Finite Group Databasebeta

← #20275765

Subgroups — #20275765

Order 6784.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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×#13133242C211yesmaximal, characteristic, complemented
2#16811187C211yesmaximal, characteristic, complemented
3#16810640C211yesmaximal, characteristic, complemented
4#16810629C211yesmaximal, characteristic, complemented
5#16810597C211yesmaximal, characteristic, complemented
6#16810394C211yesmaximal, characteristic, complemented
7C53×#575C211yesmaximal, characteristic, complemented
8C22×D212C2211yescharacteristic
9#13133242—22no—
10#13133242—22no—
11C2×#11719924—21no—
12C2×#11719922—21no—
13C2×#11719917C2211yescharacteristic
14#13133213—21no—
15#13133206—21no—
16#13133191—22no—
17#13133185—22no—
18#13133176C2211yescharacteristic
19#13133176C2211yescharacteristic
20#13133144—22no—
21#13133139—22no—
22#13133132—22no—
23C2×#41×C53C2211yescharacteristic
24C2×#41×C53C2211yescharacteristic
25#137×C53—21no—
26#137×C53—21no—
27AGL(1,ℤ/8)×C53—21no—
28AGL(1,ℤ/8)×C53—21no—
29#135×C53—22no—
30C2×M16×C53C2211yescharacteristic
31C23×D53—21no—
32#41×C53—21no—
33#41×C53—21no—
34#41×C53—21no—
35#41×C53—21no—
36#41×C53—21no—
37#41×C53—21no—
38#41×C53—21no—
39#41×C53—21no—
40C2×Q8×C53D411yescharacteristic
41C2×Q8×C53D411yescharacteristic, complemented
42C2×D4×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53D411yescharacteristic
45C2×D4×C53D411yescharacteristic, complemented
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C2311yescharacteristic
49C2×C22⋊D53—24no—
50C2×D212—21no—
51C2×D212—21no—
52C2×D212—21no—
53C2×D212—21no—
54C2×D212—21no—
55C2×D212D412yes—
56Q16×C53—22no—
57SD16×C53—22no—
58SD16×C53—22no—
59D8×C53—22no—
60M16×C53—21no—
61M16×C53—21no—
62C2×C8×C53D412yescomplemented
63C22⋊Dic53—21no—
64C22⋊Dic53D412yes—
65#11719924—24no—
66#11719922—24no—
67#11719921—24no—
68C4⋊Dic53—21no—
69C4⋊Dic53—21no—
70C4×Dic53—21no—
71C4×Dic53—21no—
72#11719917—21no—
73#11719917D412yescomplemented
74C2×#88322—21no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53C2×D411yescharacteristic
78C22×D53—22no—
79C22×D53—41no—
80C22×D53—41no—
81C22×D53—82no—
82Q8×C53—21no—
83Q8×C53—21no—
84Q8×C53—21no—
85Q8×C53—21no—
86D4×C53—21no—
87D4×C53—21no—
88D4×C53—21no—
89D4×C53—21no—
90D4×C53—41no—
91D4×C53—41no—
92D4×C53—41no—
93D4×C53—41no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2×D411yescharacteristic
102C2×C4×C53C2×D411yescharacteristic
103C2×C4×C53C2×D411yescharacteristic
104C2×C4×C53C2×D411yescharacteristic
105C2×C4×C53#4111yescharacteristic
106C22⋊D53—84no—
107C2×Dic53—22no—
108C2×Dic53—41no—
109D212—22no—
110D212—22no—
111D212—41no—
112D212—41no—
113C8×C53—22no—
114#88322—22no—
115C22×C53—41no—
116C22×C53—42no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—81no—
120C22×C53#12011yescharacteristic
121C22×C53#12111yescharacteristic
122C22×C53#12711yescharacteristic
123C2×D53—44no—
124C2×D53—42no—
125C2×D53—82no—
126C2×D53—82no—
127C4×C53—41no—
128C4×C53—41no—
129C4×C53#12011yescharacteristic
130C4×C53#12011yescharacteristic
131C4×C53#12111yescharacteristic
132C4×C53#12111yescharacteristic
133Dic53—42no—
134Dic53—81no—
135#2008—531nomaximal, Sylow
136C2×C53—21no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53—81no—
