Finite Group Databasebeta

← #20284513

Subgroups — #20284513

Order 6784.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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
2C4×D4×D53C211yesmaximal, characteristic
3#16810922C212yesmaximal, complemented
4#16810921C212yesmaximal, complemented
5#16810910C211yesmaximal, characteristic, complemented
6#16810904C211yesmaximal, characteristic, complemented
7#16810903C211yesmaximal, characteristic, complemented
8#16810662C212yesmaximal, complemented
9#16810645C211yesmaximal, characteristic, complemented
10#16810626C212yesmaximal
11C53×#476C211yesmaximal, characteristic, complemented
12C2×D4×D53C2211yescharacteristic
13C22×C4×D53—21no—
14C2×#11719912C2212yes—
15#13133217C2211yescharacteristic, complemented
16#13133208C2212yescomplemented
17#13133197—21no—
18#13133193—21no—
19Q8×Dic53—21no—
20#13133182—21no—
21C4×D212C2211yescharacteristic
22D4×Dic53C2211yescharacteristic
23#13133179C2211yescharacteristic
24C4×C22⋊D53—21no—
25#13133177—21no—
26#13133175C2212yes—
27C4⋊C4×D53C2211yescharacteristic
28C22⋊C4×D53—21no—
29C42×D53C2211yescharacteristic
30#13133151C2212yescomplemented
31#13133150C2212yescomplemented
32#13133148C2212yescomplemented
33C4⋊#88322C2211yescharacteristic, complemented
34#13133146C2212yescomplemented
35#13133139C2212yes—
36#13133138C2212yes—
37#13133137C2212yes—
38C4⋊Q8×C53C2211yescharacteristic, complemented
39C4×D4×C53C2211yescharacteristic
40#106×C53C2212yescomplemented
41C4⋊C8×C53C2211yescharacteristic, complemented
42#102×C53C2212yes—
43C23×D53—21no—
44C2×Q8×C53—21no—
45C2×D4×C53C2311yescharacteristic
46C22×C4×C53—21no—
47C2×C22⋊D53—21no—
48C22×Dic53—21no—
49D4×D53—22no—
50D4×D53Q814yescomplemented
51C2×D212C2311yescharacteristic
52C2×C4×D53—21no—
53C2×C4×D53—21no—
54C2×C4×D53—21no—
55C2×C4×D53—44no—
56C2×C4×D53D411yescharacteristic
57C2×C4×D53D411yescharacteristic
58C2×C4×D53C2311yescharacteristic
59C2×Dic106—21no—
60C2×C8×C53C2312yes—
61C4⋊C4×C53—21no—
62C4⋊C4×C53C2312yes—
63C4⋊C4×C53C2311yescharacteristic
64C22⋊C4×C53—21no—
65C42×C53C2311yescharacteristic
66C22⋊Dic53—21no—
67#11719921—22no—
68#11719921—21no—
69#11719921—21no—
70C4⋊Dic53—21no—
71C4⋊Dic53—22no—
72C4⋊Dic53C2312yes—
73C4⋊Dic53C2311yescharacteristic
74#11719919—21no—
75#11719919—22no—
76C4×Dic53—22no—
77C4×Dic53D411yescharacteristic
78C4×Dic53D411yescharacteristic
79C4×Dic53C2311yescharacteristic
80C2×#88322C2312yes—
81#11719912—24no—
82C23×C53—21no—
83C22×D53—21no—
84C22×D53—24no—
85C22×D53—42no—
86C22×D53—42no—
87C22×D53C2×D411yescharacteristic
88Q8×C53—42no—
89D4×C53—21no—
90D4×C53C2×Q812yes—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—42no—
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2411yescharacteristic
97C22⋊D53—24no—
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—42no—
101C2×Dic53C2×D411yescharacteristic
102C2×Dic53C2×D411yescharacteristic
103C2×Dic53C2×D411yescharacteristic
104D212—21no—
105D212C2×Q812yes—
106C4×D53—22no—
107C4×D53—41no—
108C4×D53—42no—
109C4×D53—42no—
110C4×D53—42no—
111C4×D53C2×Q812yes—
112C4×D53#4112yes—
113Dic106—42no—
114C8×C53—22no—
115#88322—22no—
116C22×C53—22no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53C22×D411yescharacteristic
120C2×D53—21no—
121C2×D53—21no—
122C2×D53—22no—
123C2×D53—48no—
124C2×D53—41no—
125C2×D53—41no—
126C2×D53C22⋊Q812yes—
127C4×C53—21no—
128C4×C53—41no—
129C4×C53—42no—
130C4×C53AGL(1,ℤ/8)12yes—
131C4×C53C22×Q811yescharacteristic
132C4×C53C2×#4111yescharacteristic
133Dic53—21no—
134Dic53—21no—
135Dic53—42no—
136Dic53—41no—
137Dic53C22⋊Q812yes—
138#3077—531nomaximal, Sylow
