Finite Group Databasebeta

← #44550897

Subgroups — #44550897

Order 32128.

362 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#30632254C211yesmaximal, characteristic
2#30631705C212yesmaximal, complemented
3C2×#22548632C211yesmaximal, characteristic, complemented
4#30631443C212yesmaximal, complemented
5#518×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C2211yescharacteristic
7C2×#16860794C2212yes—
8#22548711C412yescomplemented
9#22548702C412yescomplemented
10#22548686—22no—
11#22548684—22no—
12#22548674C2211yescharacteristic
13#22548643—24no—
14#22548632—22no—
15C22×D4×C251C2211yescharacteristic
16C4×D4×C251—24no—
17#117×C251C2212yes—
18C2×C22⋊C4×C251—22no—
19#16860810—22no—
20#16860810—22no—
21C2×Dic502C2×C411yescharacteristic, complemented
22C2×D1004C2×C411yescharacteristic, complemented
23C2×C22⋊D251—24no—
24C2×C22⋊D251—21no—
25C2×C4×D251—21no—
26#16860794—22no—
27#16860794—22no—
28C2×C251⋊C8—22no—
29C4⋊Dic251—21no—
30C22⋊Dic251—24no—
31C22⋊Dic251—21no—
32C4×Dic251C2×C412yescomplemented
33C24×C251D412yes—
34C2×D4×C251—22no—
35C2×D4×C251—22no—
36C2×D4×C251C2×C412yes—
37C2×D4×C251D412yes—
38C22×C4×C251—24no—
39C22×C4×C251C2311yescharacteristic
40C4⋊C4×C251—22no—
41C22⋊C4×C251—24no—
42C22⋊C4×C251—22no—
43C42×C251D414yescomplemented
44C22×D251—21no—
45Dic502—22no—
46D1004—22no—
47C22⋊D251—41no—
48C22⋊D251—41no—
49C22⋊D251—84no—
50C2×Dic251—22no—
51C2×Dic251—21no—
52C4×D251—42no—
53C251⋊C8—42no—
54C23×C251—22no—
55C23×C251—22no—
56C23×C251—82no—
57C23×C251C22⋊C414yes—
58C23×C251C2×D411yescharacteristic
59D4×C251—24no—
60D4×C251—24no—
61C2×C4×C251—24no—
62C2×C4×C251—84no—
63C2×C4×C251C22×C411yescharacteristic
64C2×C4×C251C2×D412yes—
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251#4112yes—
67C22×C251—24no—
68C22×C251—41no—
69C22×C251—44no—
70C22×C251—44no—
71C22×C251—82no—
72C22×C251C2×C22⋊C411yescharacteristic
73C22×C251#12011yescharacteristic
74C22×C251#12311yescharacteristic
75C2×D251—42no—
76C2×D251—41no—
77C4×C251—44no—
78C4×C251C4×D412yes—
79C4×C251#12112yes—
80Dic251—42no—
81Dic251—41no—
82C2×C251—21no—
83C2×C251—21no—
84C2×C251—21no—
85C2×C251—44no—
86C2×C251#38611yescharacteristic
87D251—81no—
88C251#191111yescharacteristic, complemented, Sylow
89#1911—2511nomaximal, Sylow
90#535—2511no—
91#518D25111yescharacteristic, complemented
92#421—2512no—
93C2×#100—2511no—
94H(3,ℤ/4)—2512no—
95C2×#41—2511no—
96C22×D4C2×D25111yescharacteristic
97C2×M16—2512no—
98#127—2512no—
99#124—2512no—
100#121—5022no—
101#120—5022no—
102C4×D4—24no—
103#117—2511no—
104#117C2×D25112yes—
105C2×C22⋊C4—22no—
106#104—5024no—
107#100—5022no—
108C24C22⋊D25112yes—
109#41—5022no—
110#41—5022no—
111C2×Q8—2511no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—5024no—
115C2×D4—5021no—
116C2×D4—2511no—
117C2×D4C4×D25112yes—
118C2×D4D100412yes—
119C22×C4—24no—
120C22×C4—5021no—
121C22×C4C22×D25111yescharacteristic
122M16—5022no—
123M16—5022no—
124C2×C8—5022no—
125C4⋊C4—22no—
126C4⋊C4—5021no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—5024no—
130C22⋊C4—5021no—
131C42—2512no—
132C42C22⋊D25114yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—5021no—
137C23(C2×D251)⋊C414yes—
138C23C2×C22⋊D25111yescharacteristic
139Q8—5022no—
140D4—24no—
141D4—24no—
142D4—20084no—
143D4—10041no—
144D4—10041no—
145D4—5022no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—10042no—
149C2×C4—5022no—
150C2×C4—5021no—
151C2×C4C2×C4×D25111yescharacteristic
152C2×C4C2×C22⋊D25112yes—
153C2×C4C2×D100411yescharacteristic
154C2×C4#1686081012yes—
155C8—10042no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—10042no—
162C22—10041no—
163C22C2×(C2×D251)⋊C411yescharacteristic
164C22#2254868411yescharacteristic
165C22#2254869811yescharacteristic
166C4—44no—
167C4—10042no—
168C4—10041no—
169C4C4×C22⋊D25112yes—
170C4#2254868912yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—20081no—
176C2#3063156611yescharacteristic
subgroupG/H
centerC2#30631566
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#518×C251C2
socleC2×C251#386
prime radical#30632254C2
cosocleC22×C4×C251C23
hypercenter#518D251
hypocenterC251#1911
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #30631705C2semidirect
3. C2×#22548632C2semidirect
4. #30631443C2semidirect
5. #518×C251C2semidirect
8. #22548711C4semidirect
9. #22548702C4semidirect
21. C2×Dic502C2×C4semidirect
22. C2×D1004C2×C4semidirect
32. C4×Dic251C2×C4semidirect
43. C42×C251D4semidirect
88. C251#1911semidirect
91. #518D251semidirect
132. C42C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306322542
#306317052
C2×#225486322
#306314432
#518×C2512
#1911251