Finite Group Databasebeta

← #44558684

Subgroups — #44558684

Order 32128.

582 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×C4×Dic251C211yesmaximal, characteristic, complemented
2#30631575C214yesmaximal, complemented
3#30631568C214yesmaximal, complemented
4#30631567C212yesmaximal
5#30631559C212yesmaximal, complemented
6#30631550C211yesmaximal, characteristic, complemented
7C2×#123×C251C211yesmaximal, characteristic
8C23×Dic251C2212yes—
9C2×Dic251⋊C4C2214yes—
10C2×C22⋊Dic251C2212yes—
11C2×C22⋊Dic251C2214yes—
12C2×C4×Dic251—24no—
13C2×C4×Dic251C2214yes—
14C251⋊#95C2214yes—
15C251⋊#95C2214yes—
16C251⋊#95C2214yes—
17C22×D4×C251C2211yescharacteristic
18C23×C4×C251C2211yescharacteristic
19#123×C251—24no—
20C2×C4⋊C4×C251C2212yes—
21C2×C22⋊C4×C251C2211yescharacteristic
22C2×C22⋊C4×C251C2212yes—
23C22×Dic251—28no—
24C22×Dic251—24no—
25C22×Dic251C2314yes—
26C22×Dic251C2314yes—
27Dic251⋊C4—28no—
28C22⋊Dic251—28no—
29C22⋊Dic251—24no—
30C4×Dic251—28no—
31C24×C251C2311yescharacteristic
32C24×C251C2311yescharacteristic
33C2×D4×C251—22no—
34C2×D4×C251—24no—
35C22×C4×C251—24no—
36C22×C4×C251D414yes—
37C22×C4×C251C2312yes—
38C22×C4×C251C2311yescharacteristic
39C22×C4×C251C2312yes—
40C4⋊C4×C251—24no—
41C22⋊C4×C251—22no—
42C22⋊C4×C251—24no—
43C2×Dic251—216no—
44C2×Dic251—24no—
45C2×Dic251—24no—
46C2×Dic251—24no—
47C2×Dic251—24no—
48C2×Dic251—24no—
49C2×Dic251#4118yes—
50C23×C251—24no—
51C23×C251—21no—
52C23×C251—22no—
53C23×C251—24no—
54C23×C251C2×D412yes—
55C23×C251#4114yes—
56C23×C251C2411yescharacteristic
57D4×C251—44no—
58C2×C4×C251—28no—
59C2×C4×C251—24no—
60C2×C4×C251—22no—
61C2×C4×C251—21no—
62C2×C4×C251—22no—
63C2×C4×C251—24no—
64C2×C4×C251C2×D414yes—
65C22×C251—24no—
66C22×C251—28no—
67C22×C251—41no—
68C22×C251—42no—
69C22×C251—44no—
70C22×C251#12314yes—
71C22×C251C22×D411yescharacteristic
72C22×C251C2×#4114yes—
73C22×C251C2×#4112yes—
74C4×C251—24no—
75C4×C251—41no—
76C4×C251—42no—
77Dic251—28no—
78Dic251—44no—
79C2×C251—24no—
80C2×C251—81no—
81C2×C251C2×#12311yescharacteristic
82C2×C251#52512yes—
83C2×C251#52914yes—
84C251#285311yescharacteristic, complemented, Sylow
85#2853—2511nomaximal, Sylow
86C2×#123D25111yescharacteristic
87C22×C42—2511no—
88#388—2514no—
89#387—2514no—
90#386—2512no—
91#385—2512no—
92#381—2511no—
93C22×D4C2×D25111yescharacteristic
94C23×C4—2512no—
95C23×C4C2×D25111yescharacteristic
96#123—24no—
97C2×C4⋊C4—2514no—
98C2×C4⋊C4C2×D25112yes—
99C2×C22⋊C4—2512no—
100C2×C22⋊C4—2514no—
101C2×C22⋊C4C2×D25111yescharacteristic
102C2×C22⋊C4C2×D25112yes—
103C2×C42—5024no—
104C2×C42—2514no—
105#95—2514no—
106#95—2514no—
107#95—2514no—
108C24C22×D25111yescharacteristic
109C24C22×D25111yescharacteristic
110C2×D4—22no—
111C2×D4—24no—
112C22×C4—24no—
113C22×C4—5024no—
114C22×C4—5028no—
115C22×C4—2514no—
116C22×C4—2514no—
117C22×C4C22⋊D25114yes—
118C22×C4C22×D25111yescharacteristic
119C22×C4C22×D25112yes—
120C22×C4C22×D25112yes—
121C4⋊C4—24no—
122C4⋊C4—5028no—
123C22⋊C4—22no—
124C22⋊C4—24no—
125C22⋊C4—5024no—
126C22⋊C4—5028no—
127C42—5028no—
128C23—24no—
129C23—21no—
130C23—24no—
131C23—22no—
132C23C2×C22⋊D25112yes—
133C23#1686080914yes—
134C23C23×D25111yescharacteristic
135D4—44no—
136C2×C4—24no—
137C2×C4—28no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—5024no—
143C2×C4—5024no—
144C2×C4—5024no—
145C2×C4—50216no—
146C2×C4—5024no—
147C2×C4—5024no—
148C2×C4—2518no—
149C2×C4C2×C22⋊D25114yes—
150C22—24no—
151C22—28no—
152C22—42no—
153C22—44no—
154C22—41no—
155C22#2254869614yes—
156C22C22×C22⋊D25111yescharacteristic
157C22#41×D25114yes—
158C22C2×#1686080912yes—
159C4—24no—
160C4—42no—
161C4—41no—
162C4—10044no—
163C4—5028no—
164C2—24no—
165C2—81no—
166C2C2×#2254869611yescharacteristic
167C2#3063219412yes—
168C2#3063221414yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#123×C251C2
prime radicalC22×D4×C251C22
hypercenterC2×#123D251
hypocenterC251#2853
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic251C2semidirect
2. #30631575C2semidirect
3. #30631568C2semidirect
5. #30631559C2semidirect
6. #30631550C2semidirect
84. C251#2853semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic2512
#306315752
#306315682
#306315672
#306315592
#306315502
C2×#123×C2512
#2853251