Finite Group Databasebeta

← #20290972

Subgroups — #20290972

Order 6784.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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×#13133277C211yesmaximal, characteristic, complemented
2C2×#13133275C211yesmaximal, characteristic, complemented
3#16811526C212yesmaximal, complemented
4#16811524C212yesmaximal, complemented
5#16811185C211yesmaximal, characteristic
6C2×#11719940—21no—
7C2×C22⋊#6157—21no—
8#13133277—21no—
9#13133277C2212yes—
10#13133275—21no—
11#13133275C2212yes—
12C22×C22⋊D53C411yescharacteristic, complemented
13C22×C22⋊D53C2211yescharacteristic
14#13133201—21no—
15#13133201C411yescharacteristic, complemented
16#13133201C2211yescharacteristic
17#13133191—21no—
18#13133191—21no—
19#13133187—21no—
20#13133187C411yescharacteristic, complemented
21#13133187C2211yescharacteristic
22C2×C22⋊C4×C53C411yescharacteristic, complemented
23C24×C53C2×C411yescharacteristic
24C23×D53C2×C411yescharacteristic, complemented
25C23×D53D411yescharacteristic
26C22×#6157—41no—
27C22×C4×C53—21no—
28C2×C22⋊D53—21no—
29C2×C22⋊D53—21no—
30C2×C22⋊D53—41no—
31C2×C22⋊D53—41no—
32C2×C22⋊D53—41no—
33C2×C22⋊D53—41no—
34C2×C22⋊D53C2×C411yescharacteristic
35C2×C22⋊D53D411yescharacteristic
36C2×C22⋊D53D411yescharacteristic
37C2×C22⋊D53C2311yescharacteristic
38C22×Dic53C2×C411yescharacteristic, complemented
39C22×Dic53D411yescharacteristic
40C2×D212—21no—
41C2×D212—21no—
42C22⋊#6157—22no—
43C22⋊#6157—41no—
44#11719940—22no—
45#11719940—41no—
46C2×#88324—41no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53C2×C411yescharacteristic, complemented
49C22⋊C4×C53C2×C411yescharacteristic, complemented
50#11719921—21no—
51#11719921—21no—
52#11719921—21no—
53#11719921—21no—
54C4⋊Dic53—21no—
55C4⋊Dic53—21no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—81no—
59C23×C53C22⋊C411yescharacteristic
60C23×C53C22⋊C411yescharacteristic
61C23×C53C22×C411yescharacteristic
62C22×D53—21no—
63C22×D53—21no—
64C22×D53—21no—
65C22×D53—41no—
66C22×D53—81no—
67C22×D53—81no—
68C22×D53C22⋊C411yescharacteristic
69C22×D53C2×D411yescharacteristic
70C2×#6157—42no—
71C2×#6157—82no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—81no—
75C22⋊D53—41no—
76C22⋊D53—41no—
77C22⋊D53—41no—
78C22⋊D53—41no—
79C22⋊D53—42no—
80C22⋊D53—41no—
81C22⋊D53—41no—
82C2×Dic53—21no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—41no—
86C2×Dic53C22⋊C411yescharacteristic
87C2×Dic53C2×D411yescharacteristic
88D212—81no—
89D212—81no—
90#88324—42no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—41no—
95C22×C53—41no—
96C22×C53—81no—
97C22×C53—81no—
98C22×C53#9911yescharacteristic
99C22×C53#9911yescharacteristic
100C22×C53C2×C22⋊C411yescharacteristic
101C2×D53—21no—
102C2×D53—21no—
103C2×D53—42no—
104C2×D53—41no—
105C2×D53—41no—
106C2×D53—41no—
107C2×D53—41no—
108C2×D53—41no—
109C2×D53—81no—
110C2×D53—82no—
111#6157—82no—
112C4×C53—41no—
113C4×C53—41no—
114Dic53—21no—
115Dic53—21no—
116Dic53—41no—
117#2121—531nomaximal, Sylow
118C2×C53—21no—
