Finite Group Databasebeta

← #44559595

Subgroups — #44559595

Order 32128.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D1004C211yesmaximal, characteristic, complemented
2C2×#22548743C212yesmaximal, complemented
3#30632198C211yesmaximal, characteristic, complemented
4#30631840C214yesmaximal, complemented
5#30631823C214yesmaximal, complemented
6#30631693C212yesmaximal, complemented
7#407×C251C211yesmaximal, characteristic, complemented
8C24×D251—21no—
9C2×#16860810C2211yescharacteristic, complemented
10C22×D1004—28no—
11C22×D1004—22no—
12C22×D1004—21no—
13C22×D1004C2214yes—
14C22×D1004C2212yes—
15#22548743—28no—
16C2×#16860799C2214yescomplemented
17C2×D2008C2214yescomplemented
18#22548698—21no—
19#22548694C2212yescomplemented
20#22548689C2212yescomplemented
21#22548684—21no—
22C4×C22⋊D251—22no—
23#22548674C2211yescharacteristic, complemented
24#22548637C2218yescomplemented
25C23×C4×C251C2211yescharacteristic
26C2×M16×C251C2212yescomplemented
27C22⋊C8×C251C2214yescomplemented
28C23×D251—28no—
29C23×D251—22no—
30C23×D251—24no—
31C23×D251—48no—
32#16860810—24no—
33C2×Dic502C2311yescharacteristic, complemented
34C2×D1004—28no—
35C2×D1004—216no—
36C2×D1004—216no—
37C2×D1004—24no—
38C2×D1004—24no—
39C2×D1004—22no—
40C2×D1004D418yescomplemented
41C2×D1004C2314yes—
42C2×D1004C2311yescharacteristic, complemented
43C2×C22⋊D251—21no—
44C2×C22⋊D251—22no—
45C2×C4×D251—21no—
46#16860799—28no—
47D2008—28no—
48C4⋊Dic251C2312yescomplemented
49Dic251⋊C4—22no—
50C22⋊Dic251—21no—
51C22⋊Dic251—22no—
52(C2×D251)⋊C4—22no—
53C4×Dic251—21no—
54C24×C251D411yescharacteristic
55C22×C4×C251—24no—
56C22×C4×C251D412yes—
57C22×C4×C251D411yescharacteristic
58C22×C4×C251C2312yes—
59C22×C4×C251C2311yescharacteristic
60M16×C251—24no—
61C2×C8×C251C2314yescomplemented
62C22×D251—216no—
63C22×D251—28no—
64C22×D251—24no—
65C22×D251—21no—
66C22×D251—432no—
67C22×D251—48no—
68C22×D251—48no—
69C22×D251—48no—
70Dic502—22no—
71D1004—216no—
72D1004—28no—
73D1004—24no—
74D1004—22no—
75D1004C2×D418yes—
76C22⋊D251—42no—
77C22⋊D251—44no—
78C2×Dic251—22no—
79C2×Dic251—21no—
80C4×D251—42no—
81C23×C251—22no—
82C23×C251—24no—
83C23×C251C2×D411yescharacteristic
84C23×C251C2×D412yes—
85C2×C4×C251—22no—
86C2×C4×C251—24no—
87C2×C4×C251—24no—
88C2×C4×C251C2×D414yes—
89C2×C4×C251C2×D412yes—
90C2×C4×C251C2×D411yescharacteristic
91C2×C4×C251C2411yescharacteristic
92C8×C251—24no—
93C22×C251—28no—
94C22×C251—22no—
95C22×C251—22no—
96C22×C251—22no—
97C22×C251#12012yes—
98C22×C251AGL(1,ℤ/8)14yescomplemented
99C22×C251C22×D411yescharacteristic
100C2×D251—28no—
101C2×D251—416no—
102C2×D251—432no—
103C2×D251—48no—
104C2×D251—41no—
105C2×D251—42no—
106C4×C251—22no—
107C4×C251#12012yes—
108C4×C251C22×D412yes—
109Dic251—41no—
110Dic251—42no—
111C2×C251—22no—
112C2×C251—24no—
113C2×C251C2×#12011yescharacteristic
114C2×C251C2×AGL(1,ℤ/8)12yes—
115D251—48no—
116D251—81no—
117C251#300311yescharacteristic, complemented, Sylow
118#3003—2511nomaximal, Sylow
119C23×D4—2511no—
120C2×AGL(1,ℤ/8)—2512no—
121#525—2511no—
122#450—2514no—
123#447—2514no—
124#418—2512no—
125#407D25111yescharacteristic, complemented
