Finite Group Databasebeta

← #38744713

Subgroups — #38744713

Order 25216.

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#27161265C212yesmaximal, complemented
2#27161256C211yesmaximal, characteristic, complemented
3#27160874C211yesmaximal, characteristic, complemented
4#27160866C212yesmaximal, complemented
5H(3,𝔽4)×C197C211yesmaximal, characteristic, complemented
6#20263371C2211yescharacteristic
7#20263357—21no—
8#20263295—22no—
9#20263254—21no—
10#20263243C2211yescharacteristic
11#20263241C2212yes—
12#20263240—22no—
13#124×C197—21no—
14#124×C197—22no—
15#124×C197—21no—
16#124×C197C2211yescharacteristic
17#120×C197—22no—
18#120×C197C2212yes—
19#15691358—21no—
20#15691357—41no—
21C2×C22⋊D197—22no—
22C2×C22⋊D197—42no—
23D4×D197—41no—
24#15691348—41no—
25#15691345—41no—
26#15691342—21no—
27C22⋊Dic197—22no—
28C22⋊Dic197—42no—
29C24×C197D412yescomplemented
30C2×Q8×C197—21no—
31C2×Q8×C197D411yescharacteristic, complemented
32C2×D4×C197—21no—
33C2×D4×C197—21no—
34C2×D4×C197—22no—
35C2×D4×C197C2311yescharacteristic
36C22⋊C4×C197—22no—
37C22⋊C4×C197—22no—
38C22⋊C4×C197—22no—
39C22⋊C4×C197—22no—
40C22⋊C4×C197D412yescomplemented
41C42×C197—21no—
42C42×C197D411yescharacteristic, complemented
43C22×D197—22no—
44Dic394—21no—
45D788—21no—
46C22⋊D197—42no—
47C22⋊D197—41no—
48C22⋊D197—84no—
49C2×Dic197—22no—
50C2×Dic197—42no—
51C4×D197—41no—
52#13096802—41no—
53C23×C197—22no—
54C23×C197—42no—
55C23×C197—82no—
56C23×C197C2×D412yes—
57Q8×C197—41no—
58Q8×C197—81no—
59D4×C197—41no—
60D4×C197—82no—
61D4×C197—81no—
62D4×C197—81no—
63C2×C4×C197—21no—
64C2×C4×C197—22no—
65C2×C4×C197—21no—
66C2×C4×C197C2×D411yescharacteristic
67C22×C197—22no—
68C22×C197—42no—
69C22×C197—42no—
70C22×C197—42no—
71C22×C197—44no—
72C22×C197—42no—
73C22×C197—82no—
74C22×C197#12011yescharacteristic
75C2×D197—41no—
76C2×D197—82no—
77C4×C197—21no—
78C4×C197—42no—
79C4×C197—41no—
80C4×C197—41no—
81Dic197—41no—
82Dic197—82no—
83C2×C197—21no—
84C2×C197—42no—
85C2×C197—82no—
86C2×C197B(4,𝔽2)11yescharacteristic
87D197—81no—
88C197#219911yescharacteristic, complemented, Sylow
89#2199—1971nomaximal, Sylow
90H(3,𝔽4)D19711yescharacteristic, complemented
91B(4,𝔽2)—1972no—
92#455—1971no—
93#354—1971no—
94AΣL(1,𝔽16)—1972no—
95H(5,𝔽2)—1971no—
96#137—3941no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D19711yescharacteristic
101#120—22no—
102#120—3942no—
103#120C2×D19712yes—
104#104—3941no—
105#100—1971no—
106#99—3942no—
107#99—1972no—
108C24C22⋊D19712yescomplemented
109#41—7881no—
110#41—3941no—
111C2×Q8—21no—
112C2×Q8C22⋊D19711yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—7881no—
117C2×D4—7882no—
118C2×D4—3942no—
119C2×D4C22×D19711yescharacteristic
120Q16—7881no—
121SD16—7881no—
122M16—3941no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—7882no—
128C22⋊C4—3942no—
129C22⋊C4C22⋊D19712yescomplemented
130C42—21no—
131C42C22⋊D19711yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—3942no—
136C23C2×C22⋊D19712yes—
137Q8—41no—
138Q8—81no—
139Q8—3941no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—7882no—
145D4—7881no—
146D4—15764no—
147D4—3941no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—7881no—
152C2×C4—7882no—
153C2×C4—3942no—
154C2×C4C2×C22⋊D19711yescharacteristic
155C8—7881no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—7881no—
164C22—15762no—
165C22#2026329511yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—7881no—
171C4—15762no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—15761no—
176C2#2716126611yescharacteristic
subgroupG/H
centerC2#27161266
commutator, cosocleC2×D4×C197C23
FrattiniC2×D4C22×D197
FittingH(3,𝔽4)×C197C2
socleC2×C197B(4,𝔽2)
hypercenterH(3,𝔽4)D197
hypocenterC197#2199
LCSG ▷ C2×D4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #27161265C2semidirect
2. #27161256C2semidirect
3. #27160874C2semidirect
4. #27160866C2semidirect
5. H(3,𝔽4)×C197C2semidirect
29. C24×C197D4semidirect
31. C2×Q8×C197D4semidirect
40. C22⋊C4×C197D4semidirect
42. C42×C197D4semidirect
88. C197#2199semidirect
90. H(3,𝔽4)D197semidirect
108. C24C22⋊D197semidirect
112. C2×Q8C22⋊D197semidirect
129. C22⋊C4C22⋊D197semidirect
131. C42C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271612652
#271612562
#271608742
#271608662
H(3,𝔽4)×C1972
#2199197