Finite Group Databasebeta

← #20290620

Subgroups — #20290620

Order 6784.

300 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 74 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#16811496C212yesmaximal, complemented
4#16811496C212yesmaximal, complemented
5#117×D53C211yesmaximal, characteristic
6C2×D53⋊C8C2212yes—
7C2×C4⋊#6157C2212yes—
8C2×C22⋊#6157—21no—
9#13133274—21no—
10C22⋊#88324—21no—
11(C2×#6157)⋊C4—21no—
12C22×C4×D53C2211yescharacteristic
13#13133176C411yescharacteristic, complemented
14#13133175C411yescharacteristic, complemented
15#13133175C411yescharacteristic, complemented
16#13133174—21no—
17#13133173—21no—
18C4⋊C4×D53C2211yescharacteristic
19C4⋊C4×D53C2211yescharacteristic
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—42no—
35C2×#88324—22no—
36C4⋊#6157—24no—
37D53⋊C8—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—21no—
56C22×D53—21no—
57C22×D53C22⋊C411yescharacteristic
58C22×D53C2×D411yescharacteristic
59C2×#6157—42no—
60C2×#6157—42no—
61C2×#6157—42no—
62C2×C4×C53—22no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53C22⋊C411yescharacteristic
66C2×C4×C53C22×C411yescharacteristic
67C2×Dic53—22no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53C22⋊C411yescharacteristic
71C2×Dic53C2×D411yescharacteristic
72C4×D53—22no—
73C4×D53—22no—
74C4×D53—22no—
75C4×D53—42no—
76C4×D53—41no—
77C4×D53—41no—
78C4×D53#4114yes—
79#88324—42no—
80C22×C53—21no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53C2×C22⋊C411yescharacteristic
84C2×D53—21no—
85C2×D53—21no—
86C2×D53—21no—
87C2×D53—42no—
88C2×D53—42no—
89C2×D53#12312yes—
90C2×D53#13512yes—
91C2×D53AGL(1,ℤ/8)11yescharacteristic
92C2×D53#13711yescharacteristic
93#6157—82no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53#11712yes—
98Dic53—21no—
99Dic53—41no—
100Dic53—41no—
101Dic53#12312yes—
102#1825—531nomaximal, Sylow
103C2×C53—41no—
104C2×C53#38111yescharacteristic
105C2×C53#41611yescharacteristic
106C2×C53#41711yescharacteristic
107D53—41no—
108D53#48212yes—
109D53#48512yes—
110C2×#117—531no—
111C2×C22⋊C8—531no—
112#380—531no—
113#340—532no—
114#340—532no—
115C53#182511yescharacteristic, complemented, Sylow
116C23×C4—531no—
117C22×C8—532no—
118#117—531no—
119#117—531no—
120#117—531no—
121#117—1061no—
122#117—1061no—
123#117#615711yescharacteristic, complemented
124C2×C4⋊C4—531no—
125C2×C4⋊C4—531no—
126C2×C4⋊C4—532no—
127C2×C22⋊C4—1061no—
128C2×C22⋊C4—1061no—
129C2×C42—1061no—
130C22⋊C8—1061no—
131C22⋊C8—1061no—
132#95—1061no—
133C24—531no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—531no—
137C22×C4—531no—
138C22×C4—531no—
139C22×C4—1064no—
140C22×C4—1061no—
141C22×C4—1061no—
142C22×C4—1062no—
143C22×C4C2×#615711yescharacteristic
144C2×C8—1062no—
145C2×C8—1064no—
146C4⋊C4—531no—
147C4⋊C4—531no—
148C4⋊C4—1061no—
149C4⋊C4—1061no—
150C4⋊C4—1064no—
151C4⋊C4C2×#615711yescharacteristic
152C4⋊C4C2×#615711yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—1061no—
155C22⋊C4—1061no—
156C22⋊C4—1061no—
157C22⋊C4—2122no—
158C42—21no—
159C42—1061no—
160C42—1061no—
161C42—1061no—
162C23—531no—
163C23—531no—
164C23—1061no—
165C23—1062no—
166C23—1062no—
167C23—1061no—
168C23C22⋊#615711yescharacteristic
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—531no—
173C2×C4—534no—
174C2×C4—531no—
175C2×C4—1062no—
176C2×C4—1062no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—1061no—
180C2×C4—1061no—
181C2×C4—2122no—
182C2×C4—2121no—
183C2×C4—2121no—
184C2×C4—2122no—
185C2×C4—2122no—
186C2×C4—2122no—
187C2×C4C22⋊#615711yescharacteristic
188C2×C4C22×#615711yescharacteristic
189C8—2122no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—532no—
194C22—532no—
195C22—531no—
196C22—531no—
197C22—1061no—
198C22—1061no—
199C22—1061no—
200C22—2122no—
201C22—2122no—
202C22C2×C22⋊#615711yescharacteristic
203C4—21no—
204C4—41no—
205C4—41no—
206C4—532no—
207C4—1061no—
208C4—4242no—
209C4—2121no—
210C4—2121no—
211C4#1313329612yes—
212C2—41no—
213C2—532no—
214C2—532no—
215C2—2121no—
216C2#1681156611yescharacteristic
217C2#1681161311yescharacteristic
218C2#1681161411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#117×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#117#6157
hypocenterC53#1825
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. #16811496C2semidirect
4. #16811496C2semidirect
13. #13133176C4semidirect
14. #13133175C4semidirect
15. #13133175C4semidirect
22. #117×C53C4semidirect
115. C53#1825semidirect
123. #117#6157semidirect