Finite Group Databasebeta

← #20275132

Subgroups — #20275132

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#16811135C211yesmaximal, characteristic
3#16810605C211yesmaximal, characteristic, complemented
4#16810586C211yesmaximal, characteristic, complemented
5#16810586C211yesmaximal, characteristic, complemented
6#16810393C211yesmaximal, characteristic, complemented
7C53×#344C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C22×C4×D53C2211yescharacteristic
10#13133225—22no—
11#13133223—22no—
12C2×#11719917C2211yescharacteristic
13C2×#11719917C2211yescharacteristic
14M16×D53—22no—
15#13133201C412yes—
16#13133197—21no—
17#13133192—21no—
18#13133187C412yes—
19C4×D212—22no—
20#13133135—21no—
21#13133133—21no—
22#13133129—22no—
23#13133129—22no—
24C2×M16×C53C2211yescharacteristic
25C2×M16×C53C2211yescharacteristic
26#117×C53C2211yescharacteristic
27C23×D53C2×C411yescharacteristic
28C22×C4×C53C2311yescharacteristic
29C2×C22⋊D53C2×C412yes—
30C22×Dic53C2×C411yescharacteristic
31#11719945—21no—
32#11719945—21no—
33#11719945—21no—
34#11719945—21no—
35C2×D212—22no—
36C2×D212D411yescharacteristic
37C2×C4×D53—21no—
38C2×C4×D53—21no—
39C2×C4×D53—21no—
40C2×C4×D53—21no—
41C2×C4×D53—21no—
42C2×C4×D53—21no—
43C2×C4×D53D412yes—
44C2×Dic106D411yescharacteristic
45M16×C53—21no—
46M16×C53—21no—
47M16×C53—21no—
48M16×C53—21no—
49C2×C8×C53—21no—
50C2×C8×C53D412yescomplemented
51C4⋊C4×C53—21no—
52C22⋊C4×C53C2×C412yes—
53C42×C53—21no—
54#11719921—22no—
55#11719921—22no—
56C4⋊Dic53—22no—
57#11719917—21no—
58#11719917—21no—
59#11719917—21no—
60#11719917—21no—
61C2×#88322—21no—
62C2×#88322D412yescomplemented
63#11719912—24no—
64C8×D53—24no—
65C23×C53C22×C411yescharacteristic
66C22×D53—21no—
67C22×D53—21no—
68C22×D53—21no—
69C22×D53—42no—
70C22×D53C22⋊C412yes—
71C2×C4×C53—22no—
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53#4111yescharacteristic
77C2×C4×C53#4111yescharacteristic
78C22⋊D53—22no—
79C22⋊D53—22no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53C22⋊C412yes—
84D212—21no—
85D212—21no—
86D212—44no—
87C4×D53—22no—
88C4×D53—22no—
89C4×D53—41no—
90C4×D53—41no—
91C4×D53—41no—
92C4×D53—41no—
93Dic106—21no—
94Dic106—21no—
95C8×C53—22no—
96C8×C53—41no—
97#88322—22no—
98#88322—41no—
99C22×C53—41no—
100C22×C53C2×C22⋊C411yescharacteristic
101C22×C53C4×D411yescharacteristic
102C22×C53C4×D411yescharacteristic
103C2×D53—22no—
104C2×D53—22no—
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—41no—
108C2×D53—41no—
109C2×D53—41no—
110C2×D53—41no—
111C2×D53—42no—
112C4×C53—42no—
113C4×C53#12011yescharacteristic
114C4×C53#12111yescharacteristic
115C4×C53#12111yescharacteristic
116C4×C53#12311yescharacteristic
117Dic53—22no—
118Dic53—41no—
119#1891—531nomaximal, Sylow
120C2×C53—21no—
121C2×C53—21no—
122C2×C53—21no—
123C2×C53#38611yescharacteristic
124D53—42no—
125D53—42no—
126#568—531no—
127#525—531no—
128#413—531no—
129#407—531no—
130#407—531no—
131#344—531no—
132#344D5311yescharacteristic, complemented
133C53#189111yescharacteristic, complemented, Sylow
134C2×#41—531no—
135C23×C4—531no—
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#117C2×D5311yescharacteristic
149#101—1061no—
150#100—1061no—
151C22⋊C8—1062no—
152C22⋊C8—1062no—
153C24—531no—
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—532no—
163C22×C4—531no—
164C22×C4—1061no—
165C22×C4—1061no—
166C22×C4—1061no—
167C22×C4—1061no—
168C22×C4—1061no—
169C22×C4—1061no—
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×C8D21212yescomplemented
185C4⋊C4—21no—
186C4⋊C4—1062no—
187C22⋊C4—1062no—
188C22⋊C4—1062no—
189C22⋊C4C4×D5312yes—
190C42—21no—
191C23—532no—
192C23—1061no—
193C23—1061no—
194C23—1061no—
195C23—2122no—
196C23C2×C4×D5311yescharacteristic
197Q8—1061no—
198Q8—1061no—
199D4—1062no—
200D4—1062no—
201D4—1061no—
202D4—1061no—
203D4—2124no—
204C2×C4—22no—
205C2×C4—532no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4—2121no—
214C2×C4—2121no—
215C2×C4C2×D21211yescharacteristic
216C2×C4#1171994511yescharacteristic
217C2×C4D4×D5311yescharacteristic
218C2×C4D4×D5311yescharacteristic
219C2×C4D4×D5311yescharacteristic
220C2×C4#1171994911yescharacteristic
221C8—22no—
222C8—41no—
223C8—1062no—
224C8—2121no—
225C22—41no—
226C22—1062no—
227C22—1062no—
228C22—1062no—
229C22—1062no—
230C22—2122no—
231C22—2121no—
232C22—2121no—
233C22—2121no—
234C22—2121no—
235C22C22⋊C4×D5311yescharacteristic
236C22#1313317911yescharacteristic
237C22C4×D21211yescharacteristic
238C4—42no—
239C4—1062no—
240C4—2121no—
241C4#1313318711yescharacteristic
242C4#1313319011yescharacteristic
243C4#1313319211yescharacteristic
244C4#1313320211yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—2122no—
249C2—2122no—
250C2#1681050111yescharacteristic
subgroupG/H
centerC4#13133187
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#344C2
socleC2×C53#386
hypercenter#344D53
hypocenterC53#1891
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #16811389C2semidirect
3. #16810605C2semidirect
4. #16810586C2semidirect
5. #16810586C2semidirect
6. #16810393C2semidirect
7. C53×#344C2semidirect
50. C2×C8×C53D4semidirect
62. C2×#88322D4semidirect
132. #344D53semidirect
133. C53#1891semidirect
184. C2×C8D212semidirect