—
73Dic106—44no—
74Dic106D812yescomplemented
75C8×C53—41no—
76C8×C53—41no—
77#88322—41no—
78C22×C53#12011yescharacteristic
79C2×D53—22no—
80C2×D53—41no—
81C2×D53—41no—
82C2×D53—81no—
83C2×D53—81no—
84C2×D53—81no—
85C4×C53—21no—
86C4×C53—42no—
87C4×C53C2×D811yescharacteristic
88C4×C53C2×SD1611yescharacteristic
89C4×C53AGL(1,ℤ/8)11yescharacteristic
90C4×C53#13711yescharacteristic
91Dic53—22no—
92Dic53—41no—
93Dic53—41no—
94#1627—531nomaximal, Sylow
95C2×C53#44711yescharacteristic
96C2×C53#44811yescharacteristic
97C2×C53#45411yescharacteristic
98D53—42no—
99D53—161no—
100D4×Q8—531no—
101#486—531no—
102#460—531no—
103#459—531no—
104#332D5311yescharacteristic, complemented
105Q8⋊C8—531no—
106D4⋊C8—531no—
107C53#162711yescharacteristic, complemented, Sylow
108C22×Q8—1061no—
109C2×SD16—1061no—
110C2×D8—1061no—
111C4⋊Q8—1061no—
112C4⋊Q8C2×D5311yescharacteristic
113#127—531no—
114C22⋊Q8—1061no—
115C22⋊Q8—1062no—
116C4×Q8—531no—
117C4×D4—531no—
118C4×D4—1061no—
119C4⋊C8C2×D5311yescharacteristic
120C4⋊C8C2×D5311yescharacteristic
121#102—1061no—
122#102—1061no—
123#102—1062no—
124C4×C8—531no—
125C2×Q8—21no—
126C2×Q8—531no—
127C2×Q8—1062no—
128C2×Q8—2122no—
129C2×Q8—2122no—
130C2×D4—531no—
131C2×D4—1061no—
132C2×D4—1061no—
133C2×D4—1061no—
134C22×C4—1062no—
135C22×C4—1061no—
136SD16—2122no—
137D8—2122no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—1061no—
141C4⋊C4—21no—
142C4⋊C4—531no—
143C4⋊C4—531no—
144C4⋊C4—1062no—
145C4⋊C4—1061no—
146C4⋊C4D21212yes—
147C22⋊C4—1062no—
148C22⋊C4—1061no—
149C42—1061no—
150C42C22×D5311yescharacteristic
151C23—1061no—
152C23—2121no—
153Q8—41no—
154Q8—41no—
155Q8—532no—
156Q8—1061no—
157Q8—2124no—
158D4—532no—
159D4—1061no—
160D4—2122no—
161D4—2121no—
162D4—2121no—
163D4—2121no—
164D4—2121no—
165C2×C4—22no—
166C2×C4—1061no—
167C2×C4—1061no—
168C2×C4—1061no—
169C2×C4—2124no—
170C2×C4—2121no—
171C2×C4—2121no—
172C2×C4C2×D21211yescharacteristic
173C2×C4D4×D5311yescharacteristic
174C2×C4D4×D5311yescharacteristic
175C8—41no—
176C8—41no—
177C8—2121no—
178C22—1062no—
179C22—4241no—
180C22—4241no—
181C22—4241no—
182C22—2121no—
183C22—2121no—
184C22#1313318711yescharacteristic
185C4—21no—
186C4—42no—
187C4—1062no—
188C4—2121no—
189C4—2121no—
190C4D8×D5311yescharacteristic
191C4SD16×D5311yescharacteristic
192C4#1313324411yescharacteristic
193C4#1313324911yescharacteristic
194C2—8481no—
195C2—2122no—
196C2#1681075811yescharacteristic
197C2#1681076311yescharacteristic
198C2#1681079811yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#332C2
socleC22×C53#120
hypercenter#332D53
hypocenterC53#1627
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #16811288C2semidirect
2. #16810971C2semidirect
3. #16810824C2semidirect
5. #16810353C2semidirect
6. #16810350C2semidirect
7. C53×#332C2semidirect
68. D212SD16semidirect
74. Dic106D8semidirect
104. #332D53semidirect
107. C53#1627semidirect