Finite Group Databasebeta

← #44559868

Subgroups — #44559868

Order 32128.

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

normal subgroups only

This group has 526 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×#22548741C211yesmaximal, characteristic, complemented
2C22×#16860795C211yesmaximal, characteristic, complemented
3#30632256C211yesmaximal, characteristic, complemented
4#30631902C214yesmaximal, complemented
5#30631882C214yesmaximal, complemented
6#30631714C211yesmaximal, characteristic, complemented
7#30631691C211yesmaximal, characteristic, complemented
8#30631681C211yesmaximal, characteristic, complemented
9#518×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12#22548741—24no—
13C2×#16860798C2212yescomplemented
14C2×#16860795—24no—
15C2×#16860795C2214yes—
16C2×#16860795C2212yescomplemented
17C2×#16860794C2211yescharacteristic
18C22×C251⋊C8C2211yescharacteristic
19#22548712C2212yescomplemented
20#22548703C2212yescomplemented
21#22548690—22no—
22#22548685—22no—
23C4⋊(C251⋊C8)C2214yescomplemented
24#22548638C2212yescomplemented
25#22548635C2212yescomplemented
26#22548635C2214yescomplemented
27C22×D4×C251C2211yescharacteristic
28C4×D4×C251—22no—
29C4×D4×C251C2214yescomplemented
30#117×C251C2211yescharacteristic
31#117×C251C2211yescharacteristic
32C2×C22⋊C4×C251—22no—
33C23×D251—21no—
34#16860810—22no—
35#16860810—22no—
36C2×Dic502C2311yescharacteristic
37C2×D1004—22no—
38C2×D1004—21no—
39C2×D1004—24no—
40C2×D1004—22no—
41C2×D1004C2312yes—
42C2×D1004C2311yescharacteristic
43C2×C22⋊D251—21no—
44C2×C4×D251—21no—
45#16860798—24no—
46#16860795—24no—
47#16860795—28no—
48#16860794—22no—
49C2×C251⋊C8—22no—
50C2×C251⋊C8C2312yes—
51C2×C251⋊C8C2312yes—
52(C2×D251)⋊C4—24no—
53C24×C251—21no—
54C2×D4×C251—21no—
55C2×D4×C251—22no—
56C2×D4×C251D414yescomplemented
57C2×D4×C251C2312yes—
58C22×C4×C251—24no—
59C22×C4×C251C2311yescharacteristic
60C4⋊C4×C251D412yes—
61C4⋊C4×C251C2312yes—
62C22⋊C4×C251—21no—
63C22⋊C4×C251—24no—
64C22⋊C4×C251D412yes—
65C42×C251—21no—
66C42×C251C2312yes—
67C22×D251—22no—
68C22×D251—21no—
69C22×D251—42no—
70C22×D251—82no—
71Dic502—21no—
72Dic502—21no—
73D1004—22no—
74D1004—21no—
75D1004—22no—
76D1004—21no—
77D1004—42no—
78D1004—84no—
79C22⋊D251—42no—
80C2×Dic251—21no—
81C4×D251—42no—
82C251⋊C8—22no—
83C251⋊C8—22no—
84C23×C251—22no—
85C23×C251—24no—
86C23×C251—82no—
87C23×C251C2×D411yescharacteristic
88D4×C251—24no—
89D4×C251—22no—
90D4×C251C2×D414yes—
91C2×C4×C251—22no—
92C2×C4×C251—84no—
93C2×C4×C251C2×D414yes—
94C2×C4×C251C2×D411yescharacteristic
95C2×C4×C251C2×D412yes—
96C2×C4×C251#4112yes—
97C2×C4×C251C2411yescharacteristic
98C22×C251—22no—
99C22×C251—24no—
100C22×C251—44no—
101C22×C251—84no—
102C22×C251#12112yes—
103C22×C251C22×D411yescharacteristic
104C2×D251—41no—
105C2×D251—42no—
106C2×D251—42no—
107C2×D251—44no—
108C2×D251—84no—
109C4×C251—24no—
110C4×C251—42no—
111C4×C251#12112yes—
112C4×C251C22×D411yescharacteristic
113C4×C251C2×#4111yescharacteristic
114Dic251—41no—
115C2×C251—22no—
116C2×C251—44no—
117C2×C251C2×#12111yescharacteristic
118C2×C251#57612yes—
119D251—81no—
120D251—82no—
121C251#304111yescharacteristic, complemented, Sylow
122#3041—2511nomaximal, Sylow
