Finite Group Databasebeta

← #44401737

Subgroups — #44401737

Order 29056.

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

normal subgroups only

This group has 270 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#28409774C211yesmaximal, characteristic, complemented
2#28409220C212yesmaximal, complemented
3#28409209C211yesmaximal, characteristic, complemented
4#28408818C212yesmaximal, complemented
5#354×C227C211yesmaximal, characteristic, complemented
6#20300333C2211yescharacteristic
7#20300319—21no—
8#20300277—21no—
9#20300276—21no—
10#20300274—22no—
11#20300258—22no—
12#20300258—22no—
13#20300256C2212yes—
14#20300216—21no—
15#20300205C2211yescharacteristic
16#20300202—22no—
17#124×C227C2211yescharacteristic
18#99×C227C2212yes—
19C23×D227D412yescomplemented
20#16813884—21no—
21#16813883—41no—
22C2×Dic454—21no—
23C2×D908—21no—
24C2×D908—21no—
25C2×D908—42no—
26C2×C22⋊D227—22no—
27C2×C22⋊D227—22no—
28D4×D227—41no—
29#16813874—41no—
30#16813871—41no—
31#16813868—21no—
32C22⋊Dic227D412yescomplemented
33(C2×D227)⋊C4—22no—
34(C2×D227)⋊C4—22no—
35(C2×D227)⋊C4—22no—
36(C2×D227)⋊C4—42no—
37C4×Dic227—21no—
38C2×Q8×C227D411yescharacteristic, complemented
39C2×D4×C227C2311yescharacteristic
40C22⋊C4×C227—22no—
41C22⋊C4×C227—22no—
42C42×C227D411yescharacteristic, complemented
43C22×D227—22no—
44C22×D227—22no—
45C22×D227—42no—
46C22×D227—82no—
47Dic454—21no—
48Dic454—81no—
49D908—21no—
50D908—81no—
51D908—81no—
52D908—84no—
53C22⋊D227—42no—
54C22⋊D227—41no—
55C22⋊D227—82no—
56C2×Dic227—22no—
57C2×Dic227—22no—
58C4×D227—41no—
59C227⋊C8—41no—
60C23×C227C2×D412yes—
61Q8×C227—41no—
62D4×C227—41no—
63C2×C4×C227—21no—
64C2×C4×C227—21no—
65C2×C4×C227—42no—
66C2×C4×C227C2×D411yescharacteristic
67C22×C227—22no—
68C22×C227—42no—
69C22×C227#12011yescharacteristic
70C2×D227—42no—
71C2×D227—42no—
72C2×D227—44no—
73C2×D227—41no—
74C2×D227—42no—
75C2×D227—82no—
76C2×D227—82no—
77C4×C227—21no—
78C4×C227—41no—
79C4×C227—41no—
80C4×C227—82no—
81Dic227—42no—
82Dic227—41no—
83C2×C227—21no—
84C2×C227—42no—
85C2×C227B(4,𝔽2)11yescharacteristic
86D227—82no—
87D227—81no—
88C227#219911yescharacteristic, complemented, Sylow
89#2199—2271nomaximal, Sylow
90H(3,𝔽4)—2271no—
91B(4,𝔽2)—2272no—
92#455—2271no—
93#354D22711yescharacteristic, complemented
94AΣL(1,𝔽16)—2272no—
95H(5,𝔽2)—2271no—
96#137—4541no—
97#124—4542no—
98#124—4541no—
99#124—4541no—
100#124C2×D22711yescharacteristic
101#120—4542no—
102#120—4542no—
103#120—2272no—
104#104—4541no—
105#100—2271no—
106#99—4542no—
107#99C2×D22712yes—
108C24—2272no—
109#41—9081no—
110#41—4541no—
111C2×Q8—4541no—
112C2×Q8D90811yescharacteristic, complemented
113C2×D4—9082no—
114C2×D4—9081no—
115C2×D4—4541no—
116C2×D4—4541no—
117C2×D4—4542no—
118C2×D4—4542no—
119C2×D4C22×D22711yescharacteristic
120Q16—9081no—
121SD16—9081no—
122M16—4541no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—9082no—
126C22⋊C4—4542no—
127C22⋊C4—4542no—
128C22⋊C4—4542no—
129C22⋊C4—2272no—
130C42—4541no—
131C42D90811yescharacteristic, complemented
132C23—18162no—
133C23—9082no—
134C23—4542no—
135C23—4542no—
136C23D4×D22712yes—
137Q8—41no—
138Q8—18161no—
139Q8—4541no—
140D4—41no—
141D4—18161no—
142D4—18161no—
143D4—18164no—
144D4—18162no—
145D4—9082no—
146D4—9081no—
147D4—4541no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—9081no—
152C2×C4—4542no—
153C2×C4—4542no—
154C2×C4C2×D90811yescharacteristic
155C8—9081no—
156C22—22no—
157C22—42no—
158C22—18162no—
159C22—18162no—
160C22—9082no—
161C22—9082no—
162C22—9084no—
163C22—9081no—
164C22—9082no—
165C22#2030025811yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—82no—
170C4—9082no—
171C4—9081no—
172C2—21no—
173C2—42no—
174C2—18161no—
175C2—18162no—
176C2#2840922011yescharacteristic
subgroupG/H
centerC2#28409220
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
Fitting#354×C227C2
socleC2×C227B(4,𝔽2)
hypercenter#354D227
hypocenterC227#2199
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. #28409774C2semidirect
2. #28409220C2semidirect
3. #28409209C2semidirect
4. #28408818C2semidirect
5. #354×C227C2semidirect
19. C23×D227D4semidirect
32. C22⋊Dic227D4semidirect
38. C2×Q8×C227D4semidirect
42. C42×C227D4semidirect
88. C227#2199semidirect
93. #354D227semidirect
112. C2×Q8D908semidirect
131. C42D908semidirect

Maximal subgroups

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

subgroupindex
#284097742
#284092202
#284092092
#284088182
#354×C2272
#2199227