Finite Group Databasebeta

← #20286090

Subgroups — #20286090

Order 6784.

348 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 113 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, complemented
2#16811155C211yesmaximal, characteristic
3#16810933C212yesmaximal, complemented
4#16810915C211yesmaximal, characteristic, complemented
5#16810882C212yesmaximal, complemented
6#16810868C212yesmaximal
7#16810844C212yesmaximal, complemented
8#16810834C211yesmaximal, characteristic, complemented
9#16810828C211yesmaximal, characteristic, complemented
10#16810645C211yesmaximal, characteristic, complemented
11C53×#476C211yesmaximal, characteristic, complemented
12C2×Dic212C2212yes—
13C2×#11719923C2212yes—
14C2×#11719912C2212yes—
15#13133217C2211yescharacteristic
16#13133210—21no—
17#13133208—21no—
18#13133208C2211yescharacteristic
19#13133204C2211yescharacteristic
20#13133199—21no—
21#13133197C2212yes—
22#13133197C2211yescharacteristic
23#13133193—21no—
24#13133188C2211yescharacteristic
25Q8×Dic53—21no—
26C4×Dic106C2211yescharacteristic
27#13133182C2212yes—
28C4×C22⋊D53—21no—
29#13133176—21no—
30#13133175—21no—
31C42×D53C2211yescharacteristic
32#13133150C2212yes—
33C4⋊#88322C2211yescharacteristic
34#13133146C2212yes—
35#13133142C2212yes—
36#13133140C2212yes—
37#13133140C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39C4×D4×C53C2211yescharacteristic
40#106×C53C2212yes—
41C4⋊C8×C53C2211yescharacteristic
42#102×C53C2212yes—
43C2×Q8×C53—21no—
44C2×D4×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C2×C22⋊D53—21no—
47C2×C4×D53D411yescharacteristic
48C2×C4×D53D411yescharacteristic
49C2×C4×D53C2311yescharacteristic
50C2×Dic106C2311yescharacteristic
51C2×Dic106C2312yes—
52C2×C8×C53C2312yes—
53C4⋊C4×C53—21no—
54C4⋊C4×C53C2312yes—
55C4⋊C4×C53C2311yescharacteristic
56C22⋊C4×C53—21no—
57C42×C53C2311yescharacteristic
58C22⋊Dic53—21no—
59C22⋊Dic53—21no—
60C22⋊Dic53—21no—
61#11719923—42no—
62#11719921—22no—
63#11719921—21no—
64C4⋊Dic53—21no—
65C4⋊Dic53—21no—
66C4⋊Dic53—21no—
67C4⋊Dic53—21no—
68C4⋊Dic53—22no—
69C4⋊Dic53C2311yescharacteristic
70#11719919—21no—
71#11719919—21no—
72#11719919—22no—
73C4×Dic53—21no—
74C4×Dic53—22no—
75C4×Dic53D411yescharacteristic
76C4×Dic53D411yescharacteristic
77C4×Dic53C2311yescharacteristic
78C2×#88322C2312yes—
79Dic212—42no—
80#11719912—42no—
81C23×C53—21no—
82C22×D53C2×D411yescharacteristic
83Q8×C53—42no—
84D4×C53—21no—
85D4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—42no—
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53C2×D411yescharacteristic
91C2×C4×C53C2411yescharacteristic
92C22⋊D53—42no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53C2×D411yescharacteristic
97C2×Dic53C2×D411yescharacteristic
98C2×Dic53C2×D411yescharacteristic
99C4×D53—22no—
100C4×D53—21no—
101C4×D53—21no—
102C4×D53—41no—
103Dic106—21no—
104Dic106—22no—
105Dic106—21no—
106Dic106—22no—
107C8×C53—22no—
108#88322—22no—
109C22×C53—41no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53C22×D411yescharacteristic
113C2×D53—21no—
114C2×D53—21no—
115C2×D53—21no—
116C4×C53—21no—
117C4×C53—41no—
118C4×C53—42no—
119C4×C53#13712yes—
120C4×C53H(5,𝔽2)11yescharacteristic
