Finite Group Databasebeta

← #44559602

Subgroups — #44559602

Order 32128.

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

normal subgroups only

This group has 738 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×Dic502C211yesmaximal, characteristic, complemented
2C2×#22548748C212yesmaximal, complemented
3#30632198C211yesmaximal, characteristic
4#30631845C214yesmaximal, complemented
5#30631828C214yesmaximal, complemented
6#30631697C212yesmaximal, complemented
7#407×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×Dic502—28no—
10C22×Dic502—22no—
11C22×Dic502—21no—
12C22×Dic502C2214yes—
13C22×Dic502C2212yes—
14C23×Dic251—21no—
15#22548748—28no—
16C2×Dic1004C2214yescomplemented
17C2×#16860799C2214yescomplemented
18#22548698—21no—
19#22548694C2212yes—
20#22548689C2212yes—
21#22548684—21no—
22C4×C22⋊D251—22no—
23#22548674C2211yescharacteristic
24#22548640C2218yescomplemented
25C23×C4×C251C2211yescharacteristic
26C2×M16×C251C2212yes—
27C22⋊C8×C251C2214yes—
28#16860810—24no—
29C2×Dic502—28no—
30C2×Dic502—216no—
31C2×Dic502—216no—
32C2×Dic502—24no—
33C2×Dic502—24no—
34C2×Dic502—22no—
35C2×Dic502D418yescomplemented
36C2×Dic502C2311yescharacteristic
37C2×Dic502C2314yes—
38C2×D1004C2311yescharacteristic
39C2×C22⋊D251—21no—
40C2×C22⋊D251—22no—
41C22×Dic251—28no—
42C22×Dic251—22no—
43C22×Dic251—24no—
44C2×C4×D251—21no—
45Dic1004—28no—
46#16860799—28no—
47C4⋊Dic251C2312yes—
48Dic251⋊C4—22no—
49C22⋊Dic251—21no—
50C22⋊Dic251—22no—
51(C2×D251)⋊C4—22no—
52C4×Dic251—21no—
53C24×C251D411yescharacteristic
54C22×C4×C251—24no—
55C22×C4×C251D412yes—
56C22×C4×C251D411yescharacteristic
57C22×C4×C251C2312yes—
58C22×C4×C251C2311yescharacteristic
59M16×C251—24no—
60C2×C8×C251C2314yes—
61C22×D251—21no—
62Dic502—216no—
63Dic502—28no—
64Dic502—24no—
65Dic502—22no—
66Dic502C2×D418yes—
67D1004—22no—
68C22⋊D251—42no—
69C22⋊D251—44no—
70C2×Dic251—216no—
71C2×Dic251—28no—
72C2×Dic251—24no—
73C2×Dic251—22no—
74C2×Dic251—21no—
75C4×D251—42no—
76C23×C251—22no—
77C23×C251—24no—
78C23×C251C2×D411yescharacteristic
79C23×C251C2×D412yes—
80C2×C4×C251—22no—
81C2×C4×C251—24no—
82C2×C4×C251—24no—
83C2×C4×C251C2×D414yes—
84C2×C4×C251C2×D412yes—
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2411yescharacteristic
87C8×C251—24no—
88C22×C251—28no—
89C22×C251—22no—
90C22×C251—22no—
91C22×C251—22no—
92C22×C251#12012yes—
93C22×C251#13714yescomplemented
94C22×C251C22×D411yescharacteristic
95C2×D251—41no—
96C2×D251—42no—
97C4×C251—22no—
98C4×C251#12012yes—
99C4×C251C22×D412yes—
100Dic251—28no—
101Dic251—41no—
102Dic251—42no—
103C2×C251—22no—
104C2×C251—24no—
105C2×C251C2×#12011yescharacteristic
106C2×C251C2×#13712yes—
107D251—81no—
108C251#300411yescharacteristic, complemented, Sylow
109#3004—2511nomaximal, Sylow
