Finite Group Databasebeta

← #20273661

Subgroups — #20273661

Order 6784.

320 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#16810773C211yesmaximal, characteristic, complemented
2#16810756C211yesmaximal, characteristic, complemented
3#16810532C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic, complemented
6#16810347C211yesmaximal, characteristic, complemented
7C53×#538C211yesmaximal, characteristic, complemented
8C22×D212C2211yescharacteristic
9C22×C22⋊D53—21no—
10C2×#11719924—21no—
11C2×#11719922—21no—
12C2×#11719921—21no—
13#13133137—21no—
14#13133137—21no—
15C22⋊#88322C2211yescharacteristic
16C22⋊#88322C2211yescharacteristic
17#13133124C2211yescharacteristic
18#126×C53—21no—
19#124×C53—21no—
20#123×C53—21no—
21C22⋊Q8×C53C2211yescharacteristic
22#121×C53—21no—
23#121×C53C2211yescharacteristic
24#120×C53—21no—
25C4×D4×C53—21no—
26C2×C22⋊C4×C53C2211yescharacteristic
27C24×C53D411yescharacteristic
28C23×D53—21no—
29C2×Q8×C53D411yescharacteristic, complemented
30C2×D4×C53—21no—
31C2×D4×C53—21no—
32C2×D4×C53—21no—
33C2×D4×C53D411yescharacteristic, complemented
34C22×C4×C53—21no—
35C22×C4×C53D411yescharacteristic
36C22×C4×C53C2311yescharacteristic
37C2×C22⋊D53—42no—
38C2×C22⋊D53—42no—
39C2×C22⋊D53—42no—
40C22×Dic53—21no—
41C2×D212—21no—
42C2×D212—21no—
43C2×D212—42no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53—21no—
46C4⋊C4×C53D411yescharacteristic, complemented
47C4⋊C4×C53D411yescharacteristic, complemented
48C22⋊C4×C53—21no—
49C22⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53—21no—
52C22⋊C4×C53—21no—
53C22⋊C4×C53—21no—
54C22⋊C4×C53—21no—
55C42×C53—21no—
56#11719924—42no—
57#11719922—42no—
58#11719921—42no—
59C2×#88322—21no—
60C2×#88322—21no—
61C23×C53—21no—
62C23×C53—21no—
63C23×C53—21no—
64C23×C53—21no—
65C23×C53—41no—
66C23×C53—41no—
67C23×C53C2×D411yescharacteristic
68C22×D53—22no—
69C22×D53—41no—
70C22×D53—41no—
71C22×D53—41no—
72C22×D53—41no—
73C22×D53—41no—
74C22×D53—41no—
75Q8×C53—41no—
76D4×C53—41no—
77D4×C53—41no—
78D4×C53—41no—
79D4×C53—81no—
80D4×C53—81no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—81no—
88C2×C4×C53C2×D411yescharacteristic
89C2×C4×C53C2×D411yescharacteristic
90C22⋊D53—48no—
91C2×Dic53—22no—
92C2×Dic53—41no—
93C2×Dic53—41no—
94D212—22no—
95D212—22no—
96D212—41no—
97D212—41no—
98#88322—41no—
99#88322—41no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—42no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—81no—
110C22×C53#12011yescharacteristic
111C22×C53#13511yescharacteristic
112C22×C53AGL(1,ℤ/8)11yescharacteristic
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C2×D53—81no—
117C2×D53—81no—
118C2×D53—81no—
119C2×D53—81no—
120C4×C53—21no—
121C4×C53—21no—
122C4×C53—41no—
123C4×C53—41no—
124C4×C53—41no—
125C4×C53—41no—
126Dic53—42no—
127#1600—531nomaximal, Sylow
128C2×C53—21no—
129C2×C53—21no—
130C2×C53—81no—
131C2×C53—81no—
132C2×C53#44911yescharacteristic
133C2×C53#45311yescharacteristic
134C2×C53B(4,𝔽2)11yescharacteristic
135D53—82no—
136#538D5311yescharacteristic, complemented
137#450—531no—
138#447—531no—
139#392—531no—
140#328—531no—
141#327—531no—
142#323—531no—
143C53#160011yescharacteristic, complemented, Sylow
144C22×D4—531no—
145C22×D4—1061no—
146C2×SD16—1061no—
147C2×D8—1061no—
148#126—21no—
149#124—21no—
150#123—21no—
151C22⋊Q8C2×D5311yescharacteristic
152#121—21no—
153#121C2×D5311yescharacteristic
154#120—21no—
155C4×D4—21no—
156C2×C22⋊C4—1061no—
157C2×C22⋊C4C2×D5311yescharacteristic
158#102—1061no—
159#102—1061no—
160C22⋊C8—531no—
161C22⋊C8—531no—
162#95—531no—
163C24—1061no—
164C24C22⋊D5311yescharacteristic
165C2×Q8C22⋊D5311yescharacteristic, complemented
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—1061no—
170C2×D4—1061no—
171C2×D4—2122no—
172C2×D4—2122no—
173C2×D4—2122no—
174C2×D4—2122no—
175C2×D4C22⋊D5311yescharacteristic, complemented
176C22×C4—21no—
177C22×C4—1061no—
178C22×C4C22⋊D5311yescharacteristic
179C22×C4C22×D5311yescharacteristic
180SD16—2122no—
181D8—2122no—
182C2×C8—1061no—
183C2×C8—1061no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4C22⋊D5311yescharacteristic, complemented
187C4⋊C4C22⋊D5311yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—2122no—
196C42—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—41no—
203C23—1062no—
204C23—2121no—
205C23—2121no—
206C23—2121no—
207C23—2121no—
208C23—2121no—
209C23—2121no—
210C23C2×C22⋊D5311yescharacteristic
211Q8—41no—
212D4—41no—
213D4—41no—
214D4—41no—
215D4—81no—
216D4—81no—
217D4—1062no—
218D4—1062no—
219D4—2121no—
220D4—2121no—
221D4—2128no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4—1062no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4C2×C22⋊D5311yescharacteristic
233C2×C4C2×C22⋊D5311yescharacteristic
234C8—2121no—
235C8—2121no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—81no—
246C22—4241no—
247C22—4241no—
248C22—4241no—
249C22—4241no—
250C22—2122no—
251C22—2122no—
252C22—2122no—
253C22#1313318611yescharacteristic
254C22#1313323711yescharacteristic
255C22#1313324211yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—2122no—
263C2—21no—
264C2—21no—
265C2—81no—
266C2—81no—
267C2—4242no—
268C2#1681076811yescharacteristic
269C2#1681079111yescharacteristic
270C2#1681081011yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#538C2
socleC22×C53#120
hypercenter#538D53
hypocenterC53#1600
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #16810773C2semidirect
2. #16810756C2semidirect
3. #16810532C2semidirect
4. #16810360C2semidirect
5. #16810357C2semidirect
6. #16810347C2semidirect
7. C53×#538C2semidirect
29. C2×Q8×C53D4semidirect
33. C2×D4×C53D4semidirect
46. C4⋊C4×C53D4semidirect
47. C4⋊C4×C53D4semidirect
136. #538D53semidirect
143. C53#1600semidirect
165. C2×Q8C22⋊D53semidirect
175. C2×D4C22⋊D53semidirect
186. C4⋊C4C22⋊D53semidirect
187. C4⋊C4C22⋊D53semidirect