121C4×C532−1+411yescharacteristic
122Dic53—21no—
123Dic53—21no—
124Dic53—21no—
125Dic53—41no—
126Dic53—42no—
127Dic53—41no—
128#3252—531nomaximal, Sylow
129C2×C53—81no—
130C2×C53#53711yescharacteristic
131C2×C53C2×#13711yescharacteristic
132C2×C53#57511yescharacteristic
133D53—41no—
134#533—531no—
135#529—531no—
136#479—532no—
137#477—531no—
138#476D5311yescharacteristic, complemented
139#470—532no—
140#468—532no—
141#464—532no—
142#462—531no—
143#461—531no—
144#423—531no—
145C53#325211yescharacteristic, complemented, Sylow
146C2×Q16—532no—
147C2×SD16—532no—
148C2×M16—532no—
149C4⋊Q8—531no—
150C4⋊Q8C2×D5311yescharacteristic
151#126—1061no—
152#125—531no—
153#125—1061no—
154#124—531no—
155#123—1061no—
156C22⋊Q8—531no—
157C22⋊Q8—532no—
158C22⋊Q8—1061no—
159#121—531no—
160C4×Q8—532no—
161C4×Q8—531no—
162C4×Q8—1061no—
163C4×D4—1061no—
164C4×D4C2×D5311yescharacteristic
165#117—1061no—
166#117—1061no—
167C2×C42—531no—
168#107—532no—
169#106C2×D5312yes—
170C4⋊C8—532no—
171C4⋊C8—531no—
172C4⋊C8C2×D5311yescharacteristic
173#103—532no—
174#103—532no—
175#103—532no—
176#102C2×D5312yes—
177C2×Q8—21no—
178C2×Q8—531no—
179C2×Q8—532no—
180C2×D4—1061no—
181C2×D4C22×D5311yescharacteristic
182C22×C4—21no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—531no—
186Q16—2122no—
187SD16—2122no—
188M16—2122no—
189C2×C8—532no—
190C2×C8C22×D5312yes—
191C4⋊C4—21no—
192C4⋊C4—531no—
193C4⋊C4—1061no—
194C4⋊C4—1061no—
195C4⋊C4—1061no—
196C4⋊C4—1062no—
197C4⋊C4—1062no—
198C4⋊C4—1061no—
199C4⋊C4—1061no—
200C4⋊C4—1061no—
201C4⋊C4C22×D5312yes—
202C4⋊C4C22×D5311yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—1062no—
205C22⋊C4—1061no—
206C22⋊C4—1061no—
207C22⋊C4—1061no—
208C22⋊C4—1061no—
209C42—531no—
210C42—531no—
211C42—531no—
212C42—1062no—
213C42—1061no—
214C42C22×D5311yescharacteristic
215C23—21no—
216C23—531no—
217Q8—42no—
218Q8—1062no—
219Q8—1061no—
220Q8—1061no—
221Q8—1062no—
222D4—21no—
223D4—21no—
224D4—2122no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—531no—
229C2×C4—531no—
230C2×C4—531no—
231C2×C4—1062no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1061no—
236C2×C4—1062no—
237C2×C4—2121no—
238C2×C4D4×D5311yescharacteristic
239C2×C4D4×D5311yescharacteristic
240C2×C4C23×D5311yescharacteristic
241C8—22no—
242C8—1062no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—1061no—
247C22—1061no—
248C22—1061no—
249C22C2×D4×D5311yescharacteristic
250C4—21no—
251C4—41no—
252C4—42no—
253C4—1061no—
254C4—1061no—
255C4—1061no—
256C4—2121no—
257C4—2121no—
258C4—2122no—
259C4#1313324612yes—
260C4#1313326311yescharacteristic
261C4#1313326511yescharacteristic
262C2—81no—
263C2—2121no—
264C2#1681120311yescharacteristic
265C2C2×#1313324611yescharacteristic
266C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#476C2
socleC22×C53C22×D4
hypercenter#476D53
hypocenterC53#3252
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16810933C2semidirect
4. #16810915C2semidirect
5. #16810882C2semidirect
7. #16810844C2semidirect
8. #16810834C2semidirect
9. #16810828C2semidirect
10. #16810645C2semidirect
11. C53×#476C2semidirect
138. #476D53semidirect
145. C53#3252semidirect