Finite Group Databasebeta

← #352×D53 #20276349

Subgroups — #352×D53 #20276349

Order 6784.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#123×D53C211yesmaximal, characteristic
2#100×D53C211yesmaximal, characteristic
3#99×D53C211yesmaximal, characteristic
4#16810414C211yesmaximal, characteristic, complemented
5#16810413C211yesmaximal, characteristic, complemented
6#16810412C211yesmaximal, characteristic, complemented
7C53×#352C211yesmaximal, characteristic, complemented
8C2×D4×D53C2211yescharacteristic
9C22×C4×D53C411yescharacteristic, complemented
10M16×D53—21no—
11#13133202—21no—
12#13133201C411yescharacteristic, complemented
13#13133200C411yescharacteristic, complemented
14#13133199—21no—
15#13133198C2211yescharacteristic
16C4⋊C4×D53—21no—
17C22⋊C4×D53—21no—
18C22⋊C4×D53—21no—
19C22⋊C4×D53C411yescharacteristic, complemented
20#13133134C2211yescharacteristic
21#13133133—21no—
22#13133132C2211yescharacteristic
23#13133131—21no—
24#123×C53C2211yescharacteristic
25#100×C53C2211yescharacteristic
26#99×C53C2211yescharacteristic
27C23×D53C2×C411yescharacteristic
28C23×D53D411yescharacteristic
29C2×D4×C53C2311yescharacteristic
30C22×C4×C53C2×C411yescharacteristic, complemented
31C2×C22⋊D53C2×C411yescharacteristic
32C2×C22⋊D53D411yescharacteristic
33C22×Dic53C2×C411yescharacteristic, complemented
34D4×D53—41no—
35D4×D53—41no—
36C2×D212D411yescharacteristic
37C2×C4×D53—21no—
38C2×C4×D53—21no—
39C2×C4×D53—21no—
40C2×C4×D53—41no—
41C2×C4×D53—41no—
42C2×C4×D53—41no—
43C2×C4×D53D411yescharacteristic
44M16×C53—21no—
45C4⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53C2×C411yescharacteristic, complemented
49C22⋊Dic53—21no—
50C22⋊Dic53—21no—
51C22⋊Dic53C2×C411yescharacteristic, complemented
52#11719921—21no—
53#11719921—21no—
54#11719921—21no—
55#11719921—41no—
56C4⋊Dic53—21no—
57#11719919—21no—
58#11719919—21no—
59#11719917—21no—
60#11719912—41no—
61C8×D53—41no—
62C23×C53C22×C411yescharacteristic
63C23×C53C2×D411yescharacteristic
64C22×D53—21no—
65C22×D53—21no—
66C22×D53—21no—
67C22×D53—21no—
68C22×D53—42no—
69C22×D53—42no—
70C22×D53C22⋊C411yescharacteristic
71C22×D53C22⋊C411yescharacteristic
72C22×D53C22⋊C411yescharacteristic
73D4×C53—41no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—41no—
77C2×C4×C53—41no—
78C2×C4×C53C2×D411yescharacteristic
79C22⋊D53—41no—
80C22⋊D53—41no—
81C2×Dic53—21no—
82C2×Dic53—21no—
83C2×Dic53—41no—
84C2×Dic53—41no—
85C2×Dic53C22⋊C411yescharacteristic
86D212—41no—
87C4×D53—21no—
88C4×D53—21no—
89C4×D53—42no—
90C4×D53—42no—
91C4×D53—41no—
92C4×D53—41no—
93C4×D53—82no—
94C8×C53—41no—
95#88322—41no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53C2×C22⋊C411yescharacteristic
101C2×D53—22no—
102C2×D53—21no—
103C2×D53—21no—
104C2×D53—42no—
105C2×D53—42no—
106C2×D53—42no—
107C2×D53—41no—
108C2×D53—41no—
109C2×D53—81no—
110C2×D53#9911yescharacteristic
111C2×D53#9911yescharacteristic
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—41no—
115C4×C53—81no—
116Dic53—21no—
117Dic53—41no—
118Dic53—41no—
119Dic53—81no—
120C2×#352—531nomaximal, Sylow
