Finite Group Databasebeta

← #20286108

Subgroups — #20286108

Order 6784.

342 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 115 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#16811178C211yesmaximal, characteristic
2#16811177C211yesmaximal, characteristic
3#16810907C212yesmaximal, complemented
4#16810905C211yesmaximal, characteristic, complemented
5#16810882C212yesmaximal
6#16810880C212yesmaximal
7#16810834C211yesmaximal, characteristic, complemented
8#16810833C211yesmaximal, characteristic, complemented
9#16810832C212yesmaximal, complemented
10#16810645C211yesmaximal, characteristic
11C53×#475C211yesmaximal, characteristic, complemented
12C2×Dic212C2212yes—
13C2×#11719925C2212yes—
14C2×#11719912C2212yes—
15#13133217C2211yescharacteristic
16#13133216C2211yescharacteristic
17#13133215C2211yescharacteristic
18#13133208—21no—
19#13133207—21no—
20#13133197C2212yes—
21#13133197C2211yescharacteristic
22#13133195—21no—
23#13133193—21no—
24#13133193C2211yescharacteristic
25Q8×Dic53—21no—
26C4×Dic106—21no—
27C4×Dic106C2211yescharacteristic
28#13133182C2212yes—
29#13133175—21no—
30#13133174—21no—
31C42×D53C2211yescharacteristic
32#13133148C2212yes—
33C4⋊#88322C2211yescharacteristic
34#13133146C2212yes—
35#13133142C2212yes—
36#13133140C2212yes—
37#13133140C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39C4×Q8×C53C2211yescharacteristic
40#106×C53C2212yes—
41C4⋊C8×C53C2211yescharacteristic
42#103×C53C2212yes—
43C2×Q8×C53—21no—
44C2×Q8×C53C2311yescharacteristic
45C2×C4×D53D411yescharacteristic
46C2×C4×D53D411yescharacteristic
47C2×C4×D53C2311yescharacteristic
48C2×Dic106—21no—
49C2×Dic106C2311yescharacteristic
50C2×Dic106C2312yes—
51C2×C8×C53C2312yes—
52C4⋊C4×C53—21no—
53C4⋊C4×C53—21no—
54C4⋊C4×C53C2312yes—
55C4⋊C4×C53C2311yescharacteristic
56C42×C53—21no—
57C42×C53C2311yescharacteristic
58#11719925—42no—
59#11719921—21no—
60#11719921—21no—
61#11719921—22no—
62C4⋊Dic53—21no—
63C4⋊Dic53—21no—
64C4⋊Dic53—21no—
65C4⋊Dic53—22no—
66C4⋊Dic53C2311yescharacteristic
67#11719919—21no—
68#11719919—21no—
69#11719919—21no—
70#11719919—22no—
71#11719919—21no—
72#11719919—21no—
73C4×Dic53—21no—
74C4×Dic53—22no—
75C4×Dic53D411yescharacteristic
76C4×Dic53D411yescharacteristic
77C4×Dic53C2311yescharacteristic
78C2×#88322C2312yes—
79Dic212—42no—
80#11719912—42no—
81C22×D53C2×D411yescharacteristic
82Q8×C53—21no—
83Q8×C53—21no—
84Q8×C53—42no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53C2×D411yescharacteristic
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53C2411yescharacteristic
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53C2×D411yescharacteristic
95C2×Dic53C2×D411yescharacteristic
96C2×Dic53C2×D411yescharacteristic
97C4×D53—22no—
98C4×D53—21no—
99C4×D53—21no—
100C4×D53—22no—
101Dic106—21no—
102Dic106—22no—
103Dic106—21no—
104Dic106—22no—
105Dic106—42no—
106C8×C53—22no—
107#88322—22no—
108C22×C53C22×D411yescharacteristic
109C2×D53—21no—
110C2×D53—21no—
111C2×D53—21no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—41no—