110C23×Q8—2511no—
111C2×#137—2512no—
112#525—2511no—
113#451—2514no—
114#448—2514no—
115#419—2512no—
116#407D25111yescharacteristic, complemented
117C2×#41—2511no—
118C22×Q8—5028no—
119C22×Q8—5022no—
120C22×Q8—5021no—
121C22×Q8—2514no—
122C22×Q8—2512no—
123C23×C4—5021no—
124C23×C4C2×D25111yescharacteristic
125#137—5028no—
126C2×Q16—2514no—
127C2×SD16—2514no—
128C2×M16C2×D25112yes—
129#123—5021no—
130C22⋊Q8—2512no—
131#121—2512no—
132#120—5021no—
133C4×D4—5022no—
134#117—2511no—
135#103—2518no—
136C22⋊C8C2×D25114yes—
137C24D100411yescharacteristic
138#41—5024no—
139C2×Q8—5028no—
140C2×Q8—50216no—
141C2×Q8—50216no—
142C2×Q8—5024no—
143C2×Q8—5024no—
144C2×Q8—5022no—
145C2×Q8—2518no—
146C2×Q8—2511no—
147C2×Q8—2514no—
148C2×D4—5022no—
149C2×D4—5021no—
150C2×D4—2511no—
151C22×C4—24no—
152C22×C4—5021no—
153C22×C4—5028no—
154C22×C4—5024no—
155C22×C4—5022no—
156C22×C4D100412yes—
157C22×C4D100411yescharacteristic
158C22×C4C22×D25112yes—
159C22×C4C22×D25111yescharacteristic
160Q16—5028no—
161SD16—5028no—
162M16—24no—
163C2×C8C22×D25114yes—
164C4⋊C4—5022no—
165C4⋊C4—2512no—
166C22⋊C4—5021no—
167C22⋊C4—5022no—
168C22⋊C4—5022no—
169C42—5021no—
170C23—24no—
171C23—22no—
172C23—5021no—
173C23C2×D100412yes—
174C23C2×D100411yescharacteristic
175Q8—50216no—
176Q8—5022no—
177Q8—5024no—
178Q8—5028no—
179Q8—2518no—
180D4—10044no—
181D4—10042no—
182D4—5022no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—10042no—
187C2×C4—5022no—
188C2×C4—5021no—
189C2×C4—50216no—
190C2×C4—5024no—
191C2×C4—5028no—
192C2×C4D4×D25114yes—
193C2×C4C2×D100412yes—
194C2×C4C2×D100411yescharacteristic
195C2×C4C23×D25111yescharacteristic
196C8—24no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—10042no—
202C22—10041no—
203C22#2254868512yes—
204C22#2254874814yescomplemented
205C22C22×D100411yescharacteristic
206C4—22no—
207C4—10042no—
208C4—10041no—
209C4—5028no—
210C4#2254868512yes—
211C4C2×D4×D25112yes—
212C2—24no—
213C2—22no—
214C2—20081no—
215C2C2×#2254868511yescharacteristic
216C2C2×#2254874812yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#407×C251C2
socleC22×C251C22×D4
prime radical#30632198C2
hypercenter#407D251
hypocenterC251#3004
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×Dic502C2semidirect
2. C2×#22548748C2semidirect
4. #30631845C2semidirect
5. #30631828C2semidirect
6. #30631697C2semidirect
7. #407×C251C2semidirect
16. C2×Dic1004C22semidirect
17. C2×#16860799C22semidirect
24. #22548640C22semidirect
35. C2×Dic502D4semidirect
93. C22×C251#137semidirect
108. C251#3004semidirect
116. #407D251semidirect
204. C22#22548748semidirect

Maximal subgroups

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

subgroupindex
C23×Dic5022
C2×#225487482
#306321982
#306318452
#306318282
#306316972
#407×C2512
#3004251