Finite Group Databasebeta

← #20290154

Subgroups — #20290154

Order 6784.

288 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 70 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
1D53⋊(C22⋊C8)C211yesmaximal, characteristic
2#16811565C211yesmaximal, characteristic
3#16811476C214yesmaximal, complemented
4C2×#13133197C211yesmaximal, characteristic
5C2×D53⋊C8—21no—
6C2×C4⋊#6157—21no—
7C2×C22⋊#6157—21no—
8#13133274C2212yes—
9C22⋊#88324—21no—
10(C2×#6157)⋊C4C2212yes—
11C22×Dic106C411yescharacteristic, complemented
12C22×C4×D53C2211yescharacteristic
13#13133197—21no—
14#13133197—21no—
15#13133197C412yescomplemented
16#13133197C2212yes—
17C2×C4⋊Dic53—21no—
18C2×#11719921—21no—
19C2×C4⋊C4×C53C411yescharacteristic, complemented
20C23×D53D411yescharacteristic
21C22×#6157—22no—
22C22×C4×C53—21no—
23C22×C4×C53C2×C411yescharacteristic
24C22×Dic53—21no—
25C22×Dic53D411yescharacteristic
26C2×C4×D53—42no—
27C2×C4×D53C2×C411yescharacteristic
28C2×C4×D53D411yescharacteristic
29C2×C4×D53D411yescharacteristic
30C2×C4×D53C2311yescharacteristic
31C2×Dic106—21no—
32C2×Dic106—42no—
33C2×Dic106C2×C412yes—
34C22⋊#6157—42no—
35C2×#88324—22no—
36C4⋊#6157—42no—
37D53⋊C8—42no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53C2×C412yes—
40#11719921—22no—
41#11719921—22no—
42C4⋊Dic53—22no—
43C4⋊Dic53—22no—
44C23×C53C22⋊C411yescharacteristic
45C22×D53—41no—
46C22×D53—41no—
47C22×D53C22⋊C411yescharacteristic
48C22×D53SD1612yes—
49C22×D53Q1612yes—
50C22×D53C2×D411yescharacteristic
51C2×#6157—42no—
52C2×#6157—42no—
53C2×#6157—42no—
54C2×C4×C53—22no—
55C2×C4×C53—22no—
56C2×C4×C53—42no—
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22×C411yescharacteristic
59C2×Dic53—22no—
60C2×Dic53—41no—
61C2×Dic53—42no—
62C2×Dic53C22⋊C411yescharacteristic
63C2×Dic53C2×D411yescharacteristic
64C4×D53—22no—
65C4×D53—22no—
66C4×D53—41no—
67C4×D53—41no—
68Dic106—42no—
69Dic106—42no—
70#88324—42no—
71C22×C53—21no—
72C22×C53—21no—
73C22×C53#10312yes—
74C22×C53C2×C22⋊C411yescharacteristic
75C2×D53—21no—
76C2×D53—21no—
77C2×D53—21no—
78C2×D53—21no—
79C2×D53—42no—
80C2×D53—42no—
81C2×D53#10312yes—
82C2×D53C2×SD1611yescharacteristic
83C2×D53C2×Q1611yescharacteristic
84#6157—82no—
85C4×C53—21no—
86C4×C53—42no—
87C4×C53#9912yes—
88Dic53—21no—
89Dic53—21no—
90Dic53—42no—
91#1520—531nomaximal, Sylow
92C2×C53—22no—
93C2×C53C2×#9911yescharacteristic
94C2×C53C2×#10311yescharacteristic
95C2×C53#42111yescharacteristic
96D53—21no—
97D53—22no—
98D53—21no—
99C2×C22⋊Q8—531no—
100C2×C22⋊C8—531no—
101#380—531no—
102#328—534no—
103C53#152011yescharacteristic, complemented, Sylow
104C22×Q8—531no—
105C23×C4—531no—
106C22×C8—1061no—
107C22⋊Q8—532no—
108C22⋊Q8—532no—
109C22⋊Q8—1061no—
110C22⋊Q8—1061no—
111C2×C4⋊C4—1061no—
112C2×C4⋊C4—1061no—
113C2×C4⋊C4#615711yescharacteristic, complemented
114C2×C22⋊C4—1061no—
115C2×C22⋊C4—1061no—
116C22⋊C8—532no—
117C22⋊C8—1061no—
118#95—532no—
119C24—531no—
120C2×Q8—532no—
121C2×Q8—1061no—
122C2×Q8—2122no—
123C22×C4—21no—
124C22×C4—531no—
125C22×C4—531no—
126C22×C4—531no—
127C22×C4—531no—
128C22×C4—531no—
129C22×C4—1061no—
130C22×C4—1062no—
131C22×C4—2122no—
132C22×C4C2×#615711yescharacteristic
133C2×C8—1062no—
134C2×C8—2122no—
135C4⋊C4—21no—
136C4⋊C4—1062no—
137C4⋊C4—1062no—
138C4⋊C4—2122no—
139C4⋊C4C2×#615712yes—
140C22⋊C4—1062no—
141C22⋊C4—1062no—
142C22⋊C4—2122no—
143C23—531no—
144C23—531no—
145C23—532no—
146C23—532no—
147C23—2121no—
148C23—2121no—
149C23C22⋊#615711yescharacteristic
150Q8—2122no—
151Q8—2122no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—531no—
156C2×C4—531no—
157C2×C4—1062no—
158C2×C4—1062no—
159C2×C4—1062no—
160C2×C4—2121no—
161C2×C4—2121no—
162C2×C4—2121no—
163C2×C4—2122no—
164C2×C4—2122no—
165C2×C4—2122no—
166C2×C4—2122no—
167C2×C4C22⋊#615711yescharacteristic
168C2×C4C22×#615711yescharacteristic
169C8—2122no—
170C22—21no—
171C22—21no—
172C22—531no—
173C22—532no—
174C22—531no—
175C22—1061no—
176C22—1061no—
177C22—1061no—
178C22—1061no—
179C22—2122no—
180C22—2122no—
181C22#1313328012yes—
182C22C2×C22⋊#615711yescharacteristic
183C4—21no—
184C4—42no—
185C4—1061no—
186C4—1061no—
187C4—4242no—
188C4—2122no—
189C4#1313327612yes—
190C2—22no—
191C2—1061no—
192C2—1062no—
193C2—1061no—
194C2C2×#1313327611yescharacteristic
195C2C2×#1313328011yescharacteristic
196C2#1681162111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1520
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #16811476C2semidirect
11. C22×Dic106C4semidirect
15. #13133197C4semidirect
19. C2×C4⋊C4×C53C4semidirect
103. C53#1520semidirect
113. C2×C4⋊C4#6157semidirect