115C4×C53—42no—
116C4×C53#13712yes—
117C4×C53H(5,𝔽2)11yescharacteristic
118C4×C532−1+411yescharacteristic
119Dic53—21no—
120Dic53—21no—
121Dic53—41no—
122Dic53—42no—
123Dic53—41no—
124Dic53#13712yes—
125#3255—531nomaximal, Sylow
126C2×C53#53711yescharacteristic
127C2×C53C2×#13711yescharacteristic
128C2×C53C2×#13711yescharacteristic
129D53—41no—
130#533—531no—
131#533—531no—
132#475—531no—
133#475—532no—
134#475D5311yescharacteristic, complemented
135#470—532no—
136#470—532no—
137#462—531no—
138#462—531no—
139#462—532no—
140#423—531no—
141C53#325511yescharacteristic, complemented, Sylow
142C2×Q16—532no—
143C2×Q16—532no—
144C2×M16—532no—
145C4⋊Q8—531no—
146C4⋊Q8—531no—
147C4⋊Q8—531no—
148C4⋊Q8C2×D5311yescharacteristic
149#125—1061no—
150#125—1061no—
151C22⋊Q8—531no—
152C22⋊Q8—531no—
153C22⋊Q8—532no—
154C22⋊Q8—1061no—
155C22⋊Q8—1061no—
156C4×Q8—532no—
157C4×Q8—531no—
158C4×Q8—1061no—
159C4×Q8—1061no—
160C4×Q8C2×D5311yescharacteristic
161#117—1061no—
162#117—1061no—
163C2×C42—531no—
164#106—532no—
165#106C2×D5312yes—
166C4⋊C8—532no—
167C4⋊C8—531no—
168C4⋊C8C2×D5311yescharacteristic
169#103—532no—
170#103—532no—
171#103—532no—
172#103C2×D5312yes—
173C2×Q8—21no—
174C2×Q8—531no—
175C2×Q8—532no—
176C2×Q8—1061no—
177C2×Q8C22×D5311yescharacteristic
178C22×C4—531no—
179C22×C4—531no—
180C22×C4—531no—
181Q16—2122no—
182Q16—2122no—
183M16—2122no—
184C2×C8—532no—
185C2×C8C22×D5312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—531no—
189C4⋊C4—1061no—
190C4⋊C4—1061no—
191C4⋊C4—1062no—
192C4⋊C4—1061no—
193C4⋊C4—1061no—
194C4⋊C4—1062no—
195C4⋊C4—1061no—
196C4⋊C4—1061no—
197C4⋊C4—1061no—
198C4⋊C4—1061no—
199C4⋊C4C22×D5311yescharacteristic
200C4⋊C4C22×D5312yes—
201C22⋊C4—1061no—
202C22⋊C4—1061no—
203C22⋊C4—1062no—
204C42—21no—
205C42—531no—
206C42—531no—
207C42—531no—
208C42—1062no—
209C42—1061no—
210C42C22×D5311yescharacteristic
211C23—531no—
212Q8—21no—
213Q8—21no—
214Q8—42no—
215Q8—1061no—
216Q8—1062no—
217Q8—1061no—
218Q8—1062no—
219Q8—2122no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—531no—
224C2×C4—531no—
225C2×C4—531no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4D4×D5311yescharacteristic
234C2×C4D4×D5311yescharacteristic
235C2×C4C23×D5311yescharacteristic
236C8—22no—
237C8—1062no—
238C22—1061no—
239C22—1061no—
240C22—1061no—
241C22C2×D4×D5311yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—41no—
246C4—532no—
247C4—1061no—
248C4—1061no—
249C4—2121no—
250C4—2121no—
251C4—2122no—
252C4#1313324612yes—
253C4#1313326311yescharacteristic
254C4#1313326511yescharacteristic
255C2—2121no—
256C2#1681120311yescharacteristic
257C2#137×D5311yescharacteristic
258C2C2×#1313324611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#475C2
socleC22×C53C22×D4
hypercenter#475D53
hypocenterC53#3255
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #16810907C2semidirect
4. #16810905C2semidirect
7. #16810834C2semidirect
8. #16810833C2semidirect
9. #16810832C2semidirect
11. C53×#475C2semidirect
134. #475D53semidirect
141. C53#3255semidirect