Finite Group Databasebeta

← #20290141

Subgroups — #20290141

Order 6784.

284 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#13133274C211yesmaximal, characteristic, complemented
2#16811566C211yesmaximal, characteristic, complemented
3#16811477C212yesmaximal, complemented
4#16811474C212yesmaximal, complemented
5#16811130C211yesmaximal, characteristic
6C22×#88324—21no—
7C2×C22⋊#6157—21no—
8#13133274—21no—
9#13133274C2212yes—
10(C2×#6157)⋊C4—21no—
11(C2×#6157)⋊C4C2212yes—
12C22×C4×D53C2211yescharacteristic
13#13133198—21no—
14#13133193C411yescharacteristic, complemented
15#13133193C2211yescharacteristic
16#13133188C411yescharacteristic, complemented
17#13133188C2211yescharacteristic
18#13133185—21no—
19C4×C22⋊D53—21no—
20C4×C22⋊D53—21no—
21#13133176C411yescharacteristic, complemented
22C2×#41×C53C411yescharacteristic, complemented
23C23×D53D411yescharacteristic
24C22×#6157—22no—
25#41×C53—42no—
26C2×Q8×C53C2×C411yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53C2×C411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2×C411yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C22×Dic53D411yescharacteristic
34C2×C4×D53—42no—
35C2×C4×D53C2×C411yescharacteristic
36C2×C4×D53D411yescharacteristic
37C2×C4×D53D411yescharacteristic
38C2×C4×D53C2311yescharacteristic
39C22⋊#6157—42no—
40C2×#88324—22no—
41C2×#88324—42no—
42C22⋊Dic53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53C2×C411yescharacteristic
48C4⋊Dic53C2×C411yescharacteristic
49#11719919—21no—
50#11719919—21no—
51C4×Dic53—21no—
52C23×C53—21no—
53C23×C53C22⋊C411yescharacteristic
54C22×D53—22no—
55C22×D53—22no—
56C22×D53—41no—
57C22×D53C22⋊C411yescharacteristic
58C22×D53C2×D411yescharacteristic
59C2×#6157—42no—
60C2×#6157—42no—
61C2×#6157—42no—
62Q8×C53—41no—
63D4×C53—41no—
64D4×C53—42no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—42no—
68C2×C4×C53C22⋊C411yescharacteristic
69C2×C4×C53C22×C411yescharacteristic
70C22⋊D53—42no—
71C22⋊D53—42no—
72C2×Dic53—22no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53C22⋊C411yescharacteristic
76C2×Dic53C2×D411yescharacteristic
77C4×D53—41no—
78C4×D53—41no—
79C4×D53—41no—
80C4×D53—41no—
81#88324—42no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—41no—
87C22×C53#9912yes—
88C22×C53C2×C22⋊C411yescharacteristic
89C2×D53—21no—
90C2×D53—21no—
91C2×D53—21no—
92C2×D53—21no—
93C2×D53—21no—
94C2×D53—44no—
95C2×D53#13512yes—
96#6157—82no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—41no—
100Dic53—21no—
101Dic53—21no—
102Dic53—41no—
103Dic53—41no—
104#1516—531nomaximal, Sylow
105C2×C53—22no—
106C2×C53—81no—
107C2×C53C2×#9911yescharacteristic
108C2×C53#41611yescharacteristic
109C2×C53#42111yescharacteristic
110D53—22no—
111D53—41no—
112#525—531no—
113C2×C22⋊C8—531no—
114#381—531no—
115#328—532no—
116#327—532no—
117C53#151611yescharacteristic, complemented, Sylow
118C2×#41#615711yescharacteristic, complemented
119C23×C4—531no—
120C22×C8—1061no—
121#123—1061no—
122C22⋊Q8—531no—
123C22⋊Q8—531no—
124#121—531no—
125#121—531no—
126#120—1061no—
127C4×D4—1061no—
128C4×D4—1061no—
129#117—531no—
130C2×C22⋊C4—1061no—
131C22⋊C8—532no—
132C22⋊C8—1061no—
133#95—532no—
134#95—1061no—
135C24—531no—
136#41—42no—
137C2×Q8C2×#615711yescharacteristic
138C2×D4—21no—
139C2×D4—1061no—
140C2×D4—1061no—
141C2×D4C2×#615711yescharacteristic
142C22×C4—21no—
143C22×C4—531no—
144C22×C4—531no—
145C22×C4—531no—
146C22×C4—531no—
147C22×C4—531no—
148C22×C4—1062no—
149C22×C4—2122no—
150C22×C4C2×#615711yescharacteristic
151C2×C8—1062no—
152C2×C8—2122no—
153C4⋊C4—531no—
154C4⋊C4—531no—
155C4⋊C4—1061no—
156C4⋊C4—1061no—
157C22⋊C4—1061no—
158C22⋊C4—1061no—
159C22⋊C4—1061no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—2122no—
163C42—1061no—
164C23—21no—
165C23—531no—
166C23—531no—
167C23—1062no—
168C23—1062no—
169C23—2121no—
170C23C22⋊#615711yescharacteristic
171Q8—41no—
172D4—41no—
173D4—42no—
174D4—2122no—
175D4—2122no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—531no—
180C2×C4—531no—
181C2×C4—1062no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—2121no—
185C2×C4—2121no—
186C2×C4—2121no—
187C2×C4—2121no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4C22⋊#615711yescharacteristic
192C2×C4C22×#615711yescharacteristic
193C8—2122no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—532no—
200C22—1061no—
201C22—1061no—
202C22—1061no—
203C22—1061no—
204C22—1061no—
205C22—2124no—
206C22#1313327512yes—
207C22C2×C22⋊#615711yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—1061no—
212C4—1061no—
213C4—4242no—
214C4—2121no—
215C4—2121no—
216C2—22no—
217C2—81no—
218C2—1062no—
219C2—2121no—
220C2C2×#1313327511yescharacteristic
221C2#1681161311yescharacteristic
222C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×#41×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×#41#6157
hypocenterC53#1516
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#13133274C2semidirect
2. #16811566C2semidirect
3. #16811477C2semidirect
4. #16811474C2semidirect
14. #13133193C4semidirect
16. #13133188C4semidirect
21. #13133176C4semidirect
22. C2×#41×C53C4semidirect
117. C53#1516semidirect
118. C2×#41#6157semidirect