139C2×C53—42no—
140C2×C53C2×C22⋊Q811yescharacteristic
141C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C53#57511yescharacteristic
143D53—22no—
144D53—42no—
145#533—531no—
146C2×C4×D4—531no—
147#478—532no—
148#478—532no—
149#476—531no—
150#476D5311yescharacteristic, complemented
151#474—531no—
152#474—531no—
153#428—532no—
154#423—531no—
155#418—532no—
156C53#307711yescharacteristic, complemented, Sylow
157C22×D4—531no—
158C23×C4—1061no—
159C2×M16—532no—
160C4⋊Q8—531no—
161C4⋊Q8C2×D5311yescharacteristic, complemented
162#125—532no—
163C22⋊Q8—1061no—
164C22⋊Q8—1061no—
165C4×Q8—1061no—
166C4×Q8—1061no—
167C4×D4—531no—
168C4×D4—531no—
169C4×D4—531no—
170C4×D4—1061no—
171C4×D4—1061no—
172C4×D4C2×D5311yescharacteristic
173#117—532no—
174C2×C4⋊C4—531no—
175C2×C22⋊C4—1061no—
176C2×C42—531no—
177#107—532no—
178#107—532no—
179#106—532no—
180#106C2×D5312yescomplemented
181C4⋊C8—532no—
182C4⋊C8—531no—
183C4⋊C8C2×D5311yescharacteristic, complemented
184#102—532no—
185#102—532no—
186#102—532no—
187#102C2×D5312yes—
188C24—1061no—
189C2×Q8—21no—
190C2×Q8—1061no—
191C2×D4—534no—
192C2×D4—531no—
193C2×D4—1062no—
194C2×D4—1061no—
195C2×D4C22×D5311yescharacteristic
196C22×C4—21no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—1061no—
201C22×C4—1061no—
202C22×C4—1061no—
203C22×C4—1061no—
204C22×C4—2124no—
205M16—1064no—
206C2×C8—532no—
207C2×C8C22×D5312yes—
208C4⋊C4—21no—
209C4⋊C4—532no—
210C4⋊C4—531no—
211C4⋊C4—1061no—
212C4⋊C4—1062no—
213C4⋊C4—1062no—
214C4⋊C4—1061no—
215C4⋊C4C22×D5312yes—
216C4⋊C4C22×D5311yescharacteristic
217C22⋊C4—21no—
218C22⋊C4—1061no—
219C22⋊C4—1062no—
220C22⋊C4—1061no—
221C22⋊C4—1061no—
222C42—531no—
223C42—531no—
224C42—531no—
225C42—1062no—
226C42C22×D5311yescharacteristic
227C23—21no—
228C23—531no—
229C23—1064no—
230C23—1061no—
231C23—2122no—
232C23—2122no—
233Q8—42no—
234Q8—2122no—
235D4—21no—
236D4—532no—
237D4—1064no—
238D4—1061no—
239D4Q8×D5312yes—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—531no—
244C2×C4—532no—
245C2×C4—532no—
246C2×C4—531no—
247C2×C4—531no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1061no—
251C2×C4—2121no—
252C2×C4—2122no—
253C2×C4—2122no—
254C2×C4—2122no—
255C2×C4—2122no—
256C2×C4D4×D5311yescharacteristic
257C2×C4D4×D5311yescharacteristic
258C2×C4C23×D5311yescharacteristic
259C8—22no—
260C8—1062no—
261C22—22no—
262C22—41no—
263C22—41no—
264C22—532no—
265C22—1061no—
266C22—1061no—
267C22—1062no—
268C22—2128no—
269C22—2121no—
270C22—2121no—
271C22C2×D4×D5311yescharacteristic
272C4—21no—
273C4—41no—
274C4—42no—
275C4—532no—
276C4—1061no—
277C4—1061no—
278C4—2121no—
279C4—2122no—
280C4#1313324412yes—
281C4C2×Q8×D5311yescharacteristic
282C4#41×D5311yescharacteristic
283C2—42no—
284C2—1062no—
285C2—2122no—
286C2C22⋊Q8×D5311yescharacteristic
287C2C2×#1313324411yescharacteristic
288C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#476C2
socleC22×C53C22×D4
hypercenter#476D53
hypocenterC53#3077
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. #16810922C2semidirect
4. #16810921C2semidirect
5. #16810910C2semidirect
6. #16810904C2semidirect
7. #16810903C2semidirect
8. #16810662C2semidirect
9. #16810645C2semidirect
11. C53×#476C2semidirect
15. #13133217C22semidirect
16. #13133208C22semidirect
30. #13133151C22semidirect
31. #13133150C22semidirect
32. #13133148C22semidirect
33. C4⋊#88322C22semidirect
34. #13133146C22semidirect
38. C4⋊Q8×C53C22semidirect
40. #106×C53C22semidirect
41. C4⋊C8×C53C22semidirect
50. D4×D53Q8semidirect
150. #476D53semidirect
156. C53#3077semidirect
161. C4⋊Q8C2×D53semidirect
180. #106C2×D53semidirect
183. C4⋊C8C2×D53semidirect