Finite Group Databasebeta

← #44399030

Subgroups — #44399030

Order 29056.

206 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 45 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#28409782C211yesmaximal, characteristic, complemented
2#28409396C212yesmaximal, complemented
3#28408773C212yesmaximal, complemented
4#28408772C211yesmaximal, characteristic, complemented
5#496×C227C211yesmaximal, characteristic
6#20300286—21no—
7#20300281—22no—
8#20300280C2212yes—
9#20300269—21no—
10#20300265—22no—
11#20300209—22no—
12#20300209—22no—
13#20300199C2212yes—
14C2×SD16×C227—21no—
15C2×D8×C227—21no—
16#127×C227C2211yescharacteristic
17#124×C227C2211yescharacteristic
18#97×C227C2211yescharacteristic
19C2×Dic454—21no—
20C22×Dic227—21no—
21C2×C227⋊C8—22no—
22C4⋊Dic227D412yes—
23C4⋊Dic227D412yes—
24Dic227⋊C4—22no—
25Dic227⋊C4—22no—
26Dic227⋊C4—22no—
27C22⋊Dic227—22no—
28C22⋊Dic227—21no—
29C4×Dic227—21no—
30C2×Q8×C227D411yescharacteristic
31C2×D4×C227—21no—
32C2×D4×C227—22no—
33C2×D4×C227D411yescharacteristic
34SD16×C227—42no—
35D8×C227—42no—
36C2×C8×C227—21no—
37C22⋊C4×C227—22no—
38C42×C227C2311yescharacteristic
39Dic454—81no—
40C2×Dic227—22no—
41C2×Dic227—22no—
42C2×Dic227—81no—
43C227⋊C8—42no—
44C23×C227—21no—
45C23×C227—41no—
46Q8×C227—41no—
47D4×C227—41no—
48D4×C227—41no—
49D4×C227—41no—
50D4×C227—44no—
51C2×C4×C227—21no—
52C2×C4×C227C2×D412yes—
53C2×C4×C227C2×D411yescharacteristic
54C8×C227—22no—
55C22×C227—41no—
56C22×C227—42no—
57C22×C227—81no—
58C22×C227—82no—
59C22×C227#12011yescharacteristic
60C4×C227—21no—
61C4×C227—22no—
62C4×C227—41no—
63Dic227—42no—
64Dic227—42no—
65C2×C227—81no—
66C2×C227—161no—
67C2×C227#45311yescharacteristic
68C2×C227#45412yes—
69C227#165711yescharacteristic, complemented, Sylow
70#1657—2271nomaximal, Sylow
71#563—2271no—
72#496D22711yescharacteristic
73#490—2272no—
74#330—2272no—
75#329—2271no—
76C2×SD16—21no—
77C2×D8—21no—
78C4⋊Q8—4541no—
79#127C2×D22711yescharacteristic
80#126—4542no—
81#125—2272no—
82#124C2×D22711yescharacteristic
83#123—4541no—
84C22⋊Q8—4542no—
85#102—4542no—
86#102—4542no—
87#97—2272no—
88#97C2×D22711yescharacteristic
89C2×Q8—4541no—
90C2×Q8C22⋊D22711yescharacteristic
91C2×D4—22no—
92C2×D4—21no—
93C2×D4C22⋊D22711yescharacteristic
94C22×C4—4541no—
95SD16—42no—
96D8—42no—
97C2×C8—21no—
98C2×C8—4542no—
99C4⋊C4—4542no—
100C4⋊C4—4542no—
101C4⋊C4—4542no—
102C4⋊C4—2272no—
103C4⋊C4—2272no—
104C22⋊C4—22no—
105C22⋊C4—4542no—
106C22⋊C4—4541no—
107C42—4541no—
108C42C22×D22711yescharacteristic
109C23—21no—
110C23—41no—
111Q8—41no—
112Q8—18161no—
113D4—41no—
114D4—41no—
115D4—41no—
116D4—44no—
117C2×C4—21no—
118C2×C4—18161no—
119C2×C4—4542no—
120C2×C4—4542no—
121C2×C4D4×D22712yes—
122C2×C4C2×C22⋊D22711yescharacteristic
123C8—22no—
124C8—9082no—
125C22—41no—
126C22—42no—
127C22—81no—
128C22—82no—
129C22#2030025611yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—9082no—
134C4—9082no—
135C2—81no—
136C2—161no—
137C2#2840919911yescharacteristic
138C2#2840920312yes—
subgroupG/H
centerC22#20300256
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting, prime radical#496×C227C2
socleC22×C227#120
hypercenter#496D227
hypocenterC227#1657
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #28409782C2semidirect
2. #28409396C2semidirect
3. #28408773C2semidirect
4. #28408772C2semidirect
69. C227#1657semidirect

Maximal subgroups

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

subgroupindex
#284097822
#284093962
#284087732
#284087722
#496×C2272
#1657227