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—41no—
122C2×C53—41no—
123C2×C53C2×#9911yescharacteristic
124D53—41no—
125D53—41no—
126D53—81no—
127#534—531no—
128C2×#100—531no—
129C2×#99—531no—
130#352—532no—
131AΣL(1,𝔽16)—532no—
132C53#212111yescharacteristic, complemented, Sylow
133C22×D4—531no—
134C22×D4—531no—
135C2×M16—1061no—
136#123—531no—
137#123—531no—
138#123—1061no—
139#121—1061no—
140#121—1061no—
141#120—531no—
142#120—531no—
143#120—1061no—
144C2×C22⋊C4—1061no—
145C2×C22⋊C4#615711yescharacteristic, complemented
146#100—532no—
147#100—1061no—
148#99—532no—
149#99—1061no—
150C24—531no—
151C24—531no—
152C24C2×#615711yescharacteristic
153C2×D4—531no—
154C2×D4—531no—
155C2×D4—531no—
156C2×D4—531no—
157C2×D4—1061no—
158C2×D4—1061no—
159C2×D4—1061no—
160C2×D4—1061no—
161C2×D4—2121no—
162C2×D4—2121no—
163C2×D4—2121no—
164C2×D4—2121no—
165C22×C4—21no—
166C22×C4—531no—
167C22×C4—531no—
168C22×C4—2121no—
169M16—1062no—
170M16—2121no—
171C2×C8—2121no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C22⋊C4—21no—
175C22⋊C4—1061no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1062no—
180C22⋊C4—2121no—
181C22⋊C4C2×#615711yescharacteristic, complemented
182C22⋊C4C2×#615711yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—81no—
186C23—531no—
187C23—531no—
188C23—1061no—
189C23—1061no—
190C23—1061no—
191C23—4241no—
192C23—4241no—
193C23—2121no—
194C23C22⋊#615711yescharacteristic
195C23C22⋊#615711yescharacteristic
196C23C22×#615711yescharacteristic
197D4—4241no—
198D4—4241no—
199D4—2121no—
200D4—2121no—
201D4—2121no—
202D4—2121no—
203D4—2122no—
204D4—2121no—
205D4—2121no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—81no—
209C2×C4—531no—
210C2×C4—531no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—4242no—
215C2×C4—2122no—
216C2×C4—2121no—
217C8—2122no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22—1061no—
226C22—1061no—
227C22—4241no—
228C22—4242no—
229C22—2121no—
230C22—2121no—
231C22—2121no—
232C22—2121no—
233C22—2122no—
234C22—2121no—
235C22#1313327611yescharacteristic
236C22#1313327611yescharacteristic
237C22C2×C22⋊#615711yescharacteristic
238C4—41no—
239C4—41no—
240C4—1061no—
241C4—1061no—
242C4—4242no—
243C4—2121no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—4241no—
250C2—2121no—
251C2—2121no—
252C2C2×#1313327611yescharacteristic
subgroupG/H
centerC2C2×#13133276
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C22⋊C4×C53C4
socleC2×C53C2×#99
hypercenterC2×C22⋊C4#6157
hypocenterC53#2121
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#13133277C2semidirect
2. C2×#13133275C2semidirect
3. #16811526C2semidirect
4. #16811524C2semidirect
12. C22×C22⋊D53C4semidirect
15. #13133201C4semidirect
20. #13133187C4semidirect
22. C2×C22⋊C4×C53C4semidirect
24. C23×D53C2×C4semidirect
38. C22×Dic53C2×C4semidirect
48. C22⋊C4×C53C2×C4semidirect
49. C22⋊C4×C53C2×C4semidirect
132. C53#2121semidirect
145. C2×C22⋊C4#6157semidirect
181. C22⋊C4C2×#6157semidirect
182. C22⋊C4C2×#6157semidirect