121C2×C53—21no—
122C2×C53—41no—
123C2×C53—41no—
124C2×C53C2×#9911yescharacteristic
125D53—21no—
126D53—41no—
127D53—41no—
128D53#35212yesdirect product factor
129C2×#123—531no—
130C2×#100—531no—
131C2×#99—531no—
132#352—531no—
133#352—531no—
134#352—531no—
135#352D5311yescharacteristic, direct product factor
136C53C2×#35211yescharacteristic, complemented, Sylow
137C22×D4—531no—
138C23×C4—531no—
139C2×M16—1061no—
140#123—531no—
141#123—531no—
142#123—531no—
143#123—1061no—
144#123—1061no—
145#123C2×D5311yescharacteristic
146C2×C4⋊C4—1061no—
147C2×C22⋊C4—531no—
148C2×C22⋊C4—1061no—
149C2×C22⋊C4—1061no—
150#100—531no—
151#100—1061no—
152#100C2×D5311yescharacteristic
153#99—531no—
154#99—1061no—
155#99C2×D5311yescharacteristic
156C24—531no—
157C24—531no—
158C2×D4—531no—
159C2×D4—531no—
160C2×D4—531no—
161C2×D4—2121no—
162C2×D4—2121no—
163C2×D4C22×D5311yescharacteristic
164C22×C4—531no—
165C22×C4—531no—
166C22×C4—1061no—
167C22×C4—1061no—
168C22×C4—1061no—
169C22×C4—2121no—
170C22×C4—2121no—
171C22×C4—2121no—
172C22×C4C4×D5311yescharacteristic, complemented
173M16—21no—
174M16—1061no—
175M16—2121no—
176C2×C8—2121no—
177C4⋊C4—21no—
178C4⋊C4—1061no—
179C4⋊C4—1061no—
180C4⋊C4—1061no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—531no—
184C22⋊C4—1061no—
185C22⋊C4—1061no—
186C22⋊C4—1061no—
187C22⋊C4—1061no—
188C22⋊C4—1061no—
189C22⋊C4—2121no—
190C22⋊C4C4×D5311yescharacteristic, complemented
191C23—531no—
192C23—531no—
193C23—531no—
194C23—1061no—
195C23—1061no—
196C23—1061no—
197C23—1061no—
198C23—2122no—
199C23—2122no—
200C23C2×C4×D5311yescharacteristic
201C23D4×D5311yescharacteristic
202D4—41no—
203D4—2121no—
204D4—2121no—
205D4—2121no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—531no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—1061no—
214C2×C4—1061no—
215C2×C4—4242no—
216C2×C4—2121no—
217C2×C4—2121no—
218C2×C4—2122no—
219C2×C4—2122no—
220C2×C4—2121no—
221C2×C4—2121no—
222C2×C4D4×D5311yescharacteristic
223C8—41no—
224C8—2121no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—531no—
230C22—531no—
231C22—1062no—
232C22—1061no—
233C22—1061no—
234C22—4241no—
235C22—2121no—
236C22—2121no—
237C22—2122no—
238C22—2122no—
239C22—2122no—
240C22C22⋊C4×D5311yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—1061no—
246C4—4241no—
247C4—2121no—
248C4—2121no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2—532no—
253C2—1061no—
254C2—2121no—
255C2—2121no—
256C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#352C2
socleC2×C53C2×#99
hypercenter#352D53
hypocenterC53C2×#352
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #16810414C2semidirect
5. #16810413C2semidirect
6. #16810412C2semidirect
7. C53×#352C2semidirect
9. C22×C4×D53C4semidirect
12. #13133201C4semidirect
13. #13133200C4semidirect
19. C22⋊C4×D53C4semidirect
30. C22×C4×C53C2×C4semidirect
33. C22×Dic53C2×C4semidirect
48. C22⋊C4×C53C2×C4semidirect
51. C22⋊Dic53C2×C4semidirect
128. D53#352direct (N × H)
135. #352D53direct (N × H)
136. C53C2×#352semidirect
172. C22×C4C4×D53semidirect
190. C22⋊C4C4×D53semidirect