Finite Group Databasebeta

← #20290623

Subgroups — #20290623

Order 6784.

306 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 68 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#16811594C211yesmaximal, characteristic
2C4×C22⋊#6157C211yesmaximal, characteristic
3#16811493C212yesmaximal, complemented
4#16811493C212yesmaximal, complemented
5#117×D53C211yesmaximal, characteristic
6C2×#11719936C2212yes—
7C2×C22⋊#6157—21no—
8C2×C4×#6157C2212yes—
9#13133274—21no—
10C22⋊#88324—21no—
11(C2×#6157)⋊C4—21no—
12C22×C4×D53C2211yescharacteristic
13#13133176C411yescharacteristic, complemented
14#13133175C2211yescharacteristic
15#13133175C2211yescharacteristic
16#13133174—21no—
17#13133173—21no—
18C4⋊C4×D53C411yescharacteristic, complemented
19C4⋊C4×D53C411yescharacteristic, complemented
20C22⋊C4×D53—21no—
21C42×D53—21no—
22#117×C53C411yescharacteristic, complemented
23C23×D53D411yescharacteristic
24C22×#6157—22no—
25C22×C4×C53C2×C411yescharacteristic
26C22×Dic53D411yescharacteristic
27C2×C4×D53—24no—
28C2×C4×D53—21no—
29C2×C4×D53—21no—
30C2×C4×D53C2×C411yescharacteristic
31C2×C4×D53D411yescharacteristic
32C2×C4×D53D411yescharacteristic
33C2×C4×D53C2311yescharacteristic
34C22⋊#6157—24no—
35C2×#88324—22no—
36C4×#6157—24no—
37#11719936—24no—
38C4⋊C4×C53C2×C411yescharacteristic
39C4⋊C4×C53C2×C411yescharacteristic
40C22⋊C4×C53—21no—
41C42×C53—21no—
42C22⋊Dic53—21no—
43#11719921—21no—
44#11719921—21no—
45C4⋊Dic53C2×C411yescharacteristic
46C4⋊Dic53C2×C411yescharacteristic
47#11719919—21no—
48#11719919—21no—
49C4×Dic53—21no—
50C4×Dic53—21no—
51C4×Dic53—21no—
52C23×C53C22⋊C411yescharacteristic
53C22×D53—22no—
54C22×D53—22no—
55C22×D53—41no—
56C22×D53C22⋊C411yescharacteristic
57C22×D53C2×D411yescharacteristic
58C2×#6157—24no—
59C2×#6157—42no—
60C2×#6157—42no—
61C2×C4×C53—22no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53C22⋊C411yescharacteristic
65C2×C4×C53C22×C411yescharacteristic
66C2×Dic53—22no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53C22⋊C411yescharacteristic
70C2×Dic53C2×D411yescharacteristic
71C4×D53—22no—
72C4×D53—22no—
73C4×D53—22no—
74C4×D53—42no—
75C4×D53—42no—
76C4×D53#4114yes—
77#88324—42no—
78C22×C53—21no—
79C22×C53—21no—
80C22×C53—21no—
81C22×C53C2×C22⋊C411yescharacteristic
82C2×D53—21no—
83C2×D53—21no—
84C2×D53—21no—
85C2×D53—21no—
86C2×D53—44no—
87C2×D53#10412yes—
88C2×D53#12312yes—
89#6157—44no—
90C4×C53—21no—
91C4×C53—41no—
92C4×C53—41no—
93C4×C53#11712yes—
94Dic53—21no—
95Dic53—41no—
96Dic53—41no—
97Dic53#12312yes—
98#1826—531nomaximal, Sylow
99C2×C53—41no—
100C2×C53#38111yescharacteristic
101C2×C53C2×#10411yescharacteristic
102C2×C53#42111yescharacteristic
103D53—22no—
104D53—22no—
105C2×#117—531no—
106#407—531no—
107C4×C22⋊C4—531no—
108#339—532no—
109#339—532no—
110C53#182611yescharacteristic, complemented, Sylow
111C23×C4—531no—
112C2×M16—532no—
113#117—531no—
114#117—531no—
115#117—531no—
116#117—1061no—
117#117—1061no—
118#117#615711yescharacteristic, complemented
119C2×C4⋊C4—531no—
120C2×C4⋊C4—531no—
121C2×C22⋊C4—1061no—
122C2×C22⋊C4—1061no—
123C2×C42—532no—
124C2×C42—1061no—
125C22⋊C8—1061no—
126C22⋊C8—1061no—
127#95—1061no—
128C24—531no—
129C22×C4—531no—
130C22×C4—531no—
131C22×C4—531no—
132C22×C4—531no—
133C22×C4—531no—
134C22×C4—1064no—
135C22×C4—1061no—
136C22×C4—1061no—
137C22×C4—1062no—
138C22×C4C2×#615711yescharacteristic
139M16—1064no—
140C2×C8—1062no—
141C4⋊C4—531no—
142C4⋊C4—531no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4C2×#615711yescharacteristic
146C4⋊C4C2×#615711yescharacteristic
147C22⋊C4—21no—
148C22⋊C4—1061no—
149C22⋊C4—1061no—
150C22⋊C4—1064no—
151C22⋊C4—1061no—
152C42—21no—
153C42—1061no—
154C42—1061no—
155C42—1064no—
156C42—1061no—
157C23—531no—
158C23—531no—
159C23—1062no—
160C23—1062no—
161C23—2121no—
162C23C22⋊#615711yescharacteristic
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—531no—
167C2×C4—534no—
168C2×C4—531no—
169C2×C4—1062no—
170C2×C4—1062no—
171C2×C4—1062no—
172C2×C4—1062no—
173C2×C4—1064no—
174C2×C4—1061no—
175C2×C4—1061no—
176C2×C4—2122no—
177C2×C4—2122no—
178C2×C4—2122no—
179C2×C4—2122no—
180C2×C4C22⋊#615711yescharacteristic
181C2×C4C22×#615711yescharacteristic
182C8—2122no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—532no—
187C22—532no—
188C22—1061no—
189C22—1061no—
190C22—1061no—
191C22—1061no—
192C22—2124no—
193C22C2×C22⋊#615711yescharacteristic
194C4—21no—
195C4—41no—
196C4—41no—
197C4—532no—
198C4—1061no—
199C4—2124no—
200C4—2121no—
201C4—2121no—
202C4#1313329612yes—
203C2—41no—
204C2—1062no—
205C2—1062no—
206C2#1681156611yescharacteristic
207C2D53⋊#10411yescharacteristic
208C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#117×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#117#6157
hypocenterC53#1826
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #16811493C2semidirect
4. #16811493C2semidirect
13. #13133176C4semidirect
18. C4⋊C4×D53C4semidirect
19. C4⋊C4×D53C4semidirect
22. #117×C53C4semidirect
110. C53#1826semidirect
118. #117#6157semidirect