Finite Group Databasebeta

← #20290738

Subgroups — #20290738

Order 6784.

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

normal subgroups only

This group has 378 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
1C2×#13133296C211yesmaximal, characteristic, complemented
2C2×#13133275C212yesmaximal
3#16811566C211yesmaximal, characteristic, complemented
4#16811501C212yesmaximal, complemented
5C2×#13133199C211yesmaximal, characteristic
6#13133296—24no—
7C2×C4⋊#6157—21no—
8C2×C22⋊#6157C2212yes—
9C2×C22⋊#6157C2212yes—
10C2×C4×#6157—21no—
11#13133275—24no—
12(C2×#6157)⋊C4—21no—
13(C2×#6157)⋊C4—21no—
14C22×C22⋊D53C2211yescharacteristic
15C22×C4×D53C2211yescharacteristic
16#13133199—21no—
17#13133199—21no—
18#13133199—21no—
19#13133199—21no—
20C2×C4⋊Dic53C411yescharacteristic, complemented
21C2×#11719919C411yescharacteristic, complemented
22C2×C22⋊Dic53C411yescharacteristic, complemented
23C2×#11719921C2211yescharacteristic
24C2×C22⋊C4×C53C411yescharacteristic, complemented
25C24×C53C2×C411yescharacteristic
26C23×D53C2311yescharacteristic
27C22×#6157—22no—
28C22×#6157—22no—
29C22×C4×C53C2×C411yescharacteristic
30C22×C4×C53C2×C411yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—22no—
34C2×C22⋊D53—22no—
35C22×Dic53C2×C411yescharacteristic
36C22×Dic53C2×C411yescharacteristic
37C22×Dic53C2×C411yescharacteristic
38C2×C4×D53—22no—
39C2×C4×D53—22no—
40C2×C4×D53D411yescharacteristic
41C2×C4×D53D411yescharacteristic
42C2×C4×D53D411yescharacteristic
43C2×C4×D53D411yescharacteristic
44C22⋊#6157—22no—
45C22⋊#6157—22no—
46C22⋊#6157—22no—
47C22⋊#6157D414yes—
48C4⋊#6157—24no—
49C4×#6157—24no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53—21no—
52C22⋊Dic53—21no—
53C22⋊Dic53—21no—
54#11719921—21no—
55#11719921—21no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58#11719919—21no—
59#11719919—21no—
60C23×C53—21no—
61C23×C53—22no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53C22×C411yescharacteristic
65C22×D53—41no—
66C22×D53—41no—
67C22×D53C2×D411yescharacteristic
68C22×D53C2×D411yescharacteristic
69C22×D53C2×D412yes—
70C22×D53#4112yes—
71C2×#6157—24no—
72C2×#6157—42no—
73C2×#6157—42no—
74C2×#6157—42no—
75C2×#6157—42no—
76C2×#6157—42no—
77C2×C4×C53—22no—
78C2×C4×C53—21no—
79C2×C4×C53—41no—
80C2×C4×C53C22⋊C411yescharacteristic
81C2×C4×C53C22⋊C411yescharacteristic
82C22⋊D53—41no—
83C22⋊D53—42no—
84C22⋊D53—41no—
85C2×Dic53—22no—
86C2×Dic53—21no—
87C2×Dic53—21no—
88C2×Dic53—22no—
89C2×Dic53—41no—
90C2×Dic53C22⋊C411yescharacteristic
91C2×Dic53C22⋊C411yescharacteristic
92C4×D53—21no—
93C4×D53—21no—
94C4×D53—22no—
95C4×D53—22no—
96C4×D53—21no—
97C4×D53—21no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—82no—
104C22×C53C2×C22⋊C411yescharacteristic
105C22×C53C4×D412yes—
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—21no—
110C2×D53—42no—
111C2×D53—42no—
112C2×D53#12011yescharacteristic
113C2×D53#12112yes—
114C2×D53#12311yescharacteristic
115#6157—44no—
116#6157—82no—
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—41no—
120Dic53—21no—
