Finite Group Databasebeta

← #20283991

Subgroups — #20283991

Order 6784.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#13133263C211yesmaximal, characteristic
2C22×#11719924C211yesmaximal, characteristic, complemented
3#16811197C211yesmaximal, characteristic, complemented
4#16810784C214yesmaximal, complemented
5#16810772C214yesmaximal, complemented
6#16810629C211yesmaximal, characteristic, complemented
7#16810619C211yesmaximal, characteristic, complemented
8#16810591C211yesmaximal, characteristic, complemented
9C2×#137×C53C211yesmaximal, characteristic, complemented
10#13133263—24no—
11#13133263—24no—
12C2×#11719949C2212yes—
13C2×#11719945—21no—
14C2×#11719945C2211yescharacteristic
15C22×D212—21no—
16C22×D212C2211yescharacteristic
17C2×D4×D53—22no—
18C2×D4×D53C2212yes—
19C2×#11719924—24no—
20C2×#11719924C2214yescomplemented
21C22×#88322C2211yescharacteristic, complemented
22#13133215—21no—
23#13133206—21no—
24#13133200—22no—
25#13133193C2214yescomplemented
26#13133176C2211yescharacteristic, complemented
27#13133141C2212yescomplemented
28#13133139C2214yescomplemented
29#13133138C2212yescomplemented
30#13133129C2214yescomplemented
31C2×#41×C53C2211yescharacteristic
32C22×Q8×C53C2211yescharacteristic, complemented
33#137×C53—24no—
34C2×Q16×C53C2212yescomplemented
35C2×SD16×C53C2212yescomplemented
36C2×M16×C53C2211yescharacteristic, complemented
37C23×D53—21no—
38C23×D53—22no—
39#41×C53—22no—
40#41×C53D414yes—
41C2×Q8×C53—21no—
42C2×Q8×C53—22no—
43C2×Q8×C53—22no—
44C2×Q8×C53C2312yes—
45C2×Q8×C53C2311yescharacteristic
46C2×D4×C53—21no—
47C2×D4×C53C2311yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—22no—
52C2×C22⋊D53D412yes—
53#11719949—24no—
54#11719949—24no—
55D4×D53—216no—
56D4×D53—24no—
57D4×D53—24no—
58#11719945—28no—
59#11719945—22no—
60#11719945—22no—
61C2×D212—28no—
62C2×D212—22no—
63C2×D212—21no—
64C2×D212—22no—
65C2×D212D411yescharacteristic
66C2×D212D414yes—
67C2×D212C2312yes—
68C2×C4×D53—22no—
69C2×C4×D53C2312yes—
70C2×Dic106D411yescharacteristic
71Q16×C53—24no—
72SD16×C53—24no—
73M16×C53—22no—
74C2×C8×C53C2312yes—
75C22⋊Dic53—21no—
76#11719924—28no—
77#11719921—24no—
78C4⋊Dic53C2312yes—
79#11719919—24no—
80C4×Dic53—21no—
81C2×#88322—22no—
82C2×#88322C2312yes—
83C23×C53—21no—
84C23×C53C2×D411yescharacteristic
85C22×D53—22no—
86C22×D53—28no—
87C22×D53—24no—
88C22×D53—84no—
89C22×D53—82no—
90C22×D53C2×D412yes—
91Q8×C53—21no—
92Q8×C53—22no—
93Q8×C53—22no—
94Q8×C53—42no—
95Q8×C53C2×D412yes—
96D4×C53—24no—
97D4×C53—21no—
98D4×C53C2×D412yes—
99C2×C4×C53—21no—
100C2×C4×C53—24no—
101C2×C4×C53—22no—
102C2×C4×C53—42no—
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D412yes—
105C2×C4×C53C2×D412yes—
106C2×C4×C53C2411yescharacteristic
107C22⋊D53—28no—
108C22⋊D53—24no—
109C2×Dic53—22no—
110C2×Dic53C2×D412yes—
111D212—28no—
112D212—22no—
113D212—24no—
114D212—22no—
115D212C2×D414yes—
116C4×D53—28no—
117C4×D53—22no—
118C4×D53—22no—
119Dic106—22no—
120C8×C53—22no—
121#88322—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—42no—
125C22×C53#12012yes—
126C22×C53C22×D411yescharacteristic
127C2×D53—22no—
128C2×D53—24no—
129C2×D53—24no—
130C2×D53—44no—
131C2×D53—84no—
132C2×D53—88no—
133C4×C53—22no—
134C4×C53—42no—
135C4×C53#12012yes—
136C4×C53C22×D411yescharacteristic
137C4×C53C22×D411yescharacteristic
138Dic53—22no—
139Dic53—42no—
140#3009—531nomaximal, Sylow
141C2×C53—22no—
142C2×C53—42no—
143C2×C53C2×#12011yescharacteristic
144C2×C53#57712yes—
145D53—42no—
146D53—44no—
147C2×H(5,𝔽2)—531no—
