Finite Group Databasebeta

← #44408053

Subgroups — #44408053

Order 29056.

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×Dic227C211yesmaximal, characteristic, complemented
2#28408920C214yesmaximal, complemented
3#28408913C214yesmaximal, complemented
4#28408912C212yesmaximal
5#28408904C212yesmaximal, complemented
6#28408895C211yesmaximal, characteristic, complemented
7C2×#123×C227C211yesmaximal, characteristic
8C23×Dic227C2212yes—
9C2×Dic227⋊C4C2214yes—
10C2×C22⋊Dic227C2212yes—
11C2×C22⋊Dic227C2214yes—
12C2×C4×Dic227—24no—
13C2×C4×Dic227C2214yes—
14C227⋊#95C2214yes—
15C227⋊#95C2214yes—
16C227⋊#95C2214yes—
17C22×D4×C227C2211yescharacteristic
18C23×C4×C227C2211yescharacteristic
19#123×C227—24no—
20C2×C4⋊C4×C227C2212yes—
21C2×C22⋊C4×C227C2211yescharacteristic
22C2×C22⋊C4×C227C2212yes—
23C22×Dic227—28no—
24C22×Dic227—24no—
25C22×Dic227C2314yes—
26C22×Dic227C2314yes—
27Dic227⋊C4—28no—
28C22⋊Dic227—28no—
29C22⋊Dic227—24no—
30C4×Dic227—28no—
31C24×C227C2311yescharacteristic
32C24×C227C2311yescharacteristic
33C2×D4×C227—22no—
34C2×D4×C227—24no—
35C22×C4×C227—24no—
36C22×C4×C227D414yes—
37C22×C4×C227C2312yes—
38C22×C4×C227C2311yescharacteristic
39C22×C4×C227C2312yes—
40C4⋊C4×C227—24no—
41C22⋊C4×C227—22no—
42C22⋊C4×C227—24no—
43C2×Dic227—216no—
44C2×Dic227—24no—
45C2×Dic227—24no—
46C2×Dic227—24no—
47C2×Dic227—24no—
48C2×Dic227—24no—
49C2×Dic227#4118yes—
50C23×C227—24no—
51C23×C227—21no—
52C23×C227—22no—
53C23×C227—24no—
54C23×C227C2×D412yes—
55C23×C227#4114yes—
56C23×C227C2411yescharacteristic
57D4×C227—44no—
58C2×C4×C227—28no—
59C2×C4×C227—24no—
60C2×C4×C227—22no—
61C2×C4×C227—21no—
62C2×C4×C227—22no—
63C2×C4×C227—24no—
64C2×C4×C227C2×D414yes—
65C22×C227—24no—
66C22×C227—28no—
67C22×C227—41no—
68C22×C227—42no—
69C22×C227—44no—
70C22×C227#12314yes—
71C22×C227C22×D411yescharacteristic
72C22×C227C2×#4114yes—
73C22×C227C2×#4112yes—
74C4×C227—24no—
75C4×C227—41no—
76C4×C227—42no—
77Dic227—28no—
78Dic227—44no—
79C2×C227—24no—
80C2×C227—81no—
81C2×C227C2×#12311yescharacteristic
82C2×C227#52512yes—
83C2×C227#52914yes—
84C227#285311yescharacteristic, complemented, Sylow
85#2853—2271nomaximal, Sylow
86C2×#123D22711yescharacteristic
87C22×C42—2271no—
88#388—2274no—
89#387—2274no—
90#386—2272no—
91#385—2272no—
92#381—2271no—
93C22×D4C2×D22711yescharacteristic
94C23×C4—2272no—
95C23×C4C2×D22711yescharacteristic
96#123—24no—
97C2×C4⋊C4—2274no—
98C2×C4⋊C4C2×D22712yes—
99C2×C22⋊C4—2272no—
100C2×C22⋊C4—2274no—
101C2×C22⋊C4C2×D22711yescharacteristic
102C2×C22⋊C4C2×D22712yes—
103C2×C42—4544no—
104C2×C42—2274no—
105#95—2274no—
106#95—2274no—
107#95—2274no—
108C24C22×D22711yescharacteristic
109C24C22×D22711yescharacteristic
110C2×D4—22no—
111C2×D4—24no—
112C22×C4—24no—
113C22×C4—4544no—
114C22×C4—4548no—
115C22×C4—2274no—
116C22×C4—2274no—
117C22×C4C22⋊D22714yes—
118C22×C4C22×D22711yescharacteristic
119C22×C4C22×D22712yes—
120C22×C4C22×D22712yes—
121C4⋊C4—24no—
122C4⋊C4—4548no—
123C22⋊C4—22no—
124C22⋊C4—24no—
125C22⋊C4—4544no—
126C22⋊C4—4548no—
127C42—4548no—
128C23—24no—
129C23—21no—
130C23—24no—
131C23—22no—
132C23C2×C22⋊D22712yes—
133C23#1681388314yes—
134C23C23×D22711yescharacteristic
135D4—44no—
136C2×C4—24no—
137C2×C4—28no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—4544no—
143C2×C4—4544no—
144C2×C4—4544no—
145C2×C4—45416no—
146C2×C4—4544no—
147C2×C4—4544no—
148C2×C4—2278no—
149C2×C4C2×C22⋊D22714yes—
150C22—24no—
151C22—28no—
152C22—41no—
153C22—42no—
154C22—44no—
155C22#2030026914yes—
156C22C22×C22⋊D22711yescharacteristic
157C22#41×D22714yes—
158C22C2×#1681388312yes—
159C4—24no—
160C4—41no—
161C4—42no—
162C4—9084no—
163C4—4548no—
164C2—24no—
165C2—81no—
166C2C2×#2030026911yescharacteristic
167C2#2840953912yes—
168C2#2840955914yes—
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
FittingC2×#123×C227C2
prime radicalC22×D4×C227C22
hypercenterC2×#123D227
hypocenterC227#2853
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic227C2semidirect
2. #28408920C2semidirect
3. #28408913C2semidirect
5. #28408904C2semidirect
6. #28408895C2semidirect
84. C227#2853semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic2272
#284089202
#284089132
#284089122
#284089042
#284088952
C2×#123×C2272
#2853227