Finite Group Databasebeta

← #20290127

Subgroups — #20290127

Order 6784.

222 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 56 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#16811645C211yesmaximal, characteristic, complemented
2#16811567C211yesmaximal, characteristic, complemented
3#16811477C212yesmaximal, complemented
4#16811474C212yesmaximal
5#16811152C211yesmaximal, characteristic
6C2×C4⋊#6157—21no—
7C4⋊#88324—21no—
8#13133274C2212yes—
9#13133272—21no—
10(C2×#6157)⋊C4—21no—
11(C2×#6157)⋊C4C2212yes—
12#13133210—21no—
13#13133209C411yescharacteristic, complemented
14#13133199—21no—
15#13133193C2211yescharacteristic
16#13133188C2211yescharacteristic
17Q8×Dic53C411yescharacteristic, complemented
18D4×Dic53C411yescharacteristic, complemented
19#13133177—21no—
20#13133173—21no—
21C42×D53C2211yescharacteristic
22#124×C53C411yescharacteristic, complemented
23C22×#6157—22no—
24C2×Q8×C53C2×C411yescharacteristic
25C2×D4×C53C2×C411yescharacteristic
26C2×C22⋊D53—21no—
27C22×Dic53—21no—
28C2×C4×D53D411yescharacteristic
29C2×C4×D53D411yescharacteristic
30C2×C4×D53C2311yescharacteristic
31C2×#88324—22no—
32C4⋊#6157—42no—
33C22⋊C4×C53—21no—
34C22⋊C4×C53—21no—
35C42×C53C2×C411yescharacteristic
36C22⋊Dic53—21no—
37C22⋊Dic53—21no—
38#11719921—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53C2×C411yescharacteristic
41C4⋊Dic53C2×C411yescharacteristic
42#11719919—21no—
43#11719919—21no—
44#11719919—21no—
45C4×Dic53—21no—
46C4×Dic53C2×C411yescharacteristic
47C4×Dic53D411yescharacteristic
48C4×Dic53D411yescharacteristic
49C23×C53—21no—
50C22×D53C2×D411yescharacteristic
51C2×#6157—42no—
52C2×#6157—42no—
53C2×#6157—42no—
54Q8×C53—41no—
55D4×C53—41no—
56C2×C4×C53—21no—
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22⋊C411yescharacteristic
59C2×C4×C53C22×C411yescharacteristic
60C22⋊D53—41no—
61C22⋊D53—41no—
62C2×Dic53—21no—
63C2×Dic53—21no—
64C2×Dic53—42no—
65C2×Dic53C22⋊C411yescharacteristic
66C2×Dic53C22⋊C411yescharacteristic
67C2×Dic53C2×D411yescharacteristic
68C4×D53—21no—
69C4×D53—21no—
70C4×D53—41no—
71C4×D53—41no—
72#88324—42no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53—41no—
76C22×C53C2×C22⋊C411yescharacteristic
77C2×D53—21no—
78C2×D53—21no—
79C2×D53AGL(1,ℤ/8)11yescharacteristic
80C2×D53#13711yescharacteristic
81#6157—82no—
82C4×C53—21no—
83C4×C53—21no—
84C4×C53—21no—
85C4×C53—41no—
86Dic53—21no—
87Dic53—21no—
88Dic53—21no—
89Dic53—41no—
90Dic53—41no—
91#1512—531nomaximal, Sylow
92C2×C53—81no—
93C2×C53#41011yescharacteristic
94C2×C53#41711yescharacteristic
95C2×C53#42111yescharacteristic
96D53—21no—
97D53—21no—
98#529—531no—
99#432—531no—
100#382—531no—
101#328—532no—
102#327—532no—
103C53#151211yescharacteristic, complemented, Sylow
104#126—1061no—
105#125—531no—
106#124#615711yescharacteristic, complemented
107#123—1061no—
108C22⋊Q8—531no—
109#121—531no—
110C4×Q8—531no—
111C4×D4—531no—
112C4×D4—1061no—
113#117—1061no—
114C2×C4⋊C4—1061no—
115C2×C42—531no—
116C4⋊C8—1061no—
117C22⋊C8—532no—
118#97—1061no—
119#95—532no—
120#95—1061no—
121C2×Q8C2×#615711yescharacteristic
122C2×D4—1061no—
123C2×D4C2×#615711yescharacteristic
124C22×C4—531no—
125C22×C4—531no—
126C22×C4—531no—
127C22×C4—1061no—
128C22×C4—1062no—
129C2×C8—1062no—
130C4⋊C4—531no—
131C4⋊C4—531no—
132C4⋊C4—1061no—
133C4⋊C4—1061no—
134C4⋊C4—1061no—
135C4⋊C4—1061no—
136C4⋊C4—2122no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—1061no—
140C22⋊C4—1061no—
141C22⋊C4—1061no—
142C42—531no—
143C42—531no—
144C42—531no—
145C42—1061no—
146C42C2×#615711yescharacteristic
147C23—21no—
148C23—531no—
149Q8—41no—
150D4—41no—
151D4—2121no—
152D4—2121no—
153C2×C4—21no—
154C2×C4—531no—
155C2×C4—531no—
156C2×C4—531no—
157C2×C4—1061no—
158C2×C4—1061no—
159C2×C4—1061no—
160C2×C4—1061no—
161C2×C4—2121no—
162C2×C4—2121no—
163C2×C4—2122no—
164C2×C4—2122no—
165C2×C4—2122no—
166C2×C4—2122no—
167C2×C4C22⋊#615711yescharacteristic
168C2×C4C22⋊#615711yescharacteristic
169C2×C4C22×#615711yescharacteristic
170C8—2122no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—531no—
175C22—531no—
176C22—1061no—
177C22—1061no—
178C22C2×C22⋊#615711yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—1061no—
184C4—1061no—
185C4—1061no—
186C4—4242no—
187C4—2121no—
188C4—2121no—
189C2—81no—
190C2—1061no—
191C2—1061no—
192C2#1681160311yescharacteristic
193C2#1681161411yescharacteristic
194C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#124#6157
hypocenterC53#1512
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #16811645C2semidirect
2. #16811567C2semidirect
3. #16811477C2semidirect
13. #13133209C4semidirect
17. Q8×Dic53C4semidirect
18. D4×Dic53C4semidirect
22. #124×C53C4semidirect
103. C53#1512semidirect
106. #124#6157semidirect