Finite Group Databasebeta

← #20275130

Subgroups — #20275130

Order 6784.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#16811389C211yesmaximal, characteristic, complemented
2#16811134C211yesmaximal, characteristic
3#16810605C211yesmaximal, characteristic, complemented
4#16810587C211yesmaximal, characteristic, complemented
5#16810395C211yesmaximal, characteristic, complemented
6#16810394C211yesmaximal, characteristic, complemented
7C53×#407C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133225—22no—
10#13133223—22no—
11C2×#11719917C2211yescharacteristic
12C2×#11719917C2211yescharacteristic
13M16×D53—22no—
14#13133200C412yes—
15#13133196—21no—
16#13133191—21no—
17#13133186C412yes—
18C4×C22⋊D53—22no—
19#13133176C2211yescharacteristic
20#13133135—21no—
21#13133133—21no—
22C22⋊#88322—22no—
23C23×C4×C53C2211yescharacteristic
24C2×M16×C53C2211yescharacteristic
25C2×M16×C53C2211yescharacteristic
26C22⋊C8×C53—22no—
27C24×C53C2×C411yescharacteristic
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×C53C2×C411yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C2×C22⋊D53—22no—
37C2×C22⋊D53C2×C412yes—
38#11719945—21no—
39#11719945—21no—
40#11719945—21no—
41#11719945—21no—
42C2×D212D411yescharacteristic
43C2×C4×D53D412yes—
44C2×Dic106D411yescharacteristic
45M16×C53—21no—
46M16×C53—21no—
47M16×C53—21no—
48M16×C53—21no—
49C2×C8×C53—21no—
50C2×C8×C53D412yescomplemented
51C22⋊Dic53—22no—
52C22⋊Dic53C2×C412yes—
53#11719921—22no—
54C4⋊Dic53—21no—
55#11719919—22no—
56C4×Dic53—21no—
57#11719917—21no—
58#11719917—21no—
59#11719917—21no—
60#11719917—21no—
61C2×#88322—21no—
62C2×#88322D412yescomplemented
63#11719912—24no—
64C8×D53—24no—
65C23×C53—21no—
66C23×C53—21no—
67C23×C53—21no—
68C23×C53—42no—
69C23×C53C22×C411yescharacteristic
70C22×D53C22⋊C412yes—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53#4111yescharacteristic
83C2×C4×C53#4111yescharacteristic
84C22⋊D53—22no—
85C22⋊D53—22no—
86C22⋊D53—44no—
87C2×Dic53—22no—
88C2×Dic53C22⋊C412yes—
89D212—21no—
90D212—21no—
91C4×D53—22no—
92C4×D53—22no—
93Dic106—21no—
94Dic106—21no—
95C8×C53—22no—
96C8×C53—41no—
97#88322—22no—
98#88322—41no—
99C22×C53—22no—
100C22×C53—22no—
101C22×C53—22no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53C2×C22⋊C411yescharacteristic
108C22×C53C4×D411yescharacteristic
109C22×C53C4×D411yescharacteristic
110C2×D53—22no—
111C2×D53—42no—
112C4×C53—41no—
113C4×C53#12011yescharacteristic
114C4×C53#12111yescharacteristic
115C4×C53#12111yescharacteristic
116C4×C53#12311yescharacteristic
117Dic53—22no—
118Dic53—42no—
119#1891—531nomaximal, Sylow
120C2×C53—21no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53—42no—
124C2×C53#38611yescharacteristic
125D53—42no—
126#568—531no—
127#525—531no—
128#413—531no—
129#407—531no—
130#407D5311yescharacteristic, complemented
131#344—531no—
132#344—531no—
133C53#189111yescharacteristic, complemented, Sylow
134C2×#41—531no—
135C23×C4C2×D5311yescharacteristic
136#131—1062no—
137#131—1062no—
138C2×M16—531no—
139C2×M16—531no—
140C2×M16—1062no—
141C2×M16C2×D5311yescharacteristic
142C2×M16C2×D5311yescharacteristic
143#123—532no—
144C22⋊Q8—1061no—
145#121—1061no—
146#120—532no—
147C4×D4—1062no—
148#117—531no—
149#101—1061no—
150#100—1061no—
151C22⋊C8—22no—
152C22⋊C8—1062no—
153C24C4×D5311yescharacteristic
154#41—1061no—
155#41—1061no—
156#41—1061no—
157#41—1061no—
158C2×Q8—531no—
159C2×D4—532no—
160C2×D4—531no—
161C2×D4—1062no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—532no—
169C22×C4C4×D5311yescharacteristic
170C22×C4C22×D5311yescharacteristic
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—1064no—
176M16—1061no—
177M16—1061no—
178M16—1061no—
179M16—1061no—
180C2×C8—21no—
181C2×C8—532no—
182C2×C8—1064no—
183C2×C8—1061no—
184C2×C8C22⋊D5312yescomplemented
185C4⋊C4—1062no—
186C4⋊C4—1061no—
187C22⋊C4—532no—
188C22⋊C4—1062no—
189C22⋊C4—1062no—
190C42—1061no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—532no—
196C23C2×C4×D5311yescharacteristic
197Q8—1061no—
198Q8—1061no—
199D4—1062no—
200D4—1062no—
201D4—1061no—
202D4—1061no—
203D4—2124no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—532no—
212C2×C4—1062no—
213C2×C4—1062no—
214C2×C4—1062no—
215C2×C4#1171994511yescharacteristic
216C2×C4D4×D5311yescharacteristic
217C2×C4D4×D5311yescharacteristic
218C2×C4D4×D5311yescharacteristic
219C2×C4#1171994711yescharacteristic
220C2×C4C2×C22⋊D5311yescharacteristic
221C8—22no—
222C8—41no—
223C8—1062no—
224C8—2121no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—1062no—
234C22—2122no—
235C22C22⋊C4×D5311yescharacteristic
236C22#1313317711yescharacteristic
237C22C4×C22⋊D5311yescharacteristic
238C4—41no—
239C4—1062no—
240C4—2122no—
241C4#1313318511yescharacteristic
242C4#1313318811yescharacteristic
243C4#1313319011yescharacteristic
244C4#1313319911yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—2122no—
250C2#1681049911yescharacteristic
subgroupG/H
centerC4#13133185
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#407C2
socleC2×C53#386
hypercenter#407D53
hypocenterC53#1891
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #16811389C2semidirect
3. #16810605C2semidirect
4. #16810587C2semidirect
5. #16810395C2semidirect
6. #16810394C2semidirect
7. C53×#407C2semidirect
50. C2×C8×C53D4semidirect
62. C2×#88322D4semidirect
130. #407D53semidirect
133. C53#1891semidirect
184. C2×C8C22⋊D53semidirect