Finite Group Databasebeta

← #20273822

Subgroups — #20273822

Order 6784.

216 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811347C211yesmaximal, characteristic, complemented
2#16810820C211yesmaximal, characteristic, complemented
3#16810815C211yesmaximal, characteristic
4#16810369C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×#486C211yesmaximal, characteristic, complemented
8C2×#11719923—21no—
9C2×#11719922—21no—
10#13133212—22no—
11#13133206—21no—
12#13133197—21no—
13Q8×Dic53—21no—
14C4×Dic106C2211yescharacteristic
15C4×D212C2211yescharacteristic
16#13133175—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133138—21no—
20#13133138—21no—
21C4⋊Q8×C53C2211yescharacteristic
22#127×C53C2211yescharacteristic
23#102×C53—22no—
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D212D411yescharacteristic
30C2×C4×D53—21no—
31C2×Dic106D411yescharacteristic
32C2×C8×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53D412yes—
35C42×C53C2311yescharacteristic
36#11719923—42no—
37#11719922—42no—
38#11719921—22no—
39#11719921—22no—
40#11719921—21no—
41C4⋊Dic53—22no—
42C4⋊Dic53D411yescharacteristic
43C4⋊Dic53D411yescharacteristic
44#11719919—21no—
45C4×Dic53—22no—
46C4×Dic53—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C23×C53—41no—
50C22×D53—21no—
51Q8×C53—41no—
52Q8×C53—41no—
53D4×C53—42no—
54D4×C53—41no—
55D4×C53—41no—
56D4×C53—41no—
57D4×C53—41no—
58C2×C4×C53—22no—
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C2×Dic53—21no—
63C2×Dic53—21no—
64C2×Dic53—21no—
65D212—21no—
66D212—21no—
67C4×D53—41no—
68C4×D53—41no—
69Dic106—21no—
70Dic106—21no—
71C8×C53—41no—
72#88322—41no—
73#88322—41no—
74C22×C53—81no—
75C22×C53—81no—
76C22×C53—81no—
77C22×C53#12011yescharacteristic
78C2×D53—41no—
79C2×D53—41no—
80C2×D53—41no—
81C4×C53—21no—
82C4×C53—42no—
83C4×C53#13511yescharacteristic
84C4×C53#13511yescharacteristic
85C4×C53AGL(1,ℤ/8)11yescharacteristic
86C4×C53AGL(1,ℤ/8)11yescharacteristic
87Dic53—41no—
88Dic53—41no—
89Dic53—41no—
90#1628—531nomaximal, Sylow
91C2×C53—161no—
92C2×C53#44911yescharacteristic
93C2×C53#44911yescharacteristic
94C2×C53#45411yescharacteristic
95D53—81no—
96#556—531no—
97#486D5311yescharacteristic, complemented
98#460—531no—
99#459—531no—
100#332—531no—
101Q8⋊C8—531no—
102D4⋊C8—531no—
103C53#162811yescharacteristic, complemented, Sylow
104C2×SD16—1061no—
105C2×D8—1061no—
106C4⋊Q8C2×D5311yescharacteristic
107#127C2×D5311yescharacteristic
108#126—1062no—
109#124—1061no—
110C22⋊Q8—1061no—
111C4×Q8—531no—
112C4×Q8—1061no—
113C4×D4—531no—
114#117—1061no—
115C4⋊C8—531no—
116C4⋊C8—531no—
117#102—22no—
118#102—1061no—
119#102—1061no—
120C4×C8C2×D5311yescharacteristic
121C2×Q8—21no—
122C2×Q8—531no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—531no—
127C22×C4—1061no—
128SD16—2122no—
129D8—2122no—
130C2×C8—21no—
131C2×C8—1061no—
132C2×C8—1061no—
133C4⋊C4—21no—
134C4⋊C4—531no—
135C4⋊C4—531no—
136C4⋊C4—1062no—
137C4⋊C4—1061no—
138C4⋊C4C22⋊D5312yes—
139C22⋊C4—1062no—
140C22⋊C4—1062no—
141C22⋊C4—1061no—
142C42—1062no—
143C42—1061no—
144C42C22×D5311yescharacteristic
145C23—41no—
146C23—1061no—
147Q8—41no—
148Q8—41no—
149Q8—1061no—
150Q8—1061no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—1061no—
157D4—1061no—
158C2×C4—22no—
159C2×C4—1061no—
160C2×C4—1061no—
161C2×C4—1061no—
162C2×C4—2121no—
163C2×C4—2121no—
164C2×C4D4×D5311yescharacteristic
165C2×C4D4×D5311yescharacteristic
166C2×C4C2×C22⋊D5311yescharacteristic
167C8—41no—
168C8—2121no—
169C8—2121no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—2121no—
174C22—2121no—
175C22—2121no—
176C22#1313318511yescharacteristic
177C4—21no—
178C4—42no—
179C4—2121no—
180C4—2121no—
181C4—2121no—
182C4#1313323611yescharacteristic
183C4#1313323811yescharacteristic
184C4#1313324111yescharacteristic
185C4#1313324411yescharacteristic
186C2—161no—
187C2—4241no—
188C2#1681076511yescharacteristic
189C2#1681076711yescharacteristic
190C2#1681079511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#486C2
socleC22×C53#120
hypercenter#486D53
hypocenterC53#1628
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16811347C2semidirect
2. #16810820C2semidirect
4. #16810369C2semidirect
5. #16810355C2semidirect
6. #16810352C2semidirect
7. C53×#486C2semidirect
97. #486D53semidirect
103. C53#1628semidirect