Finite Group Databasebeta

← #20273827

Subgroups — #20273827

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#16811343C211yesmaximal, characteristic, complemented
2#16810970C211yesmaximal, characteristic, complemented
3#16810824C211yesmaximal, characteristic, complemented
4#16810818C211yesmaximal, characteristic
5#16810368C211yesmaximal, characteristic, complemented
6#16810355C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719913—21no—
9C2×#11719922—21no—
10#13133217C2211yescharacteristic
11#13133214C2211yescharacteristic
12#13133210—22no—
13#13133204—21no—
14#13133196—21no—
15C4×Dic106C2211yescharacteristic
16#13133182—21no—
17#13133176—21no—
18C4⋊#88322C2211yescharacteristic
19#13133139—21no—
20#13133139—22no—
21#13133137—21no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×D4×C53D411yescharacteristic
26C22×C4×C53—21no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212—21no—
30C2×Dic106—21no—
31C2×Dic106D411yescharacteristic
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C4⋊C4×C53D411yescharacteristic
35C22⋊C4×C53—21no—
36C42×C53C2311yescharacteristic
37C22⋊Dic53—22no—
38C22⋊Dic53—22no—
39#11719922—42no—
40C4⋊Dic53—21no—
41C4⋊Dic53D411yescharacteristic
42C4⋊Dic53D412yes—
43#11719919—22no—
44#11719919—21no—
45C4×Dic53—22no—
46C4×Dic53—21no—
47C2×#88322—21no—
48#11719913—42no—
49C23×C53—21no—
50C22×D53—41no—
51D4×C53—21no—
52D4×C53—21no—
53C2×C4×C53—21no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×Dic53—22no—
60C2×Dic53—21no—
61C2×Dic53—21no—
62D212—42no—
63D212—41no—
64D212—41no—
65D212—41no—
66D212—41no—
67Dic106—21no—
68Dic106—21no—
69Dic106—41no—
70Dic106—41no—
71C8×C53—41no—
72C8×C53—41no—
73#88322—41no—
74C22×C53—41no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53#12011yescharacteristic
78C2×D53—81no—
79C2×D53—81no—
80C2×D53—81no—
81C4×C53—21no—
82C4×C53—41no—
83C4×C53#13511yescharacteristic
84C4×C53#13511yescharacteristic
85C4×C53AGL(1,ℤ/8)11yescharacteristic
86C4×C53AGL(1,ℤ/8)11yescharacteristic
87Dic53—41no—
88Dic53—42no—
89Dic53—41no—
90#1628—531nomaximal, Sylow
91C2×C53—81no—
92C2×C53#44911yescharacteristic
93C2×C53#44911yescharacteristic
94C2×C53#45411yescharacteristic
95D53—161no—
96#556—531no—
97#486—531no—
98#460—531no—
99#459—531no—
100#332—531no—
101Q8⋊C8—531no—
102D4⋊C8D5311yescharacteristic, complemented
103C53#162811yescharacteristic, complemented, Sylow
104C2×SD16—1061no—
105C2×D8—1061no—
106C4⋊Q8—531no—
107#127—531no—
108#126—1062no—
109#124—1061no—
110C22⋊Q8—1061no—
111C4×Q8—531no—
112C4×Q8—1061no—
113C4×D4C2×D5311yescharacteristic
114#117—1061no—
115C4⋊C8—531no—
116C4⋊C8C2×D5311yescharacteristic
117#102—1061no—
118#102—1062no—
119#102—1061no—
120C4×C8C2×D5311yescharacteristic
121C2×Q8—531no—
122C2×Q8—1061no—
123C2×D4—1061no—
124C2×D4—1061no—
125C2×D4—1061no—
126C2×D4D21211yescharacteristic
127C22×C4—21no—
128SD16—2122no—
129D8—2122no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—1061no—
133C4⋊C4—531no—
134C4⋊C4—532no—
135C4⋊C4—1062no—
136C4⋊C4—1061no—
137C4⋊C4—1061no—
138C4⋊C4D21211yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—1062no—
141C22⋊C4—1062no—
142C42—1062no—
143C42—1061no—
144C42C22×D5311yescharacteristic
145C23—21no—
146C23—2121no—
147Q8—1061no—
148Q8—1061no—
149Q8—2121no—
150Q8—2121no—
151D4—21no—
152D4—21no—
153D4—2122no—
154D4—2121no—
155D4—2121no—
156D4—2121no—
157D4—2121no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—1061no—
162C2×C4—1062no—
163C2×C4—1061no—
164C2×C4C2×D21211yescharacteristic
165C2×C4D4×D5311yescharacteristic
166C2×C4D4×D5311yescharacteristic
167C8—41no—
168C8—41no—
169C8—2121no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—4241no—
174C22—4241no—
175C22—4241no—
176C22#1313318711yescharacteristic
177C4—21no—
178C4—41no—
179C4—2121no—
180C4—2122no—
181C4—2121no—
182C4#1313323811yescharacteristic
183C4#1313323911yescharacteristic
184C4#1313324111yescharacteristic
185C4#1313324511yescharacteristic
186C2—81no—
187C2—8481no—
188C2#1681076911yescharacteristic
189C2#1681077011yescharacteristic
190C2#1681079711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1628
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
1. #16811343C2semidirect
2. #16810970C2semidirect
3. #16810824C2semidirect
5. #16810368C2semidirect
6. #16810355C2semidirect
7. C53×D4⋊C8C2semidirect
102. D4⋊C8D53semidirect
103. C53#1628semidirect