Finite Group Databasebeta

← #20290970

Subgroups — #20290970

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#16811525C212yesmaximal, complemented
4#16811523C212yesmaximal, complemented
5#16811182C211yesmaximal, characteristic
6C2×#11719940—21no—
7C2×C22⋊#6157—21no—
8#13133277—21no—
9#13133277C2212yes—
10#13133275—21no—
11#13133275C2212yes—
12C22×C22⋊D53C2211yescharacteristic
13#13133200C411yescharacteristic, complemented
14#13133200C2211yescharacteristic
15#13133198—21no—
16#13133189—21no—
17#13133189—21no—
18#13133186C411yescharacteristic, complemented
19#13133186C2211yescharacteristic
20#13133185—21no—
21C2×C22⋊Dic53C411yescharacteristic, complemented
22C22×D4×C53C411yescharacteristic, complemented
23C24×C53C2×C411yescharacteristic
24C24×C53C2×C411yescharacteristic, complemented
25C23×D53D411yescharacteristic
26C22×#6157—41no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53—41no—
30C2×D4×C53—41no—
31C22×C4×C53C2×C411yescharacteristic, complemented
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C2×C22⋊D53—41no—
35C2×C22⋊D53—41no—
36C2×C22⋊D53C2×C411yescharacteristic
37C2×C22⋊D53D411yescharacteristic
38C2×C22⋊D53D411yescharacteristic
39C2×C22⋊D53C2311yescharacteristic
40C22×Dic53—21no—
41C22×Dic53D411yescharacteristic
42C22⋊#6157—22no—
43C22⋊#6157—41no—
44#11719940—22no—
45#11719940—41no—
46C2×#88324—41no—
47C22⋊Dic53—21no—
48C22⋊Dic53—21no—
49C22⋊Dic53—21no—
50C22⋊Dic53C2×C411yescharacteristic, complemented
51C22⋊Dic53C2×C411yescharacteristic, complemented
52#11719921—21no—
53#11719921—21no—
54#11719919—21no—
55#11719919—21no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—41no—
60C23×C53—81no—
61C23×C53—81no—
62C23×C53C22⋊C411yescharacteristic
63C23×C53C22⋊C411yescharacteristic
64C23×C53C22×C411yescharacteristic
65C22×D53—21no—
66C22×D53—21no—
67C22×D53—81no—
68C22×D53C22⋊C411yescharacteristic
69C22×D53C2×D411yescharacteristic
70C2×#6157—42no—
71C2×#6157—82no—
72D4×C53—42no—
73D4×C53—41no—
74D4×C53—41no—
75C2×C4×C53—21no—
76C2×C4×C53—41no—
77C22⋊D53—41no—
78C22⋊D53—41no—
79C22⋊D53—41no—
80C22⋊D53—41no—
81C22⋊D53—81no—
82C22⋊D53—81no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—81no—
88C2×Dic53C22⋊C411yescharacteristic
89C2×Dic53C2×D411yescharacteristic
90#88324—42no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—41no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—81no—
100C22×C53—81no—
101C22×C53—82no—
102C22×C53#9911yescharacteristic
103C22×C53#9911yescharacteristic
104C22×C53C2×C22⋊C411yescharacteristic
105C2×D53—21no—
106C2×D53—21no—
107C2×D53—42no—
108C2×D53—41no—
109C2×D53—41no—
110C2×D53—81no—
111#6157—82no—
112C4×C53—41no—
113Dic53—21no—
114Dic53—21no—
115Dic53—41no—
116Dic53—41no—
117#2121—531nomaximal, Sylow
118C2×C53—21no—
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—41no—
122C2×C53—41no—
123C2×C53—81no—
124C2×C53C2×#9911yescharacteristic
125D53—41no—
126D53—41no—
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#615711yescharacteristic, complemented
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—531no—
145C2×C22⋊C4—1061no—
146#100—532no—
147#100—1061no—
148#99—532no—
149#99—1061no—
150C24—531no—
151C24C2×#615711yescharacteristic
152C24C2×#615711yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—531no—
158C2×D4—531no—
159C2×D4—531no—
160C2×D4—531no—
161C2×D4—1061no—
162C2×D4—1061no—
163C2×D4—2121no—
164C2×D4—2121no—
165C22×C4—531no—
166C22×C4—1061no—
167C22×C4—2121no—
168C22×C4C2×#615711yescharacteristic, complemented
169M16—1062no—
170M16—2121no—
171C2×C8—2121no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C22⋊C4—531no—
175C22⋊C4—531no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1062no—
182C22⋊C4—2121no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—81no—
188C23—81no—
189C23—531no—
190C23—531no—
191C23—1061no—
192C23—1061no—
193C23—4241no—
194C23C22⋊#615711yescharacteristic
195C23C22⋊#615711yescharacteristic
196C23C22×#615711yescharacteristic
197D4—42no—
198D4—41no—
199D4—41no—
200D4—4241no—
201D4—4241no—
202D4—2121no—
203D4—2121no—
204D4—2121no—
205D4—2121no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—531no—
209C2×C4—531no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—4242no—
215C2×C4—4241no—
216C2×C4—2122no—
217C8—2122no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22—81no—
228C22—82no—
229C22—1061no—
230C22—1061no—
231C22—4241no—
232C22—2121no—
233C22—2121no—
234C22—2122no—
235C22#1313327511yescharacteristic
236C22#1313327511yescharacteristic
237C22C2×C22⋊#615711yescharacteristic
238C4—41no—
239C4—1061no—
240C4—1061no—
241C4—4242no—
242C4—2121no—
243C4—2121no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—81no—
250C2—2121no—
251C2—2121no—
252C2C2×#1313327511yescharacteristic
subgroupG/H
centerC2C2×#13133275
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC22×D4×C53C4
socleC2×C53C2×#99
hypercenterC22×D4#6157
hypocenterC53#2121
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133277C2semidirect
2. C2×#13133275C2semidirect
3. #16811525C2semidirect
4. #16811523C2semidirect
13. #13133200C4semidirect
18. #13133186C4semidirect
21. C2×C22⋊Dic53C4semidirect
22. C22×D4×C53C4semidirect
24. C24×C53C2×C4semidirect
31. C22×C4×C53C2×C4semidirect
50. C22⋊Dic53C2×C4semidirect
51. C22⋊Dic53C2×C4semidirect
132. C53#2121semidirect
134. C22×D4#6157semidirect
152. C24C2×#6157semidirect
168. C22×C4C2×#6157semidirect