Finite Group Databasebeta

← #38744708

Subgroups — #38744708

Order 25216.

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#27161268C212yesmaximal, complemented
2#27161246C211yesmaximal, characteristic
3#27160872C211yesmaximal, characteristic, complemented
4#27160869C212yesmaximal, complemented
5#559×C197C211yesmaximal, characteristic, complemented
6#20263371C2211yescharacteristic
7#20263352—21no—
8#20263309—22no—
9#20263254—21no—
10#20263243C2211yescharacteristic
11#20263241C2212yes—
12#20263240—22no—
13#127×C197C2211yescharacteristic
14#124×C197—22no—
15#123×C197—21no—
16#123×C197C2212yes—
17C4×Q8×C197—21no—
18C4×D4×C197—22no—
19#15691358—21no—
20#15691357—41no—
21C2×C22⋊D197—22no—
22D4×D197—41no—
23#15691346—41no—
24#15691343—41no—
25#15691342—21no—
26#15691337—42no—
27C22⋊Dic197—22no—
28#15691335—42no—
29C2×Q8×C197—21no—
30C2×D4×C197—22no—
31C2×D4×C197D411yescharacteristic
32C2×D4×C197C2311yescharacteristic
33C22×C4×C197—21no—
34C22×C4×C197D412yes—
35C4⋊C4×C197—21no—
36C4⋊C4×C197—22no—
37C22⋊C4×C197—22no—
38C22⋊C4×C197—21no—
39C22⋊C4×C197—22no—
40C22⋊C4×C197D412yes—
41C42×C197—22no—
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×C197—21no—
53C23×C197C2×D412yes—
54Q8×C197—42no—
55D4×C197—41no—
56D4×C197—41no—
57D4×C197—44no—
58C2×C4×C197—21no—
59C2×C4×C197—22no—
60C2×C4×C197—22no—
61C2×C4×C197—42no—
62C2×C4×C197—81no—
63C2×C4×C197C2×D411yescharacteristic
64C22×C197—22no—
65C22×C197—42no—
66C22×C197—42no—
67C22×C197—41no—
68C22×C197#12011yescharacteristic
69C2×D197—41no—
70C2×D197—82no—
71C4×C197—22no—
72C4×C197—21no—
73C4×C197—42no—
74C4×C197—42no—
75Dic197—41no—
76Dic197—82no—
77C2×C197—21no—
78C2×C197—42no—
79C2×C197—81no—
80C2×C197B(4,𝔽2)11yescharacteristic
81D197—81no—
82C197#219811yescharacteristic, complemented, Sylow
83#2198—1971nomaximal, Sylow
84#559D19711yescharacteristic, complemented
85#458—1972no—
86#453—1971no—
87#353—1971no—
88#352—1972no—
89H(5,𝔽2)—1971no—
90AGL(1,ℤ/8)—3941no—
91#127C2×D19711yescharacteristic
92#124—22no—
93#123—21no—
94#123—3942no—
95#123C2×D19712yes—
96C4×Q8—21no—
97C4×D4—22no—
98#104—3941no—
99#100—1971no—
100#99—3942no—
101#99—1972no—
102#41—7881no—
103#41—3941no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—7881no—
107C2×D4—3942no—
108C2×D4C22⋊D19711yescharacteristic
109C2×D4C22×D19711yescharacteristic
110C22×C4—21no—
111C22×C4C22⋊D19712yes—
112SD16—7881no—
113D8—7881no—
114M16—3941no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—7882no—
118C22⋊C4—22no—
119C22⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—7882no—
122C22⋊C4—3942no—
123C22⋊C4C22⋊D19712yes—
124C42—22no—
125C42C22⋊D19711yescharacteristic
126C23—21no—
127C23—3942no—
128C23C2×C22⋊D19712yes—
129Q8—42no—
130Q8—3941no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—7882no—
135D4—7881no—
136D4—3941no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—81no—
142C2×C4—7881no—
143C2×C4—7882no—
144C2×C4—3942no—
145C2×C4C2×C22⋊D19711yescharacteristic
146C8—7881no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—7881no—
152C22—15762no—
153C22#2026329511yescharacteristic
154C4—22no—
155C4—21no—
156C4—42no—
157C4—42no—
158C4—7881no—
159C4—15762no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—15761no—
164C2#2716126611yescharacteristic
subgroupG/H
centerC2#27161266
commutator, cosocleC2×D4×C197C23
FrattiniC2×D4C22×D197
Fitting#559×C197C2
socleC2×C197B(4,𝔽2)
prime radical#27161246C2
hypercenter#559D197
hypocenterC197#2198
LCSG ▷ C2×D4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #27161268C2semidirect
3. #27160872C2semidirect
4. #27160869C2semidirect
5. #559×C197C2semidirect
82. C197#2198semidirect
84. #559D197semidirect

Maximal subgroups

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

subgroupindex
#271612682
#271612462
#271608722
#271608692
#559×C1972
#2198197