Finite Group Databasebeta

← #44559591

Subgroups — #44559591

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×#22548740C212yesmaximal, complemented
3#30631837C214yesmaximal, complemented
4#30631820C214yesmaximal, complemented
5#30631691C212yesmaximal, complemented
6#30631651C211yesmaximal, characteristic, complemented
7#525×C251C211yesmaximal, characteristic, complemented
8C24×D251—21no—
9C22×D1004—28no—
10C22×D1004—22no—
11C22×D1004—21no—
12C22×D1004C2214yes—
13C22×D1004C2212yes—
14#22548740—28no—
15C2×#16860797C2214yescomplemented
16C2×#16860795C2214yescomplemented
17C2×#16860794C2212yescomplemented
18#22548635C2218yescomplemented
19C22⋊(C251⋊C8)C2214yescomplemented
20C2×#41×C251C2211yescharacteristic, complemented
21C23×C4×C251C2211yescharacteristic
22#123×C251—21no—
23C22⋊Q8×C251C2212yescomplemented
24#121×C251C2212yescomplemented
25#120×C251—21no—
26C4×D4×C251—22no—
27#117×C251C2211yescharacteristic, complemented
28C23×D251—28no—
29C23×D251—22no—
30C23×D251—24no—
31C23×D251—48no—
32C2×D1004—28no—
33C2×D1004—216no—
34C2×D1004—216no—
35C2×D1004—24no—
36C2×D1004—24no—
37C2×D1004—22no—
38C2×D1004D418yescomplemented
39C2×D1004C2314yes—
40#16860797—28no—
41#16860795—28no—
42#16860794—24no—
43C2×C251⋊C8C2314yescomplemented
44C24×C251D411yescharacteristic
45#41×C251—24no—
46C2×Q8×C251C2311yescharacteristic, complemented
47C2×D4×C251—22no—
48C2×D4×C251—21no—
49C2×D4×C251C2311yescharacteristic, complemented
50C22×C4×C251—24no—
51C22×C4×C251—21no—
52C22×C4×C251D412yes—
53C22×C4×C251D411yescharacteristic
54C22×C4×C251C2312yes—
55C22×C4×C251C2311yescharacteristic
56C4⋊C4×C251—22no—
57C4⋊C4×C251C2312yescomplemented
58C22⋊C4×C251—21no—
59C22⋊C4×C251—22no—
60C22⋊C4×C251—22no—
61C42×C251—21no—
62C22×D251—216no—
63C22×D251—28no—
64C22×D251—24no—
65C22×D251—432no—
66C22×D251—48no—
67C22×D251—48no—
68C22×D251—48no—
69D1004—216no—
70D1004—28no—
71D1004—24no—
72D1004C2×D418yes—
73C251⋊C8—24no—
74C23×C251—22no—
75C23×C251—24no—
76C23×C251—21no—
77C23×C251C2×D411yescharacteristic
78C23×C251C2×D412yes—
79Q8×C251—22no—
80D4×C251—22no—
81D4×C251—42no—
82D4×C251—44no—
83C2×C4×C251—22no—
84C2×C4×C251—24no—
85C2×C4×C251—24no—
86C2×C4×C251—22no—
87C2×C4×C251—21no—
88C2×C4×C251—42no—
89C2×C4×C251C2×D414yes—
90C2×C4×C251C2×D412yes—
91C2×C4×C251C2×D411yescharacteristic
92C2×C4×C251C2411yescharacteristic
93C22×C251—28no—
94C22×C251—22no—
95C22×C251—22no—
96C22×C251—22no—
97C22×C251—41no—
98C22×C251—42no—
99C22×C251#12012yes—
100C22×C251AGL(1,ℤ/8)14yescomplemented
101C22×C251C22×D411yescharacteristic
102C2×D251—28no—
103C2×D251—416no—
104C2×D251—432no—
105C2×D251—48no—
106C4×C251—22no—
107C4×C251—41no—
108C4×C251—42no—
109C4×C251#12012yes—
110C4×C251C22×D412yes—
111C2×C251—22no—
112C2×C251—24no—
113C2×C251—81no—
114C2×C251C2×#12011yescharacteristic
115C2×C251C2×AGL(1,ℤ/8)12yes—
116D251—48no—
117C251#300311yescharacteristic, complemented, Sylow
118#3003—2511nomaximal, Sylow
119C23×D4—2511no—
120C2×AGL(1,ℤ/8)—2512no—
121#525D25111yescharacteristic, complemented
122#450—2514no—
123#447—2514no—
124#418—2512no—
125#407—2511no—
126C25—5021no—
