Finite Group Databasebeta

← #44401730

Subgroups — #44401730

Order 29056.

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

normal subgroups only

This group has 246 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#28409221C212yesmaximal, complemented
2#28409199C211yesmaximal, characteristic
3#28408825C211yesmaximal, characteristic, complemented
4#28408822C212yesmaximal, complemented
5#559×C227C211yesmaximal, characteristic, complemented
6#20300333C2211yescharacteristic
7#20300314—21no—
8#20300271—22no—
9#20300216—21no—
10#20300205C2211yescharacteristic
11#20300203C2212yes—
12#20300202—22no—
13#127×C227C2211yescharacteristic
14#124×C227—22no—
15#123×C227—21no—
16#123×C227C2212yes—
17C4×Q8×C227—21no—
18C4×D4×C227—22no—
19#16813884—21no—
20#16813883—41no—
21C2×C22⋊D227—22no—
22D4×D227—41no—
23#16813872—41no—
24#16813869—41no—
25#16813868—21no—
26Dic227⋊C4—42no—
27C22⋊Dic227—22no—
28(C2×D227)⋊C4—42no—
29C2×Q8×C227—21no—
30C2×D4×C227—22no—
31C2×D4×C227D411yescharacteristic
32C2×D4×C227C2311yescharacteristic
33C22×C4×C227—21no—
34C22×C4×C227D412yes—
35C4⋊C4×C227—21no—
36C4⋊C4×C227—22no—
37C22⋊C4×C227—22no—
38C22⋊C4×C227—21no—
39C22⋊C4×C227—22no—
40C22⋊C4×C227D412yes—
41C42×C227—22no—
42C42×C227D411yescharacteristic
43C22×D227—22no—
44Dic454—21no—
45D908—21no—
46C22⋊D227—42no—
47C22⋊D227—41no—
48C2×Dic227—22no—
49C2×Dic227—42no—
50C4×D227—41no—
51C227⋊C8—41no—
52C23×C227—21no—
53C23×C227C2×D412yes—
54Q8×C227—42no—
55D4×C227—41no—
56D4×C227—44no—
57D4×C227—41no—
58C2×C4×C227—21no—
59C2×C4×C227—22no—
60C2×C4×C227—22no—
61C2×C4×C227—42no—
62C2×C4×C227—81no—
63C2×C4×C227C2×D411yescharacteristic
64C22×C227—22no—
65C22×C227—42no—
66C22×C227—42no—
67C22×C227—41no—
68C22×C227#12011yescharacteristic
69C2×D227—41no—
70C2×D227—82no—
71C4×C227—22no—
72C4×C227—21no—
73C4×C227—42no—
74C4×C227—42no—
75Dic227—41no—
76Dic227—82no—
77C2×C227—21no—
78C2×C227—42no—
79C2×C227—81no—
80C2×C227B(4,𝔽2)11yescharacteristic
81D227—81no—
82C227#219811yescharacteristic, complemented, Sylow
83#2198—2271nomaximal, Sylow
84#559D22711yescharacteristic, complemented
85#458—2272no—
86#453—2271no—
87#353—2271no—
88#352—2272no—
89H(5,𝔽2)—2271no—
90AGL(1,ℤ/8)—4541no—
91#127C2×D22711yescharacteristic
92#124—22no—
93#123—21no—
94#123—4542no—
95#123C2×D22712yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—4541no—
99#100—2271no—
100#99—4542no—
101#99—2272no—
102#41—9081no—
103#41—4541no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—9081no—
107C2×D4—4542no—
108C2×D4C22⋊D22711yescharacteristic
109C2×D4C22×D22711yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D22712yes—
112SD16—9081no—
113D8—9081no—
114M16—4541no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—9082no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—9082no—
122C22⋊C4—4542no—
123C22⋊C4C22⋊D22712yes—
124C42—22no—
125C42C22⋊D22711yescharacteristic
126C23—21no—
127C23—4542no—
128C23C2×C22⋊D22712yes—
129Q8—42no—
130Q8—4541no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—9082no—
135D4—9081no—
136D4—4541no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—9081no—
143C2×C4—9082no—
144C2×C4—4542no—
145C2×C4C2×C22⋊D22711yescharacteristic
146C8—9081no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—18162no—
152C22—9081no—
153C22#2030025711yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—18162no—
159C4—9081no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—18161no—
164C2#2840921911yescharacteristic
subgroupG/H
centerC2#28409219
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
Fitting#559×C227C2
socleC2×C227B(4,𝔽2)
prime radical#28409199C2
hypercenter#559D227
hypocenterC227#2198
LCSG ▷ C2×D4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #28409221C2semidirect
3. #28408825C2semidirect
4. #28408822C2semidirect
5. #559×C227C2semidirect
82. C227#2198semidirect
84. #559D227semidirect

Maximal subgroups

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

subgroupindex
#284092212
#284091992
#284088252
#284088222
#559×C2272
#2198227