Finite Group Databasebeta

← #20290034

Subgroups — #20290034

Order 6784.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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#16811639C211yesmaximal, characteristic, complemented
2C4×#11719940C211yesmaximal, characteristic, complemented
3#16811471C212yesmaximal, complemented
4D4⋊#88324C212yesmaximal, complemented
5#16811168C211yesmaximal, characteristic
6C2×#11719940—21no—
7C4⋊#88324C2212yes—
8C22⋊#88324—21no—
9#13133272—21no—
10C4×#88324—21no—
11C4×#88324C2212yes—
12C2×#11719947—21no—
13#13133216C411yescharacteristic
14#13133213C411yescharacteristic
15#13133203—21no—
16#13133191C411yescharacteristic
17#13133189—21no—
18D4×Dic53C2211yescharacteristic
19#13133179C2211yescharacteristic
20#13133177—21no—
21C2×C4×Dic53C2211yescharacteristic
22#121×C53C411yescharacteristic
23C2×D4×C53—21no—
24C2×D4×C53C2×C411yescharacteristic
25C22×C4×C53C2×C411yescharacteristic
26C2×C22⋊D53—21no—
27C2×C22⋊D53—21no—
28C2×C22⋊D53—21no—
29C22×Dic53—21no—
30C22×Dic53D411yescharacteristic
31C22×Dic53D411yescharacteristic
32#11719947—44no—
33C2×D212C2×C411yescharacteristic
34C2×C4×D53—21no—
35C2×Dic106—21no—
36#11719940—24no—
37C2×#88324—22no—
38C2×#88324—22no—
39C4⋊C4×C53C2×C411yescharacteristic
40C22⋊C4×C53—21no—
41C22⋊Dic53—21no—
42#11719921—21no—
43#11719921—21no—
44C4⋊Dic53C2×C411yescharacteristic
45#11719919—21no—
46C4×Dic53C2×C411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53D411yescharacteristic
49C4×Dic53C2311yescharacteristic
50C23×C53—21no—
51C23×C53C22⋊C411yescharacteristic
52C22×D53—21no—
53D4×C53—22no—
54D4×C53—42no—
55C2×C4×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22×C411yescharacteristic
59C22⋊D53—42no—
60C22⋊D53—42no—
61C22⋊D53—42no—
62C2×Dic53—22no—
63C2×Dic53—22no—
64C2×Dic53—21no—
65C2×Dic53—42no—
66C2×Dic53C22⋊C411yescharacteristic
67C2×Dic53C22⋊C411yescharacteristic
68C2×Dic53C2×D411yescharacteristic
69C2×Dic53C2×D411yescharacteristic
70D212—22no—
71C4×D53—42no—
72Dic106—42no—
73#88324—24no—
74#88324—42no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—21no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53C2×C22⋊C411yescharacteristic
82C2×D53—41no—
83C2×D53—41no—
84C2×D53—41no—
85C4×C53—21no—
86C4×C53—41no—
87C4×C53#13112yes—
88Dic53—21no—
89Dic53—21no—
90Dic53—41no—
91Dic53#10412yes—
92Dic53AGL(1,ℤ/8)12yes—
93#1484—531nomaximal, Sylow
94C2×C53—41no—
95C2×C53—81no—
96C2×C53#40811yescharacteristic
97C2×C53#41811yescharacteristic
98C2×C53C2×#10411yescharacteristic
99D53—81no—
100#532—531no—
101#431—531no—
102C4×M16—531no—
103D4⋊C8—532no—
104D4⋊C8—532no—
105C53#148411yescharacteristic, complemented, Sylow
106C2×#41—1061no—
107C2×M16—1061no—
108C4⋊Q8—531no—
109#127—531no—
110#124—1061no—
111#121—531no—
112#121—1061no—
113#121#615711yescharacteristic
114C4×D4—531no—
115C4×D4—531no—
116C4×D4—1061no—
117C2×C42—531no—
118C4⋊C8—532no—
119C22⋊C8—1061no—
120#97—1061no—
121C4×C8—532no—
122C4×C8—1061no—
123#41—2124no—
124C2×Q8—1061no—
125C2×D4—21no—
126C2×D4—531no—
127C2×D4—1061no—
128C2×D4—1061no—
129C2×D4—1061no—
130C2×D4C2×#615711yescharacteristic
131C22×C4—531no—
132C22×C4—531no—
133C22×C4—1061no—
134C22×C4—1061no—
135C22×C4C2×#615711yescharacteristic
136M16—1064no—
137C2×C8—1062no—
138C2×C8—1062no—
139C4⋊C4—531no—
140C4⋊C4—1061no—
141C4⋊C4C2×#615711yescharacteristic
142C22⋊C4—21no—
143C22⋊C4—1061no—
144C22⋊C4—1061no—
145C22⋊C4—1061no—
146C42—531no—
147C42—531no—
148C42—531no—
149C42—531no—
150C23—21no—
151C23—1061no—
152C23C22⋊#615711yescharacteristic
153Q8—2122no—
154D4—22no—
155D4—42no—
156D4—1062no—
157D4—2122no—
158D4—2122no—
159D4—2122no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—531no—
163C2×C4—531no—
164C2×C4—531no—
165C2×C4—531no—
166C2×C4—1062no—
167C2×C4—1062no—
168C2×C4—1061no—
169C2×C4—2122no—
170C2×C4—2122no—
171C2×C4C22⋊#615711yescharacteristic
172C2×C4C22×#615711yescharacteristic
173C8—1064no—
174C8—2122no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—2121no—
182C22—2121no—
183C22—2121no—
184C22C2×C22⋊#615711yescharacteristic
185C4—21no—
186C4—41no—
187C4—532no—
188C4—532no—
189C4—1061no—
190C4—1061no—
191C4—2121no—
192C4#1313330412yes—
193C2—41no—
194C2—81no—
195C2—4241no—
196C2#1681159611yescharacteristic
197C2#1681161511yescharacteristic
198C2D53⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#121×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#121#6157
hypocenterC53#1484
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #16811639C2semidirect
2. C4×#11719940C2semidirect
3. #16811471C2semidirect
4. D4⋊#88324C2semidirect
105. C53#1484semidirect