Finite Group Databasebeta

← #44550764

Subgroups — #44550764

Order 32128.

444 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 97 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#30632162C211yesmaximal, characteristic, complemented
2#30631692C212yesmaximal, complemented
3C2×#22548632C211yesmaximal, characteristic, complemented
4#30631447C212yesmaximal, complemented
5C2×#127×C251C211yesmaximal, characteristic
6C2×#16860794C2212yes—
7D4×Dic251—24no—
8#22548674C2212yes—
9C2×C22⋊Dic251—22no—
10#22548636—24no—
11#22548632—22no—
12C22×D4×C251—22no—
13C22×D4×C251C2211yescharacteristic
14C22×D4×C251C2211yescharacteristic
15#127×C251—22no—
16#127×C251C414yescomplemented
17C2×C42×C251C2211yescharacteristic
18C22×Dic251—24no—
19#16860794—22no—
20#16860794—22no—
21C2×C251⋊C8—22no—
22C4⋊Dic251D414yescomplemented
23C22⋊Dic251—24no—
24C22⋊Dic251—22no—
25C4×Dic251—22no—
26C24×C251—21no—
27C24×C251D412yes—
28C2×D4×C251—28no—
29C2×D4×C251—22no—
30C2×D4×C251—22no—
31C2×D4×C251—21no—
32C2×D4×C251—22no—
33C2×D4×C251—22no—
34C2×D4×C251—48no—
35C2×D4×C251C2×C412yes—
36C2×D4×C251C2×C412yescomplemented
37C2×D4×C251D412yes—
38C22×C4×C251—21no—
39C22×C4×C251C2311yescharacteristic
40C42×C251—21no—
41C42×C251C2×C412yescomplemented
42C2×Dic251—24no—
43C2×Dic251—84no—
44C251⋊C8—42no—
45C23×C251—22no—
46C23×C251—22no—
47C23×C251—22no—
48C23×C251—44no—
49C23×C251—44no—
50C23×C251—41no—
51C23×C251—41no—
52C23×C251C22⋊C414yes—
53C23×C251C2×D411yescharacteristic
54D4×C251—24no—
55D4×C251—24no—
56D4×C251—24no—
57D4×C251—416no—
58D4×C251—42no—
59C2×C4×C251—24no—
60C2×C4×C251—22no—
61C2×C4×C251C22×C411yescharacteristic
62C2×C4×C251C2×D412yes—
63C2×C4×C251C2×D411yescharacteristic
64C2×C4×C251#4112yes—
65C22×C251—22no—
66C22×C251—24no—
67C22×C251—44no—
68C22×C251—44no—
69C22×C251—44no—
70C22×C251—42no—
71C22×C251—82no—
72C22×C251—82no—
73C22×C251—82no—
74C22×C251C2×C22⋊C411yescharacteristic
75C22×C251#12011yescharacteristic
76C22×C251#12311yescharacteristic
77C4×C251—24no—
78C4×C251C4×D412yes—
79C4×C251#12112yes—
80Dic251—44no—
81C2×C251—21no—
82C2×C251—21no—
83C2×C251—44no—
84C2×C251—82no—
85C2×C251#38611yescharacteristic
86C2×C251#45312yes—
87C251#188711yescharacteristic, complemented, Sylow
88#1887—2511nomaximal, Sylow
89C2×#127D25111yescharacteristic
90#518—2511no—
91#418—2512no—
92C2×#100—2511no—
93#339—2512no—
94C22×D4—22no—
95C22×D4C2×D25111yescharacteristic
96C22×D4C2×D25111yescharacteristic
97C2×M16—2512no—
98#127—22no—
99#127Dic25114yescomplemented
100C4×D4—5024no—
101#117—2512no—
102C2×C22⋊C4—5022no—
103C2×C42C2×D25111yescharacteristic
104#102—5024no—
105#100—5022no—
106C24—21no—
107C24C22⋊D25112yes—
108C2×D4—28no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—48no—
115C2×D4C2×Dic25112yes—
116C2×D4C2×Dic25112yescomplemented
117C2×D4C22⋊D25112yes—
118C22×C4—21no—
119C22×C4—5024no—
120C22×C4C22×D25111yescharacteristic
121M16—5022no—
122M16—5022no—
123C2×C8—5022no—
124C4⋊C4—2514no—
125C22⋊C4—5024no—
126C22⋊C4—5022no—
127C42—21no—
128C42—5022no—
129C42C2×Dic25112yescomplemented
130C23—22no—
131C23—22no—
132C23—22no—
133C23—44no—
134C23—44no—
135C23—41no—
136C23—41no—
137C23C22⋊Dic25114yes—
138C23C2×C22⋊D25111yescharacteristic
139D4—24no—
140D4—24no—
141D4—24no—
142D4—416no—
143D4—42no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—20084no—
147C2×C4—5024no—
148C2×C4C22×Dic25111yescharacteristic
149C2×C4D4×D25112yes—
150C2×C4C2×C22⋊D25111yescharacteristic
151C2×C4#1686080912yes—
152C8—10042no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—44no—
158C22—42no—
159C22—82no—
160C22—82no—
161C22—82no—
162C22C2×C22⋊Dic25111yescharacteristic
163C22#2254868311yescharacteristic
164C22#2254869611yescharacteristic
165C4—24no—
166C4—10044no—
167C4D4×Dic25112yes—
168C4#2254868712yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—82no—
173C2#3063156711yescharacteristic
174C2#3063185412yes—
subgroupG/H
centerC22#22548683
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting, prime radicalC2×#127×C251C2
socleC22×C251#120
cosocleC22×C4×C251C23
hypercenterC2×#127D251
hypocenterC251#1887
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #30632162C2semidirect
2. #30631692C2semidirect
3. C2×#22548632C2semidirect
4. #30631447C2semidirect
16. #127×C251C4semidirect
22. C4⋊Dic251D4semidirect
36. C2×D4×C251C2×C4semidirect
41. C42×C251C2×C4semidirect
87. C251#1887semidirect
99. #127Dic251semidirect
116. C2×D4C2×Dic251semidirect
129. C42C2×Dic251semidirect

Maximal subgroups

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

subgroupindex
#306321622
#306316922
C2×#225486322
#306314472
C2×#127×C2512
#1887251