Finite Group Databasebeta

← #44550892

Subgroups — #44550892

Order 32128.

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

normal subgroups only

This group has 344 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#30631693C212yesmaximal, complemented
3#30631661C211yesmaximal, characteristic, complemented
4#30631444C212yesmaximal, complemented
5#568×C251C211yesmaximal, characteristic, complemented
6C22×D1004C2211yescharacteristic
7#22548711C412yescomplemented
8#22548704C412yescomplemented
9#22548689—22no—
10#22548683—22no—
11#22548674C2212yes—
12#22548674C2211yescharacteristic
13#22548637—24no—
14#22548630—22no—
15C2×#41×C251C2211yescharacteristic
16#131×C251—24no—
17C2×M16×C251—22no—
18C2×M16×C251C2212yes—
19C23×D251—21no—
20C2×D1004—22no—
21C2×D1004—21no—
22C2×D1004—22no—
23C2×D1004C2×C412yes—
24C2×C22⋊D251—24no—
25C4⋊Dic251—21no—
26C4⋊Dic251—22no—
27C22⋊Dic251—21no—
28C22⋊Dic251—22no—
29(C2×D251)⋊C4—24no—
30C4×Dic251—22no—
31C4×Dic251C2×C412yes—
32#41×C251—22no—
33#41×C251—22no—
34C2×Q8×C251C2×C411yescharacteristic
35C2×D4×C251C2×C411yescharacteristic
36C2×D4×C251D412yes—
37C22×C4×C251D412yes—
38C22×C4×C251C2311yescharacteristic
39M16×C251—24no—
40M16×C251—22no—
41M16×C251—22no—
42C2×C8×C251—24no—
43C2×C8×C251D414yescomplemented
44C22×D251—22no—
45C22×D251—42no—
46C22×D251—82no—
47D1004—24no—
48D1004—41no—
49D1004—41no—
50C22⋊D251—84no—
51C2×Dic251—22no—
52C2×Dic251—42no—
53C23×C251C22⋊C412yes—
54C23×C251C2×D411yescharacteristic
55Q8×C251—22no—
56D4×C251—24no—
57D4×C251—22no—
58C2×C4×C251—22no—
59C2×C4×C251—22no—
60C2×C4×C251C22⋊C412yes—
61C2×C4×C251C22×C411yescharacteristic
62C2×C4×C251C2×D412yes—
63C2×C4×C251C2×D411yescharacteristic
64C2×C4×C251#4112yes—
65C8×C251—24no—
66C22×C251—22no—
67C22×C251—41no—
68C22×C251—42no—
69C22×C251C2×C22⋊C411yescharacteristic
70C22×C251#12112yes—
71C2×D251—44no—
72C2×D251—42no—
73C2×D251—84no—
74C4×C251—22no—
75C4×C251C4×D412yes—
76C4×C251#12011yescharacteristic
77C4×C251#12311yescharacteristic
78Dic251—42no—
79Dic251—82no—
80C2×C251—21no—
81C2×C251—22no—
82C2×C251—42no—
83C2×C251#38611yescharacteristic
84D251—82no—
85C251#191011yescharacteristic, complemented, Sylow
86#1910—2511nomaximal, Sylow
87#568D25111yescharacteristic, complemented
88#535—2511no—
89#418—2512no—
90#410—2511no—
91H(3,ℤ/4)—2512no—
92C2×#41C2×D25111yescharacteristic
93C22×D4—2511no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D25112yes—
97#127—2512no—
98#124—2512no—
99#121—5022no—
100#120—5022no—
101#117—2512no—
102#117—2511no—
103#102—5024no—
104#99—5022no—
105C24—5021no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D25111yescharacteristic
109C2×D4—5024no—
110C2×D4—5021no—
111C2×D4—5022no—
112C2×D4—5022no—
113C2×D4—2512no—
114C2×D4C4×D25111yescharacteristic
115C2×D4D100412yes—
116C22×C4C22⋊D25112yes—
117C22×C4C22×D25111yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D25114yescomplemented
123C4⋊C4—5022no—
124C4⋊C4—5021no—
125C22⋊C4—5022no—
126C22⋊C4—5021no—
127C22⋊C4—5024no—
128C42—5022no—
129C42—2512no—
130C23—20082no—
131C23—10042no—
132C23—5022no—
133C23(C2×D251)⋊C412yes—
134C23C2×D100411yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—20084no—
139D4—10041no—
140D4—10041no—
141D4—5024no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—10042no—
145C2×C4—5022no—
146C2×C4(C2×D251)⋊C412yes—
147C2×C4C2×C4×D25111yescharacteristic
148C2×C4C2×C22⋊D25112yes—
149C2×C4C2×C22⋊D25111yescharacteristic
150C2×C4#1686081012yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—20084no—
156C22—10044no—
157C22—10042no—
158C22C2×(C2×D251)⋊C411yescharacteristic
159C22#2254868912yes—
160C4—22no—
161C4—20082no—
162C4—10042no—
163C4C4×C22⋊D25112yes—
164C4#2254868411yescharacteristic
165C4#2254869811yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—20082no—
170C2#3063156611yescharacteristic
subgroupG/H
centerC2#30631566
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#568×C251C2
socleC2×C251#386
prime radical#30632251C2
cosocleC22×C4×C251C23
hypercenter#568D251
hypocenterC251#1910
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #30631693C2semidirect
3. #30631661C2semidirect
4. #30631444C2semidirect
5. #568×C251C2semidirect
7. #22548711C4semidirect
8. #22548704C4semidirect
43. C2×C8×C251D4semidirect
85. C251#1910semidirect
87. #568D251semidirect
122. C2×C8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306322512
#306316932
#306316612
#306314442
#568×C2512
#1910251