127C2×#41C2×D25111yescharacteristic, complemented
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×M16—2512no—
138#123—21no—
139C22⋊Q8C2×D25112yescomplemented
140#121C2×D25112yescomplemented
141#120—21no—
142C4×D4—22no—
143#117C2×D25111yescharacteristic, complemented
144#102—2518no—
145C22⋊C8—2514no—
146C24—10048no—
147C24—5028no—
148C24—5022no—
149C24—5024no—
150C24C22⋊D25111yescharacteristic
151#41—24no—
152C2×Q8C22×D25111yescharacteristic, complemented
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—5028no—
156C2×D4—50216no—
157C2×D4—50216no—
158C2×D4—5024no—
159C2×D4—5024no—
160C2×D4—5022no—
161C2×D4—2518no—
162C2×D4—2514no—
163C2×D4C22×D25111yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4C22⋊D25112yes—
167C22×C4C22⋊D25111yescharacteristic
168C22×C4C22×D25112yes—
169C22×C4C22×D25111yescharacteristic
170SD16—5028no—
171D8—5028no—
172M16—5024no—
173C2×C8—2514no—
174C4⋊C4—22no—
175C4⋊C4C22×D25112yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—24no—
181C23—22no—
182C23—21no—
183C23—100432no—
184C23—10048no—
185C23—10048no—
186C23—10048no—
187C23—50216no—
188C23—5024no—
189C23—5028no—
190C23C2×C22⋊D25112yes—
191C23C2×C22⋊D25111yescharacteristic
192Q8—22no—
193D4—22no—
194D4—44no—
195D4—42no—
196D4—50216no—
197D4—5024no—
198D4—5028no—
199D4—2518no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4D4×D25114yes—
207C2×C4C2×C22⋊D25112yes—
208C2×C4C2×C22⋊D25111yescharacteristic
209C2×C4C23×D25111yescharacteristic
210C8—5024no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22—41no—
217C22—100432no—
218C22—100416no—
219C22—10048no—
220C22—5028no—
221C22#2254868312yes—
222C22#2254874014yescomplemented
223C22C22×C22⋊D25111yescharacteristic
224C4—22no—
225C4—42no—
226C4—41no—
227C4#2254868312yes—
228C4C2×D4×D25112yes—
229C2—24no—
230C2—22no—
231C2—81no—
232C2—10048no—
233C2C2×#2254868311yescharacteristic
234C2C2×#2254874012yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#525×C251C2
socleC22×C251C22×D4
hypercenter#525D251
hypocenterC251#3003
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×D1004C2semidirect
2. C2×#22548740C2semidirect
3. #30631837C2semidirect
4. #30631820C2semidirect
5. #30631691C2semidirect
6. #30631651C2semidirect
7. #525×C251C2semidirect
15. C2×#16860797C22semidirect
16. C2×#16860795C22semidirect
17. C2×#16860794C22semidirect
18. #22548635C22semidirect
19. C22⋊(C251⋊C8)C22semidirect
20. C2×#41×C251C22semidirect
23. C22⋊Q8×C251C22semidirect
24. #121×C251C22semidirect
27. #117×C251C22semidirect
38. C2×D1004D4semidirect
43. C2×C251⋊C8C23semidirect
46. C2×Q8×C251C23semidirect
49. C2×D4×C251C23semidirect
57. C4⋊C4×C251C23semidirect
100. C22×C251AGL(1,ℤ/8)semidirect
117. C251#3003semidirect
121. #525D251semidirect
127. C2×#41C2×D251semidirect
139. C22⋊Q8C2×D251semidirect
140. #121C2×D251semidirect
143. #117C2×D251semidirect
152. C2×Q8C22×D251semidirect
163. C2×D4C22×D251semidirect
175. C4⋊C4C22×D251semidirect
222. C22#22548740semidirect

Maximal subgroups

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

subgroupindex
C23×D10042
C2×#225487402
#306318372
#306318202
#306316912
#306316512
#525×C2512
#3003251