148C2×#137D5311yescharacteristic, complemented
149C22×SD16—531no—
150#536—531no—
151#452—534no—
152#450—534no—
153#418—531no—
154#416—531no—
155#408—531no—
156C53#300911yescharacteristic, complemented, Sylow
157H(5,𝔽2)—1064no—
158H(5,𝔽2)—1064no—
159C2×#41—531no—
160C2×#41—532no—
161C2×#41—1061no—
162C2×#41C2×D5311yescharacteristic
163C22×Q8C2×D5311yescharacteristic, complemented
164C22×D4—532no—
165C22×D4—531no—
166C22×D4—1061no—
167C22×D4—1062no—
168#137—24no—
169C2×Q16C2×D5312yescomplemented
170C2×SD16—534no—
171C2×SD16—1064no—
172C2×SD16C2×D5312yescomplemented
173C2×M16C2×D5311yescharacteristic, complemented
174C22×C8—531no—
175C4⋊Q8—1061no—
176#124—1061no—
177#123—1062no—
178C22⋊Q8—534no—
179#117—531no—
180#103—532no—
181#102—532no—
182#102—534no—
183C22⋊C8—534no—
184C24—1062no—
185C24—1061no—
186#41—22no—
187#41—1068no—
188#41—1064no—
189#41—1064no—
190#41—1062no—
191#41—1062no—
192#41C22⋊D5314yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—531no—
197C2×Q8C22×D5312yes—
198C2×Q8C22×D5311yescharacteristic
199C2×D4—21no—
200C2×D4—532no—
201C2×D4—534no—
202C2×D4—531no—
203C2×D4—532no—
204C2×D4—1062no—
205C2×D4—1068no—
206C2×D4—1062no—
207C2×D4—1064no—
208C2×D4—10616no—
209C2×D4—1064no—
210C2×D4—1061no—
211C2×D4—1062no—
212C2×D4C22×D5311yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—532no—
216C22×C4—1062no—
217C22×C4C22×D5311yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—1068no—
221M16—22no—
222C2×C8—532no—
223C2×C8—1062no—
224C2×C8C22×D5312yes—
225C4⋊C4—532no—
226C4⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—1061no—
229C42—1061no—
230C23—21no—
231C23—532no—
232C23—1068no—
233C23—1064no—
234C23—1062no—
235C23—4242no—
236C23—4244no—
237C23D4×D5311yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—1062no—
243Q8C2×C22⋊D5312yes—
244D4—24no—
245D4—21no—
246D4—534no—
247D4—1068no—
248D4—1068no—
249D4—1064no—
250D4—1064no—
251D4—1062no—
252D4—1062no—
253D4C2×C22⋊D5312yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—532no—
259C2×C4—1068no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4D4×D5311yescharacteristic
264C2×C4D4×D5312yes—
265C2×C4C2×C22⋊D5312yes—
266C2×C4C23×D5311yescharacteristic
267C8—22no—
268C8—1062no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—1064no—
273C22—1062no—
274C22—1064no—
275C22—4244no—
276C22—4248no—
277C22—2124no—
278C22#1313318512yes—
279C22C2×D4×D5311yescharacteristic
280C4—22no—
281C4—42no—
282C4—1062no—
283C4—2122no—
284C4#1313318512yes—
285C4C2×D4×D5311yescharacteristic
286C4C22×C22⋊D5311yescharacteristic
287C2—22no—
288C2—42no—
289C2—2122no—
290C2—2124no—
291C2C2×#1313318511yescharacteristic
292C2#1681143212yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#137×C53C2
socleC22×C53C22×D4
hypercenterC2×#137D53
hypocenterC53#3009
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#11719924C2semidirect
3. #16811197C2semidirect
4. #16810784C2semidirect
5. #16810772C2semidirect
6. #16810629C2semidirect
7. #16810619C2semidirect
8. #16810591C2semidirect
9. C2×#137×C53C2semidirect
20. C2×#11719924C22semidirect
21. C22×#88322C22semidirect
25. #13133193C22semidirect
26. #13133176C22semidirect
27. #13133141C22semidirect
28. #13133139C22semidirect
29. #13133138C22semidirect
30. #13133129C22semidirect
32. C22×Q8×C53C22semidirect
34. C2×Q16×C53C22semidirect
35. C2×SD16×C53C22semidirect
36. C2×M16×C53C22semidirect
148. C2×#137D53semidirect
156. C53#3009semidirect
163. C22×Q8C2×D53semidirect
169. C2×Q16C2×D53semidirect
172. C2×SD16C2×D53semidirect
173. C2×M16C2×D53semidirect