123C2×AGL(1,ℤ/8)—2511no—
124C22×D8—2511no—
125#535—2511no—
126#518D25111yescharacteristic, complemented
127#463—2514no—
128#459—2514no—
129#424—2511no—
130#418—2511no—
131#416—2511no—
132C2×#41—2511no—
133C22×D4—2511no—
134C22×D4C2×D25111yescharacteristic
135AGL(1,ℤ/8)—5024no—
136C2×SD16—2512no—
137C2×D8—5024no—
138C2×D8—2514no—
139C2×D8—2512no—
140C2×M16—2511no—
141C22×C8—2511no—
142#127—2512no—
143#124—2512no—
144#121—5022no—
145#120—5022no—
146C4×D4—22no—
147C4×D4C2×D25114yescomplemented
148#117C2×D25111yescharacteristic
149#117C2×D25111yescharacteristic
150C2×C22⋊C4—22no—
151C4⋊C8—2514no—
152#103—2512no—
153#102—2512no—
154#102—2514no—
155C24—21no—
156C24—5021no—
157#41—5022no—
158#41—5022no—
159C2×Q8—2511no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—5021no—
163C2×D4—5024no—
164C2×D4—5021no—
165C2×D4—5022no—
166C2×D4—5022no—
167C2×D4—2512no—
168C2×D4—2511no—
169C2×D4D100414yescomplemented
170C2×D4C22×D25112yes—
171C22×C4—24no—
172C22×C4—5021no—
173C22×C4C22×D25111yescharacteristic
174SD16—5024no—
175D8—5028no—
176D8—5024no—
177M16—5022no—
178C2×C8—5022no—
179C2×C8—2512no—
180C2×C8—2512no—
181C4⋊C4C22⋊D25112yes—
182C4⋊C4C22×D25112yes—
183C22⋊C4—21no—
184C22⋊C4—24no—
185C22⋊C4—5024no—
186C22⋊C4C22⋊D25112yes—
187C42—21no—
188C42C22×D25112yes—
189C23—22no—
190C23—24no—
191C23—82no—
192C23—20082no—
193C23—10042no—
194C23—5022no—
195C23—5021no—
196C23C2×C22⋊D25111yescharacteristic
197Q8—5021no—
198Q8—5021no—
199D4—22no—
200D4—24no—
201D4—20084no—
202D4—10042no—
203D4—10042no—
204D4—5022no—
205D4—5021no—
206D4—5022no—
207D4—5021no—
208D4C2×D100414yes—
209C2×C4—22no—
210C2×C4—84no—
211C2×C4—10042no—
212C2×C4—5021no—
213C2×C4C2×C22⋊D25114yes—
214C2×C4C2×C22⋊D25111yescharacteristic
215C2×C4C2×D100412yes—
216C2×C4#1686081012yes—
217C2×C4C23×D25111yescharacteristic
218C8—5022no—
219C8—5022no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—84no—
224C22—20084no—
225C22—10042no—
226C22—10044no—
227C22—10041no—
228C22—10042no—
229C22#2254868912yes—
230C22C22×C22⋊D25111yescharacteristic
231C4—24no—
232C4—42no—
233C4—10041no—
234C4#2254868912yes—
235C4C22×D100411yescharacteristic
236C4C2×#1686081011yescharacteristic
237C2—22no—
238C2—44no—
239C2—20082no—
240C2—20081no—
241C2C2×#2254868911yescharacteristic
242C2#3063249012yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#518×C251C2
socleC22×C251C22×D4
hypercenter#518D251
hypocenterC251#3041
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. C22×#16860795C2semidirect
3. #30632256C2semidirect
4. #30631902C2semidirect
5. #30631882C2semidirect
6. #30631714C2semidirect
7. #30631691C2semidirect
8. #30631681C2semidirect
9. #518×C251C2semidirect
13. C2×#16860798C22semidirect
16. C2×#16860795C22semidirect
19. #22548712C22semidirect
20. #22548703C22semidirect
23. C4⋊(C251⋊C8)C22semidirect
24. #22548638C22semidirect
25. #22548635C22semidirect
26. #22548635C22semidirect
29. C4×D4×C251C22semidirect
56. C2×D4×C251D4semidirect
121. C251#3041semidirect
126. #518D251semidirect
147. C4×D4C2×D251semidirect
169. C2×D4D1004semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
C22×#168607952
#306322562
#306319022
#306318822
#306317142
#306316912
#306316812
#518×C2512
#3041251