126C25—5021no—
127C2×#41—2511no—
128C22×D4—5028no—
129C22×D4—5022no—
130C22×D4—5021no—
131C22×D4—2514no—
132C22×D4—2512no—
133C23×C4C2×D25111yescharacteristic
134AGL(1,ℤ/8)—5028no—
135C2×SD16—2514no—
136C2×D8—2514no—
137C2×M16C2×D25112yescomplemented
138#123—5021no—
139C22⋊Q8—2512no—
140#121—2512no—
141#120—5021no—
142C4×D4—5022no—
143#117—2511no—
144#102—2518no—
145C22⋊C8C2×D25114yescomplemented
146C24—10048no—
147C24—5028no—
148C24—5022no—
149C24—5024no—
150C24D100411yescharacteristic
151#41—5024no—
152C2×Q8—2511no—
153C2×D4—5028no—
154C2×D4—50216no—
155C2×D4—50216no—
156C2×D4—5022no—
157C2×D4—5021no—
158C2×D4—5024no—
159C2×D4—5024no—
160C2×D4—5022no—
161C2×D4—2518no—
162C2×D4—2511no—
163C2×D4—2514no—
164C22×C4—24no—
165C22×C4—5021no—
166C22×C4D100412yes—
167C22×C4D100411yescharacteristic
168C22×C4C22×D25112yes—
169C22×C4C22×D25111yescharacteristic
170SD16—5028no—
171D8—5028no—
172M16—24no—
173C2×C8C22×D25114yescomplemented
174C4⋊C4—5022no—
175C4⋊C4—2512no—
176C22⋊C4—5021no—
177C22⋊C4—5022no—
178C22⋊C4—5022no—
179C42—5021no—
180C23—24no—
181C23—22no—
182C23—100432no—
183C23—10048no—
184C23—10048no—
185C23—10048no—
186C23—50216no—
187C23—5021no—
188C23—5024no—
189C23—5028no—
190C23C2×D100412yes—
191C23C2×D100411yescharacteristic
192Q8—5022no—
193D4—10044no—
194D4—10042no—
195D4—50216no—
196D4—5022no—
197D4—5024no—
198D4—5028no—
199D4—2518no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—10042no—
204C2×C4—5022no—
205C2×C4—5021no—
206C2×C4D4×D25114yes—
207C2×C4C2×D100412yes—
208C2×C4C2×D100411yescharacteristic
209C2×C4C23×D25111yescharacteristic
210C8—24no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—100432no—
216C22—100416no—
217C22—10042no—
218C22—10041no—
219C22—10048no—
220C22—5028no—
221C22#2254868512yes—
222C22#2254874314yescomplemented
223C22C22×D100411yescharacteristic
224C4—22no—
225C4—10042no—
226C4—10041no—
227C4#2254868512yes—
228C4C2×D4×D25112yes—
229C2—24no—
230C2—22no—
231C2—20081no—
232C2—10048no—
233C2C2×#2254868511yescharacteristic
234C2C2×#2254874312yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#407×C251C2
socleC22×C251C22×D4
hypercenter#407D251
hypocenterC251#3003
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C23×D1004C2semidirect
2. C2×#22548743C2semidirect
3. #30632198C2semidirect
4. #30631840C2semidirect
5. #30631823C2semidirect
6. #30631693C2semidirect
7. #407×C251C2semidirect
9. C2×#16860810C22semidirect
16. C2×#16860799C22semidirect
17. C2×D2008C22semidirect
19. #22548694C22semidirect
20. #22548689C22semidirect
23. #22548674C22semidirect
24. #22548637C22semidirect
26. C2×M16×C251C22semidirect
27. C22⋊C8×C251C22semidirect
33. C2×Dic502C23semidirect
40. C2×D1004D4semidirect
42. C2×D1004C23semidirect
48. C4⋊Dic251C23semidirect
61. C2×C8×C251C23semidirect
98. C22×C251AGL(1,ℤ/8)semidirect
117. C251#3003semidirect
125. #407D251semidirect
137. C2×M16C2×D251semidirect
145. C22⋊C8C2×D251semidirect
173. C2×C8C22×D251semidirect
222. C22#22548743semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×D10042
C2×#225487432
#306321982
#306318402
#306318232
#306316932
#407×C2512
#3003251