Finite Group Databasebeta

← #20275125

Subgroups — #20275125

Order 6784.

386 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 83 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×#13133200C211yesmaximal, characteristic
2#16811096C211yesmaximal, characteristic
3C2×#13133131C211yesmaximal, characteristic
4#16810486C211yesmaximal, characteristic, complemented
5#16810390C211yesmaximal, characteristic, complemented
6#16810389C211yesmaximal, characteristic, complemented
7C53×#381C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C2211yescharacteristic
9#13133200—22no—
10#13133200C414yescomplemented
11C4×C22⋊D53—22no—
12#13133177—22no—
13#13133173—21no—
14C2×#11719919—21no—
15C2×C22⋊Dic53C2211yescharacteristic
16C2×C22⋊Dic53C2211yescharacteristic
17C2×C22⋊Dic53C2211yescharacteristic
18C2×#11719921—21no—
19C22⋊C4×D53—21no—
20#13133131—22no—
21#13133124—22no—
22C23×C4×C53C2211yescharacteristic
23C2×C22⋊C4×C53C2211yescharacteristic
24C2×C22⋊C4×C53C2211yescharacteristic
25#95×C53—22no—
26C24×C53C2311yescharacteristic
27C23×D53D411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C22×C4×C53—21no—
35C22×C4×C53C2×C412yescomplemented
36C22×C4×C53D412yes—
37C2×C22⋊D53—21no—
38C2×C22⋊D53—21no—
39C2×C22⋊D53—21no—
40C2×C22⋊D53—21no—
41C2×C22⋊D53C2×C412yes—
42C2×C22⋊D53D412yes—
43C22×Dic53—21no—
44C22×Dic53—21no—
45C22×Dic53D411yescharacteristic
46C22×Dic53D412yes—
47C2×C4×D53—22no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53—21no—
52C22⋊Dic53—21no—
53C22⋊Dic53—21no—
54C22⋊Dic53—21no—
55C22⋊Dic53—21no—
56C22⋊Dic53—21no—
57C22⋊Dic53C2×C412yescomplemented
58#11719921—22no—
59#11719921—24no—
60#11719919—22no—
61#11719919—24no—
62C4×Dic53—22no—
63C23×C53—42no—
64C23×C53C22×C411yescharacteristic
65C23×C53C2×D411yescharacteristic
66C23×C53C2×D411yescharacteristic
67C23×C53C2×D411yescharacteristic
68C23×C53C2×D411yescharacteristic
69C23×C53#4111yescharacteristic
70C23×C53#4111yescharacteristic
71C22×D53—21no—
72C22×D53—21no—
73C22×D53—42no—
74C22×D53C22⋊C412yes—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—41no—
80C2×C4×C53—41no—
81C2×C4×C53—41no—
82C2×C4×C53—41no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—41no—
86C22⋊D53—22no—
87C22⋊D53—22no—
88C22⋊D53—22no—
89C22⋊D53—22no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—41no—
98C2×Dic53C22⋊C412yes—
99C4×D53—82no—
100C22×C53—22no—
101C22×C53—22no—
102C22×C53—22no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53C2×C22⋊C411yescharacteristic
108C22×C53C4×D411yescharacteristic
109C22×C53C4×D411yescharacteristic
110C22×C53#12011yescharacteristic
111C22×C53#12111yescharacteristic
112C22×C53#12111yescharacteristic
113C22×C53#12311yescharacteristic
114C2×D53—22no—
115C2×D53—42no—
116C2×D53—42no—
117C2×D53—81no—
118C4×C53—42no—
119C4×C53—42no—
120C4×C53—81no—
121Dic53—22no—
122Dic53—42no—
123Dic53—42no—
124Dic53—81no—
125#1890—531nomaximal, Sylow
126C2×C53—21no—
127C2×C53—21no—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53#38611yescharacteristic
133C2×C53#45812yes—
134D53—42no—
135C2×#123—531no—
136#518—531no—
137C2×#99—531no—
138#381—531no—
139#381D5311yescharacteristic, complemented
140#342—531no—
141#342—531no—
142C53#189011yescharacteristic, complemented, Sylow
143C22×D4—531no—
144C23×C4C2×D5311yescharacteristic
145#123—534no—
146#123—1062no—
147C4×D4—1062no—
148C4×D4—1062no—
149#117—1061no—
150C2×C4⋊C4—1061no—
151C2×C22⋊C4—531no—
152C2×C22⋊C4—531no—
153C2×C22⋊C4—531no—
154C2×C22⋊C4—1061no—
155C2×C22⋊C4—1061no—
156C2×C22⋊C4C2×D5311yescharacteristic
157C2×C22⋊C4C2×D5311yescharacteristic
158#99—1062no—
159#95—22no—
160#95—1062no—
161C24—531no—
162C24C22×D5311yescharacteristic
163C2×D4—532no—
164C2×D4—532no—
165C2×D4—1061no—
166C2×D4—1061no—
167C2×D4—1061no—
168C2×D4—1061no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—531no—
177C22×C4—532no—
178C22×C4—1062no—
179C22×C4—1061no—
180C22×C4—1061no—
181C22×C4C4×D5312yescomplemented
182C22×C4C22⋊D5312yes—
183C4⋊C4—1064no—
184C4⋊C4—1062no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—532no—
190C22⋊C4—1064no—
191C22⋊C4—1062no—
192C22⋊C4—1061no—
193C22⋊C4—1061no—
194C22⋊C4—1061no—
195C22⋊C4—1061no—
196C22⋊C4—1061no—
197C42—1062no—
198C23—42no—
199C23—532no—
200C23—1061no—
201C23—1061no—
202C23—2122no—
203C23C2×C4×D5311yescharacteristic
204C23#1171994511yescharacteristic
205C23D4×D5311yescharacteristic
206C23D4×D5311yescharacteristic
207C23D4×D5311yescharacteristic
208C23#1171994711yescharacteristic
209C23C2×C22⋊D5311yescharacteristic
210D4—1062no—
211D4—1062no—
212D4—1062no—
213D4—1062no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—532no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—4242no—
231C2×C4—2122no—
232C2×C4—2121no—
233C2×C4—2122no—
234C2×C4—2122no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—1062no—
243C22—4241no—
244C22—2122no—
245C22—2122no—
246C22C22⋊C4×D5311yescharacteristic
247C22#1313317711yescharacteristic
248C22C4×C22⋊D5311yescharacteristic
249C22#1313318511yescharacteristic
250C22#1313318811yescharacteristic
251C22#1313319011yescharacteristic
252C22#1313319911yescharacteristic
253C4—42no—
254C4—42no—
255C4—81no—
256C4—1062no—
257C4—4241no—
258C4—2122no—
259C4—2122no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—2122no—
267C2#1681049911yescharacteristic
268C2#1681081212yes—
subgroupG/H
centerC22#13133185
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#381C2
socleC22×C53#120
hypercenter#381D53
hypocenterC53#1890
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
4. #16810486C2semidirect
5. #16810390C2semidirect
6. #16810389C2semidirect
7. C53×#381C2semidirect
10. #13133200C4semidirect
35. C22×C4×C53C2×C4semidirect
57. C22⋊Dic53C2×C4semidirect
139. #381D53semidirect
142. C53#1890semidirect
181. C22×C4C4×D53semidirect