Finite Group Databasebeta

← #20273841

Subgroups — #20273841

Order 6784.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811294C211yesmaximal, characteristic
2#16810997C211yesmaximal, characteristic, complemented
3#16810915C211yesmaximal, characteristic, complemented
4#16810366C211yesmaximal, characteristic, complemented
5#16810353C211yesmaximal, characteristic, complemented
6D4⋊#88322C211yesmaximal, characteristic, complemented
7C53×#476C211yesmaximal, characteristic, complemented
8C2×#11719945—21no—
9C2×#11719924—22no—
10#13133214C2211yescharacteristic
11#13133213—21no—
12#13133191—21no—
13#13133190—22no—
14C4×Dic106C2211yescharacteristic
15#13133179—21no—
16#13133150—21no—
17C4⋊#88322C2211yescharacteristic
18#13133140—21no—
19C4×#88322C2211yescharacteristic
20C4⋊Q8×C53C2211yescharacteristic
21C4×D4×C53C2211yescharacteristic
22#106×C53—21no—
23C4⋊C8×C53C2211yescharacteristic
24#102×C53—21no—
25C2×Q8×C53—21no—
26C2×D4×C53D411yescharacteristic
27C22×C4×C53—21no—
28C2×C22⋊D53—22no—
29C2×C22⋊D53—22no—
30#11719945—42no—
31#11719945—42no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212D412yes—
35C2×C4×D53—22no—
36C2×Dic106D411yescharacteristic
37C2×C8×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53D411yescharacteristic
41C22⋊C4×C53—21no—
42C42×C53C2311yescharacteristic
43#11719924—44no—
44#11719921—22no—
45C4⋊Dic53D411yescharacteristic
46#11719919—21no—
47C4×Dic53—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C23×C53—21no—
51C22×D53—22no—
52Q8×C53—42no—
53D4×C53—21no—
54D4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C22⋊D53—44no—
63C22⋊D53—82no—
64C2×Dic53—21no—
65C2×Dic53—21no—
66D212—42no—
67D212—41no—
68D212—41no—
69D212—41no—
70D212—41no—
71C4×D53—44no—
72Dic106—21no—
73Dic106SD1612yescomplemented
74C8×C53—41no—
75#88322—22no—
76#88322—41no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53#12011yescharacteristic
81C2×D53—42no—
82C2×D53—42no—
83C2×D53—42no—
84C4×C53—21no—
85C4×C53—41no—
86C4×C53—81no—
87C4×C53C2×SD1611yescharacteristic
88C4×C53#13511yescharacteristic
89C4×C53AGL(1,ℤ/8)11yescharacteristic
90C4×C53#13711yescharacteristic
91Dic53—22no—
92Dic53—41no—
93#1631—531nomaximal, Sylow
94C2×C53—81no—
95C2×C53#45011yescharacteristic
96C2×C53#45211yescharacteristic
97C2×C53#45311yescharacteristic
98D53—82no—
99#550—531no—
100#492—531no—
101#477—531no—
102#476D5311yescharacteristic, complemented
103#331—531no—
104Q8⋊C8—531no—
105D4⋊C8—531no—
106C53#163111yescharacteristic, complemented, Sylow
107C2×#41—1061no—
108C2×SD16—1062no—
109C4⋊Q8C2×D5311yescharacteristic
110#127—531no—
111#127—1061no—
112#121—1062no—
113#121—1061no—
114C4×Q8—531no—
115C4×D4—1061no—
116C4×D4C2×D5311yescharacteristic
117#107—1061no—
118#106—21no—
119C4⋊C8—531no—
120C4⋊C8C2×D5311yescharacteristic
121#103—1061no—
122#102—21no—
123C4×C8—531no—
124#41—2122no—
125#41—2122no—
126C2×Q8—21no—
127C2×Q8—531no—
128C2×D4—532no—
129C2×D4—1062no—
130C2×D4—1062no—
131C2×D4—1061no—
132C2×D4—1061no—
133C2×D4C22⋊D5311yescharacteristic
134C22×C4—21no—
135C22×C4—1062no—
136SD16—2124no—
137C2×C8—21no—
138C2×C8—1061no—
139C2×C8—1061no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—531no—
143C4⋊C4—1061no—
144C4⋊C4C22⋊D5311yescharacteristic
145C22⋊C4—21no—
146C22⋊C4—1062no—
147C42—1061no—
148C42C22×D5311yescharacteristic
149C23—21no—
150C23—1062no—
151Q8—42no—
152Q8—532no—
153Q8—1061no—
154D4—21no—
155D4—21no—
156D4—4242no—
157D4—2122no—
158D4—2124no—
159D4—2121no—
160D4—2121no—
161D4—2121no—
162D4—2121no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—1061no—
168C2×C4—1061no—
169C2×C4—2124no—
170C2×C4D4×D5311yescharacteristic
171C2×C4D4×D5311yescharacteristic
172C2×C4C2×C22⋊D5311yescharacteristic
173C8—41no—
174C8—1062no—
175C8—2121no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—2122no—
180C22—2122no—
181C22—2122no—
182C22#1313318511yescharacteristic
183C4—21no—
184C4—41no—
185C4—81no—
186C4—1062no—
187C4—2121no—
188C4SD16×D5311yescharacteristic
189C4#1313323711yescharacteristic
190C4#1313324411yescharacteristic
191C4#1313324811yescharacteristic
192C2—81no—
193C2—4242no—
194C2#1681077211yescharacteristic
195C2#1681078311yescharacteristic
196C2#1681078911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#476C2
socleC22×C53#120
hypercenter#476D53
hypocenterC53#1631
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #16810997C2semidirect
3. #16810915C2semidirect
4. #16810366C2semidirect
5. #16810353C2semidirect
6. D4⋊#88322C2semidirect
7. C53×#476C2semidirect
73. Dic106SD16semidirect
102. #476D53semidirect
106. C53#1631semidirect