Finite Group Databasebeta

← #20290118

Subgroups — #20290118

Order 6784.

242 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 58 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#16811646C211yesmaximal, characteristic, complemented
2C4×C4⋊#6157C211yesmaximal, characteristic, complemented
3#16811476C212yesmaximal, complemented
4#16811475C212yesmaximal
5#16811158C211yesmaximal, characteristic
6C2×C4⋊#6157—21no—
7C2×C4×#6157—21no—
8#13133284—21no—
9#13133274C2212yes—
10#13133272—21no—
11(C2×#6157)⋊C4C2212yes—
12#13133212—21no—
13#13133205C411yescharacteristic, complemented
14#13133202—21no—
15#13133197C2211yescharacteristic
16#13133192C2211yescharacteristic
17#13133182C411yescharacteristic, complemented
18#13133179—21no—
19#13133179C411yescharacteristic, complemented
20#13133175—21no—
21C42×D53C2211yescharacteristic
22#125×C53C411yescharacteristic, complemented
23C22×#6157—22no—
24C2×D212—21no—
25C2×D212C2×C411yescharacteristic
26C2×C4×D53—21no—
27C2×C4×D53D411yescharacteristic
28C2×C4×D53D411yescharacteristic
29C2×C4×D53C2311yescharacteristic
30C2×Dic106C2×C411yescharacteristic
31C2×#88324—22no—
32C4⋊#6157—24no—
33C4×#6157—42no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53C2×C411yescharacteristic
37C4⋊C4×C53C2×C411yescharacteristic
38C42×C53C2×C411yescharacteristic
39#11719921—21no—
40#11719921—21no—
41#11719921—21no—
42#11719921—21no—
43#11719921—21no—
44C4⋊Dic53—21no—
45#11719919—21no—
46C4×Dic53—21no—
47C4×Dic53C2×C411yescharacteristic
48C4×Dic53D411yescharacteristic
49C4×Dic53D411yescharacteristic
50C22×D53—21no—
51C22×D53C2×D411yescharacteristic
52C2×#6157—24no—
53C2×#6157—42no—
54C2×#6157—42no—
55C2×C4×C53—21no—
56C2×C4×C53—21no—
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22⋊C411yescharacteristic
59C2×C4×C53C22×C411yescharacteristic
60C2×Dic53—21no—
61C2×Dic53C22⋊C411yescharacteristic
62C2×Dic53C22⋊C411yescharacteristic
63C2×Dic53C2×D411yescharacteristic
64D212—41no—
65D212—41no—
66D212—41no—
67C4×D53—21no—
68C4×D53—21no—
69C4×D53—22no—
70C4×D53—41no—
71C4×D53—42no—
72Dic106—41no—
73#88324—42no—
74C22×C53C2×C22⋊C411yescharacteristic
75C2×D53—21no—
76C2×D53—21no—
77C2×D53—41no—
78C2×D53—41no—
79C2×D53—41no—
80C2×D53AGL(1,ℤ/8)11yescharacteristic
81C2×D53#13711yescharacteristic
82#6157—44no—
83C4×C53—21no—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—41no—
87C4×C53—41no—
88Dic53—21no—
89Dic53—21no—
90Dic53—41no—
91Dic53#10412yes—
92#1509—531nomaximal, Sylow
93C2×C53#41011yescharacteristic
94C2×C53#41711yescharacteristic
95C2×C53C2×#10411yescharacteristic
96D53—21no—
97D53—21no—
98D53—81no—
99#529—531no—
100#432—531no—
101C4×C4⋊C4—531no—
102#328—532no—
103#327—532no—
104C53#150911yescharacteristic, complemented, Sylow
105#126—1061no—
106#125#615711yescharacteristic, complemented
107#124—531no—
108#123—1061no—
109C22⋊Q8—531no—
110#121—531no—
111C4×Q8—531no—
112C4×D4—531no—
113C4×D4—1061no—
114#117—1061no—
115C2×C4⋊C4—1061no—
116C2×C42—531no—
117C2×C42—1061no—
118C4⋊C8—1061no—
119C22⋊C8—532no—
120#97—1061no—
121#95—532no—
122C2×Q8—531no—
123C2×D4—531no—
124C2×D4—1061no—
125C22×C4—531no—
126C22×C4—531no—
127C22×C4—531no—
128C22×C4—1061no—
129C22×C4—1062no—
130C2×C8—1062no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—1061no—
134C4⋊C4—1061no—
135C4⋊C4—1064no—
136C4⋊C4C2×#615711yescharacteristic
137C4⋊C4C2×#615711yescharacteristic
138C22⋊C4—1061no—
139C22⋊C4—1061no—
140C22⋊C4—1061no—
141C22⋊C4—1061no—
142C22⋊C4—1061no—
143C42—531no—
144C42—531no—
145C42—531no—
146C42—1061no—
147C42—2122no—
148C42C2×#615711yescharacteristic
149C23—531no—
150C23—1061no—
151Q8—2121no—
152D4—2121no—
153D4—2121no—
154D4—2121no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—531no—
158C2×C4—531no—
159C2×C4—531no—
160C2×C4—1062no—
161C2×C4—1061no—
162C2×C4—1061no—
163C2×C4—1064no—
164C2×C4—1061no—
165C2×C4—2121no—
166C2×C4—2122no—
167C2×C4—2122no—
168C2×C4—2122no—
169C2×C4C22⋊#615711yescharacteristic
170C2×C4C22⋊#615711yescharacteristic
171C2×C4C22×#615711yescharacteristic
172C8—2122no—
173C22—531no—
174C22—531no—
175C22—1061no—
176C22—1061no—
177C22—2121no—
178C22—2121no—
179C22—2121no—
180C22C2×C22⋊#615711yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4—532no—
187C4—1061no—
188C4—1061no—
189C4—2124no—
190C4—2121no—
191C2—1061no—
192C2—1061no—
193C2—4241no—
194C2#1681160411yescharacteristic
195C2#1681161411yescharacteristic
196C2D53⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#125×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#125#6157
hypocenterC53#1509
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. #16811646C2semidirect
2. C4×C4⋊#6157C2semidirect
3. #16811476C2semidirect
13. #13133205C4semidirect
17. #13133182C4semidirect
19. #13133179C4semidirect
22. #125×C53C4semidirect
104. C53#1509semidirect
106. #125#6157semidirect