140C2×C53—81no—
141C2×C53#39211yescharacteristic
142D53—82no—
143#575D5311yescharacteristic, complemented
144C2×AGL(1,ℤ/8)—531no—
145#535—531no—
146#421—531no—
147#418—531no—
148#410—531no—
149#344—531no—
150C53#200811yescharacteristic, complemented, Sylow
151C2×#41C2×D5311yescharacteristic
152C2×#41C2×D5311yescharacteristic
153C22×D4—531no—
154#137—21no—
155#137—21no—
156AGL(1,ℤ/8)—21no—
157AGL(1,ℤ/8)—21no—
158AGL(1,ℤ/8)—1062no—
159AGL(1,ℤ/8)—1062no—
160#135—22no—
161C2×SD16—1061no—
162C2×D8—1061no—
163C2×M16—531no—
164C2×M16C2×D5311yescharacteristic
165#127—1061no—
166#124—1061no—
167#121—1062no—
168#120—1062no—
169#117—531no—
170#117—531no—
171#104—1062no—
172#102—1062no—
173#99—1062no—
174C24—1061no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—21no—
183C2×Q8C22⋊D5311yescharacteristic
184C2×Q8C22⋊D5311yescharacteristic, complemented
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—532no—
188C2×D4—1064no—
189C2×D4—1061no—
190C2×D4—1061no—
191C2×D4—1061no—
192C2×D4—1061no—
193C2×D4—1061no—
194C2×D4C22⋊D5311yescharacteristic
195C2×D4C22⋊D5311yescharacteristic, complemented
196C22×C4—21no—
197C22×C4—21no—
198C22×C4C22×D5311yescharacteristic
199Q16—22no—
200SD16—22no—
201SD16—22no—
202SD16—1064no—
203D8—22no—
204D8—1064no—
205M16—21no—
206M16—21no—
207M16—532no—
208M16—1061no—
209C2×C8—1061no—
210C2×C8C22⋊D5312yescomplemented
211C4⋊C4—1061no—
212C4⋊C4—1061no—
213C22⋊C4—532no—
214C22⋊C4—1064no—
215C22⋊C4—1061no—
216C42—1061no—
217C42—1061no—
218C23—21no—
219C23—21no—
220C23—1062no—
221C23—4242no—
222C23—2121no—
223C23—2121no—
224C23D4×D5311yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—21no—
228Q8—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—1062no—
238D4—1062no—
239D4—4244no—
240D4—2121no—
241D4—2121no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—1062no—
249C2×C4—2121no—
250C2×C4D4×D5311yescharacteristic
251C2×C4D4×D5311yescharacteristic
252C2×C4#1171994711yescharacteristic
253C2×C4C2×C22⋊D5311yescharacteristic
254C2×C4C2×C22⋊D5311yescharacteristic
255C2×C4C2×C22⋊D5311yescharacteristic
256C8—22no—
257C8—1062no—
258C22—41no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—81no—
263C22—4242no—
264C22—4242no—
265C22—2124no—
266C22—2122no—
267C22#1313318511yescharacteristic
268C22#1313318911yescharacteristic
269C22#1313321311yescharacteristic
270C4—41no—
271C4—41no—
272C4—4241no—
273C4—2122no—
274C4#1313318511yescharacteristic
275C4#1313318611yescharacteristic
276C4#1313318811yescharacteristic
277C4#1313318911yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—81no—
282C2—81no—
283C2—4242no—
284C2#1681053111yescharacteristic
subgroupG/H
centerC2#16810531
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#575C2
socleC2×C53#392
hypercenter#575D53
hypocenterC53#2008
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. C2×#13133242C2semidirect
2. #16811187C2semidirect
3. #16810640C2semidirect
4. #16810629C2semidirect
5. #16810597C2semidirect
6. #16810394C2semidirect
7. C53×#575C2semidirect
41. C2×Q8×C53D4semidirect
45. C2×D4×C53D4semidirect
62. C2×C8×C53D4semidirect
73. #11719917D4semidirect
143. #575D53semidirect
150. C53#2008semidirect
184. C2×Q8C22⋊D53semidirect
195. C2×D4C22⋊D53semidirect
210. C2×C8C22⋊D53semidirect