Finite Group Databasebeta

← #44550890

Subgroups — #44550890

Order 32128.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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#30632454C211yesmaximal, characteristic, complemented
2#30632251C211yesmaximal, characteristic
3#30631691C212yesmaximal, complemented
4#30631443C212yesmaximal, complemented
5#410×C251C211yesmaximal, characteristic, complemented
6C22×D1004C2211yescharacteristic
7#22548722—24no—
8C2×#16860794—22no—
9C2×#16860794C2212yes—
10#22548711C412yescomplemented
11#22548704C412yescomplemented
12#22548689—22no—
13#22548683—22no—
14#22548674C2211yescharacteristic
15#22548635—24no—
16C2×#41×C251C2211yescharacteristic
17#117×C251C2212yes—
18#99×C251—22no—
19C23×D251—21no—
20C2×D1004—22no—
21C2×D1004—21no—
22C2×D1004—22no—
23C2×D1004C2×C412yes—
24C2×C22⋊D251—24no—
25#16860794—24no—
26#16860794—22no—
27#16860794—22no—
28C2×C251⋊C8—24no—
29C2×C251⋊C8D414yescomplemented
30C4⋊Dic251—21no—
31C22⋊Dic251—21no—
32(C2×D251)⋊C4—24no—
33C4×Dic251C2×C412yes—
34#41×C251—22no—
35#41×C251—22no—
36C2×Q8×C251C2×C411yescharacteristic
37C2×D4×C251C2×C411yescharacteristic
38C2×D4×C251D412yes—
39C22×C4×C251D412yes—
40C22×C4×C251C2311yescharacteristic
41C4⋊C4×C251—22no—
42C22⋊C4×C251—22no—
43C42×C251—22no—
44C22×D251—22no—
45C22×D251—42no—
46C22×D251—82no—
47D1004—24no—
48D1004—41no—
49D1004—41no—
50C22⋊D251—84no—
51C2×Dic251—22no—
52C251⋊C8—24no—
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×C251—42no—
61C2×C4×C251C22⋊C412yes—
62C2×C4×C251C22×C411yescharacteristic
63C2×C4×C251C2×D412yes—
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251#4112yes—
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×C251—82no—
76C4×C251C4×D412yes—
77C4×C251#12011yescharacteristic
78C4×C251#12311yescharacteristic
79Dic251—42no—
80C2×C251—21no—
81C2×C251—22no—
82C2×C251—42no—
83C2×C251#38611yescharacteristic
84D251—82no—
85C251#191011yescharacteristic, complemented, Sylow
86#1910—2511nomaximal, Sylow
87#568—2511no—
88#535—2511no—
89#418—2512no—
90#410D25111yescharacteristic, complemented
91H(3,ℤ/4)—2512no—
92C2×#41C2×D25111yescharacteristic
93C22×D4—2511no—
94#131—5024no—
95C2×M16—5022no—
96C2×M16—2512no—
97#127—2512no—
98#124—2512no—
99#121—5022no—
100#120—5022no—
101#117—2511no—
102#117C2×D25112yes—
103#102—5024no—
104#99—22no—
105C24—5021no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D25111yescharacteristic
109C2×D4—5024no—
110C2×D4—5022no—
111C2×D4—5021no—
112C2×D4—5022no—
113C2×D4—2512no—
114C2×D4C4×D25111yescharacteristic
115C2×D4C22⋊D25112yes—
116C22×C4D100412yes—
117C22×C4C22×D25111yescharacteristic
118M16—5024no—
119M16—5022no—
120M16—5022no—
121C2×C8—5024no—
122C2×C8—2514no—
123C4⋊C4—22no—
124C4⋊C4—5021no—
125C22⋊C4—22no—
126C22⋊C4—5021no—
127C22⋊C4—5024no—
128C42—22no—
129C42—2512no—
130C23—20082no—
131C23—10042no—
132C23—5022no—
133C23(C2×D251)⋊C412yes—
134C23C2×C22⋊D25111yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—20084no—
139D4—10041no—
140D4—10041no—
141D4—5024no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—5022no—
146C2×C4(C2×D251)⋊C412yes—
147C2×C4C2×C4×D25111yescharacteristic
148C2×C4D4×D25112yes—
149C2×C4C2×D100411yescharacteristic
150C2×C4#1686080912yes—
151C8—5024no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—20084no—
156C22—10044no—
157C22—10042no—
158C22C2×(C2×D251)⋊C411yescharacteristic
159C22#2254868712yes—
160C4—22no—
161C4—82no—
162C4—10042no—
163C4(C22⋊D251)⋊C412yes—
164C4#2254868511yescharacteristic
165C4#2254869911yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—20082no—
170C2#3063156411yescharacteristic
subgroupG/H
centerC2#30631564
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#410×C251C2
socleC2×C251#386
prime radical#30632251C2
cosocleC22×C4×C251C23
hypercenter#410D251
hypocenterC251#1910
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #30632454C2semidirect
3. #30631691C2semidirect
4. #30631443C2semidirect
5. #410×C251C2semidirect
10. #22548711C4semidirect
11. #22548704C4semidirect
29. C2×C251⋊C8D4semidirect
85. C251#1910semidirect
90. #410D251semidirect

Maximal subgroups

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

subgroupindex
#306324542
#306322512
#306316912
#306314432
#410×C2512
#1910251