Finite Group Databasebeta

← #38770851

Subgroups — #38770851

Order 25472.

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

normal subgroups only

This group has 270 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#27164204C212yesmaximal, complemented
2#27164195C211yesmaximal, characteristic, complemented
3#27163813C211yesmaximal, characteristic, complemented
4#27163805C212yesmaximal, complemented
5H(3,𝔽4)×C199C211yesmaximal, characteristic, complemented
6#20264000C2211yescharacteristic
7#20263986—21no—
8#20263924—22no—
9#20263883—21no—
10#20263872C2211yescharacteristic
11#20263870C2212yes—
12#20263869—22no—
13#124×C199—21no—
14#124×C199—22no—
15#124×C199—21no—
16#124×C199C2211yescharacteristic
17#120×C199—22no—
18#120×C199C2212yes—
19#15691886—21no—
20#15691885—41no—
21C2×C22⋊D199—22no—
22C2×C22⋊D199—42no—
23D4×D199—41no—
24#15691876—41no—
25#15691873—41no—
26#15691870—21no—
27C22⋊Dic199—22no—
28C22⋊Dic199—42no—
29C24×C199D412yescomplemented
30C2×Q8×C199—21no—
31C2×Q8×C199D411yescharacteristic, complemented
32C2×D4×C199—21no—
33C2×D4×C199—21no—
34C2×D4×C199—22no—
35C2×D4×C199C2311yescharacteristic
36C22⋊C4×C199—22no—
37C22⋊C4×C199—22no—
38C22⋊C4×C199—22no—
39C22⋊C4×C199—22no—
40C22⋊C4×C199D412yescomplemented
41C42×C199—21no—
42C42×C199D411yescharacteristic, complemented
43C22×D199—22no—
44Dic398—21no—
45D796—21no—
46C22⋊D199—42no—
47C22⋊D199—41no—
48C22⋊D199—84no—
49C2×Dic199—22no—
50C2×Dic199—42no—
51C4×D199—41no—
52C199⋊C8—41no—
53C23×C199—22no—
54C23×C199—42no—
55C23×C199—82no—
56C23×C199C2×D412yes—
57Q8×C199—41no—
58Q8×C199—81no—
59D4×C199—41no—
60D4×C199—82no—
61D4×C199—81no—
62D4×C199—81no—
63C2×C4×C199—21no—
64C2×C4×C199—22no—
65C2×C4×C199—21no—
66C2×C4×C199C2×D411yescharacteristic
67C22×C199—22no—
68C22×C199—42no—
69C22×C199—42no—
70C22×C199—42no—
71C22×C199—44no—
72C22×C199—42no—
73C22×C199—82no—
74C22×C199#12011yescharacteristic
75C2×D199—41no—
76C2×D199—82no—
77C4×C199—21no—
78C4×C199—42no—
79C4×C199—41no—
80C4×C199—41no—
81Dic199—41no—
82Dic199—82no—
83C2×C199—21no—
84C2×C199—42no—
85C2×C199—82no—
86C2×C199B(4,𝔽2)11yescharacteristic
87D199—81no—
88C199#219911yescharacteristic, complemented, Sylow
89#2199—1991nomaximal, Sylow
90H(3,𝔽4)D19911yescharacteristic, complemented
91B(4,𝔽2)—1992no—
92#455—1991no—
93#354—1991no—
94AΣL(1,𝔽16)—1992no—
95H(5,𝔽2)—1991no—
96#137—3981no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D19911yescharacteristic
101#120—22no—
102#120—3982no—
103#120C2×D19912yes—
104#104—3981no—
105#100—1991no—
106#99—3982no—
107#99—1992no—
108C24C22⋊D19912yescomplemented
109#41—7961no—
110#41—3981no—
111C2×Q8—21no—
112C2×Q8C22⋊D19911yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—7961no—
117C2×D4—7962no—
118C2×D4—3982no—
119C2×D4C22×D19911yescharacteristic
120Q16—7961no—
121SD16—7961no—
122M16—3981no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—7962no—
128C22⋊C4—3982no—
129C22⋊C4C22⋊D19912yescomplemented
130C42—21no—
131C42C22⋊D19911yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—3982no—
136C23C2×C22⋊D19912yes—
137Q8—41no—
138Q8—81no—
139Q8—3981no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—7962no—
145D4—7961no—
146D4—15924no—
147D4—3981no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—7961no—
152C2×C4—7962no—
153C2×C4—3982no—
154C2×C4C2×C22⋊D19911yescharacteristic
155C8—7961no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—7961no—
164C22—15922no—
165C22#2026392411yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—7961no—
171C4—15922no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—15921no—
176C2#2716420511yescharacteristic
subgroupG/H
centerC2#27164205
commutator, cosocleC2×D4×C199C23
FrattiniC2×D4C22×D199
FittingH(3,𝔽4)×C199C2
socleC2×C199B(4,𝔽2)
hypercenterH(3,𝔽4)D199
hypocenterC199#2199
LCSG ▷ C2×D4×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #27164204C2semidirect
2. #27164195C2semidirect
3. #27163813C2semidirect
4. #27163805C2semidirect
5. H(3,𝔽4)×C199C2semidirect
29. C24×C199D4semidirect
31. C2×Q8×C199D4semidirect
40. C22⋊C4×C199D4semidirect
42. C42×C199D4semidirect
88. C199#2199semidirect
90. H(3,𝔽4)D199semidirect
108. C24C22⋊D199semidirect
112. C2×Q8C22⋊D199semidirect
129. C22⋊C4C22⋊D199semidirect
131. C42C22⋊D199semidirect

Maximal subgroups

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

subgroupindex
#271642042
#271641952
#271638132
#271638052
H(3,𝔽4)×C1992
#2199199