Finite Group Databasebeta

← #44398744

Subgroups — #44398744

Order 29056.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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#28409663C211yesmaximal, characteristic
2#28409363C212yesmaximal
3#28408962C211yesmaximal, characteristic
4#28408770C212yesmaximal
5#324×C227C211yesmaximal, characteristic
6#20300287—22no—
7#20300279—22no—
8#20300267—21no—
9#20300267C2212yes—
10#20300247—21no—
11C2×C4⋊Dic227C2211yescharacteristic
12C2×Dic227⋊C4—21no—
13#20300220—22no—
14#20300213—22no—
15C227⋊#95—21no—
16C227⋊#95C2211yescharacteristic
17C2×C4⋊C4×C227C2211yescharacteristic
18C22⋊C8×C227C2212yes—
19C2×Dic454—21no—
20C2×Dic454D412yes—
21C22×Dic227—21no—
22C22×Dic227—21no—
23C4⋊Dic227—21no—
24C4⋊Dic227—21no—
25C4⋊Dic227—21no—
26C4⋊Dic227—22no—
27C4⋊Dic227D412yes—
28Dic227⋊C4—22no—
29Dic227⋊C4—22no—
30Dic227⋊C4—42no—
31C22⋊Dic227—22no—
32C4×Dic227—22no—
33C22×C4×C227D411yescharacteristic
34C22×C4×C227D411yescharacteristic
35C22×C4×C227C2311yescharacteristic
36C2×C8×C227—22no—
37C4⋊C4×C227—22no—
38Dic454—42no—
39Dic454—81no—
40C2×Dic227—22no—
41C2×Dic227—22no—
42C2×Dic227—22no—
43C2×Dic227—41no—
44C2×Dic227—41no—
45C2×Dic227—41no—
46C2×Dic227—41no—
47C2×Dic227—41no—
48C23×C227C2×D411yescharacteristic
49C2×C4×C227—21no—
50C2×C4×C227—21no—
51C2×C4×C227—21no—
52C2×C4×C227—41no—
53C2×C4×C227—41no—
54C2×C4×C227SD1612yes—
55C2×C4×C227C2×D412yes—
56C8×C227—42no—
57C22×C227—21no—
58C22×C227—21no—
59C22×C227#12011yescharacteristic
60C22×C227C2×SD1611yescharacteristic
61C22×C227#13711yescharacteristic
62C4×C227—22no—
63C4×C227—22no—
64C4×C227—41no—
65Dic227—42no—
66Dic227—42no—
67Dic227—42no—
68Dic227—81no—
69C2×C227—21no—
70C2×C227—21no—
71C2×C227#44811yescharacteristic
72C2×C227#45611yescharacteristic
73C2×C227#45811yescharacteristic
74C227#161111yescharacteristic, complemented, Sylow
75#1611—2271nomaximal, Sylow
76#544—2271no—
77#483—2272no—
78#398—2271no—
79#328—2272no—
80#324D22711yescharacteristic
81C4⋊Q8—4542no—
82#125—4542no—
83C22⋊Q8—4541no—
84C22⋊Q8—2272no—
85#117—4541no—
86C2×C4⋊C4—4541no—
87C2×C4⋊C4—2271no—
88C2×C4⋊C4C2×D22711yescharacteristic
89#106—4542no—
90#103—4542no—
91C22⋊C8C2×D22712yes—
92#95—4541no—
93#95—2271no—
94C2×Q8—4541no—
95C2×Q8—2272no—
96C22×C4—4541no—
97C22×C4—4541no—
98C22×C4D90811yescharacteristic
99C22×C4D90811yescharacteristic
100C22×C4C22×D22711yescharacteristic
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—9082no—
104C4⋊C4—4542no—
105C4⋊C4—4541no—
106C4⋊C4—4541no—
107C4⋊C4—4541no—
108C4⋊C4—4542no—
109C4⋊C4—4542no—
110C4⋊C4—2272no—
111C22⋊C4—4542no—
112C42—4542no—
113C23C2×D90811yescharacteristic
114Q8—18161no—
115Q8—9082no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—9081no—
122C2×C4—9081no—
123C2×C4—9081no—
124C2×C4—9081no—
125C2×C4—9081no—
126C2×C4—4542no—
127C2×C4—4542no—
128C2×C4—4542no—
129C2×C4#1681387312yes—
130C2×C4D4×D22712yes—
131C8—42no—
132C22—21no—
133C22—21no—
134C22#2030025811yescharacteristic
135C22C2×#1681387311yescharacteristic
136C22#2030032111yescharacteristic
137C4—22no—
138C4—22no—
139C4—41no—
140C4—18161no—
141C4—9082no—
142C4—9082no—
143C4—9082no—
144C2—21no—
145C2—21no—
146C2#2840917311yescharacteristic
147C2#2840921711yescharacteristic
148C2#2840922311yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
Fitting#324×C227C2
socleC22×C227#120
prime radicalC23×C227C2×D4
hypercenter#324D227
hypocenterC227#1611
LCSG ▷ C22×C4×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
74. C227#1611semidirect

Maximal subgroups

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

subgroupindex
#284096632
#284093632
#284089622
#284087702
#324×C2272
#1611227