Finite Group Databasebeta

← #44550912

Subgroups — #44550912

Order 32128.

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

normal subgroups only

This group has 340 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#30632251C211yesmaximal, characteristic
2#30631671C211yesmaximal, characteristic, complemented
3C2×#22548631C212yesmaximal, complemented
4#30631459C212yesmaximal, complemented
5#568×C251C211yesmaximal, characteristic, complemented
6C22×D1004C2211yescharacteristic
7C2×#16860794C2212yes—
8#22548711—21no—
9#22548704—21no—
10#22548689—22no—
11#22548683C414yes—
12#22548674C2211yescharacteristic
13#22548634—21no—
14#22548632—21no—
15#22548631—24no—
16C2×#41×C251C2211yescharacteristic
17#131×C251—24no—
18C2×M16×C251—22no—
19C2×M16×C251C2212yes—
20C23×D251C2×C412yes—
21C2×D1004—22no—
22C2×D1004—21no—
23C2×D1004—21no—
24C2×D1004—22no—
25C2×C22⋊D251—24no—
26#16860794—22no—
27#16860794—22no—
28C2×C251⋊C8—22no—
29C4⋊Dic251—21no—
30C22⋊Dic251C2×C412yes—
31(C2×D251)⋊C4—24no—
32C4×Dic251—21no—
33#41×C251—22no—
34#41×C251—22no—
35C2×Q8×C251D411yescharacteristic
36C2×D4×C251C2×C412yes—
37C2×D4×C251D411yescharacteristic
38C22×C4×C251D412yes—
39C22×C4×C251C2311yescharacteristic
40M16×C251—24no—
41M16×C251—22no—
42M16×C251D414yescomplemented
43C2×C8×C251—24no—
44C2×C8×C251—22no—
45C22×D251—22no—
46C22×D251—24no—
47C22×D251—82no—
48D1004—41no—
49D1004—42no—
50D1004—41no—
51C22⋊D251—84no—
52C2×Dic251—22no—
53C251⋊C8—42no—
54C23×C251C22⋊C412yes—
55C23×C251C22×C411yescharacteristic
56Q8×C251—22no—
57D4×C251—24no—
58D4×C251—22no—
59C2×C4×C251—22no—
60C2×C4×C251—22no—
61C2×C4×C251C22⋊C412yes—
62C2×C4×C251C2×D412yes—
63C2×C4×C251C2×D411yescharacteristic
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251#4112yes—
66C8×C251—24no—
67C22×C251—22no—
68C22×C251—41no—
69C22×C251—42no—
70C22×C251C2×C22⋊C411yescharacteristic
71C22×C251C4×D412yes—
72C2×D251—44no—
73C2×D251—42no—
74C2×D251—84no—
75C4×C251—22no—
76C4×C251#12011yescharacteristic
77C4×C251#12112yes—
78C4×C251#12311yescharacteristic
79Dic251—42no—
80C2×C251—21no—
81C2×C251—22no—
82C2×C251—42no—
83C2×C251#38611yescharacteristic
84D251—82no—
85C251#191411yescharacteristic, complemented, Sylow
86#1914—2511nomaximal, Sylow
87#568D25111yescharacteristic, complemented
88#535—2511no—
89#413—2511no—
90C2×#100—2512no—
91#344—2512no—
92C2×#41C2×D25111yescharacteristic
93C22×D4—2511no—
94#131—24no—
95C2×M16—22no—
96C2×M16—2512no—
97C2×M16C2×D25112yes—
98#127—5021no—
99#124—5021no—
100#121—5022no—
101#120—2514no—
102#117—2511no—
103#101—5021no—
104#100—5021no—
105#100—5024no—
106C24—2512no—
107#41—22no—
108#41—22no—
109C2×Q8D100411yescharacteristic
110C2×D4—5024no—
111C2×D4—5022no—
112C2×D4—5021no—
113C2×D4—5021no—
114C2×D4—5022no—
115C2×D4C4×D25112yes—
116C2×D4D100411yescharacteristic
117C22×C4C22⋊D25112yes—
118C22×C4C22×D25111yescharacteristic
119M16—24no—
120M16—22no—
121M16—5022no—
122M16—5022no—
123M16C22⋊D25114yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—5022no—
127C4⋊C4—5021no—
128C22⋊C4—5024no—
129C22⋊C4—2512no—
130C42—5021no—
131C23—20082no—
132C23—5024no—
133C23—5022no—
134C23(C2×D251)⋊C412yes—
135C23C2×C4×D25111yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—20084no—
140D4—10041no—
141D4—10042no—
142D4—10041no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—5022no—
146C2×C4(C2×D251)⋊C412yes—
147C2×C4C2×C22⋊D25112yes—
148C2×C4C2×C22⋊D25111yescharacteristic
149C2×C4C2×D100411yescharacteristic
150C2×C4#1686081012yes—
151C8—24no—
152C8—10042no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—20084no—
157C22—10044no—
158C22—10042no—
159C22C2×(C2×D251)⋊C411yescharacteristic
160C22C4×C22⋊D25112yes—
161C4—22no—
162C4—10042no—
163C4#2254868411yescharacteristic
164C4#2254868912yes—
165C4#2254869811yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—20082no—
170C2#3063156611yescharacteristic
subgroupG/H
centerC2#30631566
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#568×C251C2
socleC2×C251#386
prime radical#30632251C2
cosocleC22×C4×C251C23
hypercenter#568D251
hypocenterC251#1914
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #30631671C2semidirect
3. C2×#22548631C2semidirect
4. #30631459C2semidirect
5. #568×C251C2semidirect
42. M16×C251D4semidirect
85. C251#1914semidirect
87. #568D251semidirect
123. M16C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306322512
#306316712
C2×#225486312
#306314592
#568×C2512
#1914251