Finite Group Databasebeta

← #44401749

Subgroups — #44401749

Order 29056.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#28409223C212yesmaximal
3#28409209C211yesmaximal, characteristic
4#28408821C212yesmaximal, complemented
5#354×C227C211yesmaximal, characteristic, complemented
6#20300333C2211yescharacteristic
7#20300319—21no—
8#20300286—21no—
9#20300281—22no—
10#20300280—21no—
11#20300272—22no—
12#20300269C2212yes—
13#20300265—22no—
14#20300216—21no—
15#20300205C2211yescharacteristic
16#20300202—22no—
17#124×C227C2211yescharacteristic
18#99×C227C2212yes—
19#16813884—21no—
20#16813883—41no—
21C2×Dic454—21no—
22C2×C22⋊D227—22no—
23D4×D227—41no—
24C22×Dic227D412yes—
25#16813874—41no—
26#16813871—41no—
27#16813868—21no—
28C4⋊Dic227—21no—
29C4⋊Dic227—21no—
30C4⋊Dic227—42no—
31Dic227⋊C4—22no—
32Dic227⋊C4—22no—
33Dic227⋊C4—22no—
34C22⋊Dic227—22no—
35C22⋊Dic227D412yes—
36(C2×D227)⋊C4—42no—
37C4×Dic227—21no—
38C2×Q8×C227D411yescharacteristic
39C2×D4×C227C2311yescharacteristic
40C22⋊C4×C227—22no—
41C22⋊C4×C227—22no—
42C42×C227D411yescharacteristic
43C22×D227—22no—
44Dic454—21no—
45Dic454—81no—
46D908—21no—
47C22⋊D227—42no—
48C22⋊D227—41no—
49C2×Dic227—22no—
50C2×Dic227—22no—
51C2×Dic227—22no—
52C2×Dic227—42no—
53C4×D227—41no—
54C227⋊C8—41no—
55C23×C227C2×D412yes—
56Q8×C227—41no—
57D4×C227—41no—
58C2×C4×C227—21no—
59C2×C4×C227—21no—
60C2×C4×C227—42no—
61C2×C4×C227C2×D411yescharacteristic
62C22×C227—22no—
63C22×C227—42no—
64C22×C227#12011yescharacteristic
65C2×D227—41no—
66C2×D227—82no—
67C4×C227—21no—
68C4×C227—41no—
69C4×C227—41no—
70C4×C227—82no—
71Dic227—42no—
72Dic227—41no—
73Dic227—42no—
74C2×C227—21no—
75C2×C227—42no—
76C2×C227B(4,𝔽2)11yescharacteristic
77D227—81no—
78C227#220111yescharacteristic, complemented, Sylow
79#2201—2271nomaximal, Sylow
80#563—2271no—
81#458—2272no—
82#455—2271no—
83#354D22711yescharacteristic, complemented
84#352—2272no—
85H(5,𝔽2)—2271no—
86#137—4541no—
87C4⋊Q8—4541no—
88#126—4542no—
89#125—4541no—
90#124C2×D22711yescharacteristic
91#123—4542no—
92#123—2272no—
93C22⋊Q8—4542no—
94#104—4541no—
95#100—2271no—
96#99—4542no—
97#99C2×D22712yes—
98#41—9081no—
99#41—4541no—
100C2×Q8—4541no—
101C2×Q8D90811yescharacteristic
102C2×D4—9081no—
103C2×D4—4542no—
104C2×D4C22×D22711yescharacteristic
105C22×C4—2272no—
106Q16—9081no—
107SD16—9081no—
108M16—4541no—
109C4⋊C4—9082no—
110C4⋊C4—4542no—
111C4⋊C4—4542no—
112C4⋊C4—4541no—
113C4⋊C4—4541no—
114C4⋊C4—4542no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—9082no—
118C22⋊C4—4542no—
119C22⋊C4—2272no—
120C42—4541no—
121C42D90811yescharacteristic
122C23—4542no—
123C23D4×D22712yes—
124Q8—41no—
125Q8—18161no—
126Q8—4541no—
127D4—41no—
128D4—9082no—
129D4—9081no—
130D4—4541no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—9081no—
135C2×C4—9082no—
136C2×C4—4542no—
137C2×C4—4542no—
138C2×C4—4542no—
139C2×C4C2×D90811yescharacteristic
140C8—9081no—
141C22—22no—
142C22—42no—
143C22—18162no—
144C22—9081no—
145C22#2030025811yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—9082no—
151C4—9081no—
152C4—9082no—
153C2—21no—
154C2—42no—
155C2—18161no—
156C2#2840922011yescharacteristic
subgroupG/H
centerC2#28409220
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
Fitting#354×C227C2
socleC2×C227B(4,𝔽2)
prime radical#20300333C22
hypercenter#354D227
hypocenterC227#2201
LCSG ▷ C2×D4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #28409782C2semidirect
4. #28408821C2semidirect
5. #354×C227C2semidirect
78. C227#2201semidirect
83. #354D227semidirect

Maximal subgroups

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

subgroupindex
#284097822
#284092232
#284092092
#284088212
#354×C2272
#2201227