Finite Group Databasebeta

← #20273784

Subgroups — #20273784

Order 6784.

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

normal subgroups only

This group has 236 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#16811334C211yesmaximal, characteristic, complemented
2#16810820C211yesmaximal, characteristic, complemented
3#16810815C211yesmaximal, characteristic
4#16810372C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×#492C211yesmaximal, characteristic, complemented
8C2×#11719923—21no—
9C2×#11719922—21no—
10#13133208—21no—
11#13133197—21no—
12#13133193—22no—
13C4×Dic106C2211yescharacteristic
14C4×D212C2211yescharacteristic
15#13133179—21no—
16C4⋊C4×D53—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133138—21no—
20#13133138—21no—
21C2×SD16×C53—22no—
22C4⋊Q8×C53C2211yescharacteristic
23#127×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53D412yes—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D212D411yescharacteristic
30C2×C4×D53—21no—
31C2×C4×D53—22no—
32C2×Dic106D411yescharacteristic
33SD16×C53—44no—
34C2×C8×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C42×C53C2311yescharacteristic
38#11719923—42no—
39#11719922—42no—
40#11719921—22no—
41#11719921—21no—
42C4⋊Dic53—22no—
43C4⋊Dic53D411yescharacteristic
44C4⋊Dic53D411yescharacteristic
45#11719919—22no—
46#11719919—21no—
47C4×Dic53—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C23×C53—41no—
51C22×D53—21no—
52Q8×C53—42no—
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—
66D212SD1612yescomplemented
67C4×D53—41no—
68C4×D53—41no—
69C4×D53—82no—
70Dic106—21no—
71Dic106—21no—
72C8×C53—22no—
73#88322—41no—
74#88322—41no—
75C22×C53—81no—
76C22×C53—81no—
77C22×C53—81no—
78C22×C53#12011yescharacteristic
79C2×D53—22no—
80C2×D53—41no—
81C2×D53—41no—
82C4×C53—21no—
83C4×C53—42no—
84C4×C53C2×SD1611yescharacteristic
85C4×C53#13511yescharacteristic
86C4×C53AGL(1,ℤ/8)11yescharacteristic
87C4×C53#13711yescharacteristic
88Dic53—41no—
89Dic53—41no—
90Dic53—41no—
91#1622—531nomaximal, Sylow
92C2×C53—161no—
93C2×C53#44811yescharacteristic
94C2×C53#44911yescharacteristic
95C2×C53#45311yescharacteristic
96D53—42no—
97#554—531no—
98#492D5311yescharacteristic, complemented
99#460—531no—
100#459—531no—
101#333—531no—
102Q8⋊C8—531no—
103D4⋊C8—531no—
104C53#162211yescharacteristic, complemented, Sylow
105C2×SD16—22no—
106C2×SD16—1061no—
107C2×D8—1061no—
108C4⋊Q8C2×D5311yescharacteristic
109#127C2×D5311yescharacteristic
110#125—1061no—
111C22⋊Q8—1061no—
112C22⋊Q8—1062no—
113C4×Q8—531no—
114C4×D4—531no—
115C4×D4—1061no—
116C2×C4⋊C4—1061no—
117C4⋊C8—531no—
118C4⋊C8—531no—
119#102—1061no—
120#102—1061no—
121C4×C8C2×D5311yescharacteristic
122C2×Q8—531no—
123C2×Q8C22⋊D5312yes—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—531no—
128C22×C4—1062no—
129C22×C4—1061no—
130SD16—44no—
131SD16—2122no—
132D8—2122no—
133C2×C8—21no—
134C2×C8—1061no—
135C2×C8—1061no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—531no—
139C4⋊C4—531no—
140C4⋊C4—1062no—
141C4⋊C4—1062no—
142C4⋊C4—1061no—
143C22⋊C4—1062no—
144C22⋊C4—1061no—
145C42—1061no—
146C42C22×D5311yescharacteristic
147C23—41no—
148C23—1061no—
149Q8—42no—
150Q8—1061no—
151Q8—1061no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—532no—
158D4—1061no—
159C2×C4—22no—
160C2×C4—1061no—
161C2×C4—1061no—
162C2×C4—1061no—
163C2×C4—4242no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4D4×D5311yescharacteristic
167C2×C4D4×D5311yescharacteristic
168C2×C4C2×C22⋊D5311yescharacteristic
169C8—22no—
170C8—2121no—
171C8—2121no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—1062no—
176C22—2121no—
177C22—2121no—
178C22#1313318511yescharacteristic
179C4—21no—
180C4—42no—
181C4—2121no—
182C4—2121no—
183C4—2121no—
184C4SD16×D5311yescharacteristic
185C4#1313323811yescharacteristic
186C4#1313324411yescharacteristic
187C4#1313324611yescharacteristic
188C2—161no—
189C2—2122no—
190C2#1681076011yescharacteristic
191C2#1681076711yescharacteristic
192C2#1681079011yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#492C2
socleC22×C53#120
hypercenter#492D53
hypocenterC53#1622
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #16811334C2semidirect
2. #16810820C2semidirect
4. #16810372C2semidirect
5. #16810355C2semidirect
6. #16810352C2semidirect
7. C53×#492C2semidirect
66. D212SD16semidirect
98. #492D53semidirect
104. C53#1622semidirect