121Dic53—21no—
122Dic53—41no—
123Dic53—41no—
124#1899—531nomaximal, Sylow
125C2×C53—22no—
126C2×C53—81no—
127C2×C53#38611yescharacteristic
128C2×C53#41011yescharacteristic
129C2×C53#41011yescharacteristic
130D53—21no—
131D53—22no—
132D53—21no—
133C2×#123—531no—
134C2×#117—531no—
135C2×#99—532no—
136#381—531no—
137#342—532no—
138C53#189911yescharacteristic, complemented, Sylow
139C22×D4—531no—
140C23×C4—531no—
141#123—1061no—
142#123—1061no—
143#123—1061no—
144#123—1061no—
145#117—1064no—
146C2×C4⋊C4—531no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—1061no—
149C2×C22⋊C4—532no—
150C2×C22⋊C4—531no—
151C2×C22⋊C4—531no—
152C2×C22⋊C4—532no—
153C2×C22⋊C4#615711yescharacteristic, complemented
154C2×C42—1061no—
155#99—1064no—
156#95—1061no—
157#95—1061no—
158C24—531no—
159C24C2×#615711yescharacteristic
160C2×D4—1062no—
161C2×D4—1061no—
162C2×D4—1061no—
163C2×D4—1062no—
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—531no—
169C22×C4—531no—
170C22×C4—531no—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4—1062no—
174C22×C4—1062no—
175C22×C4C2×#615711yescharacteristic
176C22×C4C2×#615711yescharacteristic
177C4⋊C4—1061no—
178C4⋊C4—1061no—
179C4⋊C4—1064no—
180C4⋊C4—1061no—
181C4⋊C4—1061no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—534no—
185C22⋊C4—1061no—
186C22⋊C4—1061no—
187C22⋊C4—1061no—
188C22⋊C4—1061no—
189C22⋊C4—1062no—
190C22⋊C4—1062no—
191C22⋊C4—1062no—
192C42—1064no—
193C23—21no—
194C23—22no—
195C23—41no—
196C23—41no—
197C23—531no—
198C23—531no—
199C23—532no—
200C23—532no—
201C23—2121no—
202C23—2121no—
203C23C22×#615711yescharacteristic
204D4—2121no—
205D4—2122no—
206D4—2121no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—531no—
211C2×C4—531no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1061no—
217C2×C4—1061no—
218C2×C4—1062no—
219C2×C4—1064no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—1062no—
223C2×C4—2122no—
224C2×C4—2122no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4—2121no—
229C2×C4C22⋊#615711yescharacteristic
230C2×C4C22⋊#615711yescharacteristic
231C22—21no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—82no—
237C22—531no—
238C22—532no—
239C22—531no—
240C22—1061no—
241C22—1061no—
242C22—1061no—
243C22—1061no—
244C22—2122no—
245C22—2122no—
246C22C2×C22⋊#615711yescharacteristic
247C22D4×#615712yes—
248C4—21no—
249C4—21no—
250C4—41no—
251C4—1061no—
252C4—1061no—
253C4—4242no—
254C4—2124no—
255C4—2121no—
256C4—2121no—
257C2—22no—
258C2—81no—
259C2—1062no—
260C2—1061no—
261C2—1061no—
262C2#1681157111yescharacteristic
263C2#1681160211yescharacteristic
264C2#1681160311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C22⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C22⋊C4#6157
hypocenterC53#1899
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133296C2semidirect
3. #16811566C2semidirect
4. #16811501C2semidirect
20. C2×C4⋊Dic53C4semidirect
21. C2×#11719919C4semidirect
22. C2×C22⋊Dic53C4semidirect
24. C2×C22⋊C4×C53C4semidirect
138. C53#1899semidirect
153. C2×C22⋊C4#6157semidirect