Finite Group Databasebeta

← #44408056

Subgroups — #44408056

Order 29056.

582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#20300271C211yesmaximal, characteristic
2#28408923C214yesmaximal, complemented
3#28408916C214yesmaximal, complemented
4#28408911C212yesmaximal
5#28408905C212yesmaximal, complemented
6#28408895C211yesmaximal, characteristic, complemented
7C22×C42×C227C211yesmaximal, characteristic, complemented
8C22×C22⋊D227C2211yescharacteristic
9#20300271—24no—
10C2×Dic227⋊C4C2214yes—
11C2×Dic227⋊C4C2212yes—
12C2×C22⋊Dic227C2212yes—
13C2×C22⋊Dic227C2211yescharacteristic
14C2×(C2×D227)⋊C4C2214yes—
15C2×(C2×D227)⋊C4C2212yes—
16C227⋊#95C2214yes—
17C227⋊#95C2214yes—
18C227⋊#95C2214yes—
19C23×C4×C227C2212yes—
20C23×C4×C227C2211yescharacteristic
21C2×C42×C227—24no—
22C2×C42×C227C2214yes—
23C23×D227C2311yescharacteristic
24C2×C22⋊D227—22no—
25C2×C22⋊D227—24no—
26C22×Dic227C2311yescharacteristic
27C22×Dic227C2314yes—
28C22×Dic227C2312yes—
29Dic227⋊C4—24no—
30Dic227⋊C4—28no—
31C22⋊Dic227—22no—
32C22⋊Dic227—24no—
33(C2×D227)⋊C4—24no—
34(C2×D227)⋊C4—28no—
35C24×C227C2311yescharacteristic
36C22×C4×C227—24no—
37C22×C4×C227—24no—
38C22×C4×C227—28no—
39C22×C4×C227D414yes—
40C22×C4×C227C2312yes—
41C22×C4×C227C2314yes—
42C42×C227—28no—
43C22×D227—21no—
44C22×D227—24no—
45C22×D227—22no—
46C22⋊D227—44no—
47C2×Dic227—22no—
48C2×Dic227—21no—
49C2×Dic227—24no—
50C2×Dic227—24no—
51C2×Dic227—24no—
52C2×Dic227—24no—
53C2×Dic227—22no—
54C23×C227—24no—
55C23×C227C2×D412yes—
56C23×C227#4114yes—
57C23×C227C2411yescharacteristic
58C2×C4×C227—216no—
59C2×C4×C227—28no—
60C2×C4×C227—24no—
61C2×C4×C227—24no—
62C2×C4×C227—24no—
63C2×C4×C227C2×D414yes—
64C2×C4×C227#4118yes—
65C22×C227—24no—
66C22×C227—28no—
67C22×C227#12314yes—
68C22×C227C22×D411yescharacteristic
69C22×C227C2×#4112yes—
70C22×C227C2×#4114yes—
71C2×D227—41no—
72C2×D227—42no—
73C2×D227—44no—
74C4×C227—24no—
75C4×C227—28no—
76Dic227—41no—
77Dic227—42no—
78Dic227—44no—
79C2×C227—24no—
80C2×C227C2×#12311yescharacteristic
81C2×C227#52512yes—
82C2×C227#52914yes—
83D227—81no—
84C227#285311yescharacteristic, complemented, Sylow
85#2853—2271nomaximal, Sylow
86C2×#123—2271no—
87C22×C42D22711yescharacteristic, complemented
88#388—2274no—
89#387—2274no—
90#386—2272no—
91#385—2272no—
92#381—2271no—
93C22×D4—2271no—
94C23×C4C2×D22712yes—
95C23×C4C2×D22711yescharacteristic
96#123—4544no—
97C2×C4⋊C4—2272no—
98C2×C4⋊C4—2274no—
99C2×C22⋊C4—2271no—
100C2×C22⋊C4—2272no—
101C2×C22⋊C4—2272no—
102C2×C22⋊C4—2274no—
103C2×C42—24no—
104C2×C42C2×D22714yes—
105#95—2274no—
106#95—2274no—
107#95—2274no—
108C24—2271no—
109C24C22×D22711yescharacteristic
110C2×D4—4542no—
111C2×D4—4544no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4—2274no—
116C22×C4—2271no—
117C22×C4—2272no—
118C22×C4C22⋊D22714yes—
119C22×C4C22×D22712yes—
120C22×C4C22×D22714yes—
121C4⋊C4—4544no—
122C4⋊C4—4548no—
123C22⋊C4—4542no—
124C22⋊C4—4544no—
125C22⋊C4—4544no—
126C22⋊C4—4548no—
127C42—28no—
128C23—24no—
129C23—4541no—
130C23—4544no—
131C23—4542no—
132C23C2×C22⋊D22712yes—
133C23#1681388414yes—
134C23C23×D22711yescharacteristic
135D4—9084no—
136C2×C4—216no—
137C2×C4—24no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—28no—
141C2×C4—4542no—
142C2×C4—4541no—
143C2×C4—4544no—
144C2×C4—4544no—
145C2×C4—4544no—
146C2×C4—4544no—
147C2×C4—4542no—
148C2×C4C2×C22⋊D22714yes—
149C2×C4#1681388418yes—
150C22—24no—
151C22—28no—
152C22—9081no—
153C22—9082no—
154C22—9084no—
155C22#2030027114yes—
156C22C22×C22⋊D22711yescharacteristic
157C22C2×#1681388412yes—
158C22C2×#1681388414yes—
159C4—24no—
160C4—28no—
161C4—9081no—
162C4—9082no—
163C4—9084no—
164C2—24no—
165C2—18161no—
166C2C2×#2030027111yescharacteristic
167C2#2840954312yes—
168C2#2840956614yes—
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
FittingC22×C42×C227C2
prime radicalC22×C22⋊D227C22
hypercenterC22×C42D227
hypocenterC227#2853
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #28408923C2semidirect
3. #28408916C2semidirect
5. #28408905C2semidirect
6. #28408895C2semidirect
7. C22×C42×C227C2semidirect
84. C227#2853semidirect
87. C22×C42D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002712
#284089232
#284089162
#284089112
#284089052
#284088952
C22×C42×C2272
#2853227