Finite Group Databasebeta

← #38744725

Subgroups — #38744725

Order 25216.

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

normal subgroups only

This group has 226 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#27161268C212yesmaximal
2#27161256C211yesmaximal, characteristic
3#27160874C211yesmaximal, characteristic, complemented
4#27160869C212yesmaximal, complemented
5#563×C197C211yesmaximal, characteristic, complemented
6#20263371C2211yescharacteristic
7#20263357—21no—
8#20263309—22no—
9#20263254—21no—
10#20263243C2211yescharacteristic
11#20263241C2212yes—
12#20263240—22no—
13C4⋊Q8×C197—21no—
14#126×C197—22no—
15#125×C197—21no—
16#124×C197C2211yescharacteristic
17#123×C197C2212yes—
18C22⋊Q8×C197—22no—
19#15691358—21no—
20#15691357—41no—
21C2×C22⋊D197—22no—
22D4×D197—41no—
23#15691348—41no—
24#15691345—41no—
25#15691342—21no—
26#15691337—42no—
27C22⋊Dic197—22no—
28#15691335—42no—
29C2×Q8×C197—21no—
30C2×Q8×C197D411yescharacteristic
31C2×D4×C197C2311yescharacteristic
32C22×C4×C197D412yes—
33C4⋊C4×C197—21no—
34C4⋊C4×C197—22no—
35C4⋊C4×C197—22no—
36C4⋊C4×C197—21no—
37C4⋊C4×C197—22no—
38C22⋊C4×C197—22no—
39C22⋊C4×C197—22no—
40C22⋊C4×C197D412yes—
41C42×C197—21no—
42C42×C197D411yescharacteristic
43C22×D197—22no—
44Dic394—21no—
45D788—21no—
46C22⋊D197—42no—
47C22⋊D197—41no—
48C2×Dic197—22no—
49C2×Dic197—42no—
50C4×D197—41no—
51#13096802—41no—
52C23×C197C2×D412yes—
53Q8×C197—41no—
54Q8×C197—81no—
55D4×C197—41no—
56C2×C4×C197—21no—
57C2×C4×C197—22no—
58C2×C4×C197—21no—
59C2×C4×C197—22no—
60C2×C4×C197—42no—
61C2×C4×C197C2×D411yescharacteristic
62C22×C197—22no—
63C22×C197—42no—
64C22×C197#12011yescharacteristic
65C2×D197—41no—
66C2×D197—82no—
67C4×C197—21no—
68C4×C197—42no—
69C4×C197—41no—
70C4×C197—42no—
71C4×C197—41no—
72Dic197—41no—
73Dic197—82no—
74C2×C197—21no—
75C2×C197—42no—
76C2×C197B(4,𝔽2)11yescharacteristic
77D197—81no—
78C197#220111yescharacteristic, complemented, Sylow
79#2201—1971nomaximal, Sylow
80#563D19711yescharacteristic, complemented
81#458—1972no—
82#455—1971no—
83#354—1971no—
84#352—1972no—
85H(5,𝔽2)—1971no—
86#137—3941no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D19711yescharacteristic
91#123—3942no—
92#123C2×D19712yes—
93C22⋊Q8—22no—
94#104—3941no—
95#100—1971no—
96#99—3942no—
97#99—1972no—
98#41—7881no—
99#41—3941no—
100C2×Q8—21no—
101C2×Q8C22⋊D19711yescharacteristic
102C2×D4—7881no—
103C2×D4—3942no—
104C2×D4C22×D19711yescharacteristic
105C22×C4C22⋊D19712yes—
106Q16—7881no—
107SD16—7881no—
108M16—3941no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—7882no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—7882no—
118C22⋊C4—3942no—
119C22⋊C4C22⋊D19712yes—
120C42—21no—
121C42C22⋊D19711yescharacteristic
122C23—3942no—
123C23C2×C22⋊D19712yes—
124Q8—41no—
125Q8—81no—
126Q8—3941no—
127D4—41no—
128D4—7882no—
129D4—7881no—
130D4—3941no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—7881no—
137C2×C4—7882no—
138C2×C4—3942no—
139C2×C4C2×C22⋊D19711yescharacteristic
140C8—7881no—
141C22—22no—
142C22—42no—
143C22—7881no—
144C22—15762no—
145C22#2026329511yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—7881no—
152C4—15762no—
153C2—21no—
154C2—42no—
155C2—15761no—
156C2#2716126611yescharacteristic
subgroupG/H
centerC2#27161266
commutator, cosocleC2×D4×C197C23
FrattiniC2×D4C22×D197
Fitting#563×C197C2
socleC2×C197B(4,𝔽2)
prime radical#20263371C22
hypercenter#563D197
hypocenterC197#2201
LCSG ▷ C2×D4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #27160874C2semidirect
4. #27160869C2semidirect
5. #563×C197C2semidirect
78. C197#2201semidirect
80. #563D197semidirect

Maximal subgroups

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

subgroupindex
#271612682
#271612562
#271608742
#271608692
#563×C1972
#2201197