Finite Group Databasebeta

← #38744710

Subgroups — #38744710

Order 25216.

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

normal subgroups only

This group has 246 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#27161815C211yesmaximal, characteristic, complemented
2#27161270C212yesmaximal, complemented
3#27161246C211yesmaximal, characteristic
4#27160868C212yesmaximal, complemented
5#353×C197C211yesmaximal, characteristic, complemented
6#20263371C2211yescharacteristic
7#20263352—21no—
8#20263313—22no—
9#20263310—22no—
10#20263307—21no—
11#20263307C2212yes—
12C4×Dic394—21no—
13D4×Dic197—22no—
14#20263254—21no—
15#20263243C2211yescharacteristic
16#20263240—22no—
17#127×C197C2211yescharacteristic
18#99×C197C2212yes—
19#15691358—21no—
20#15691357—41no—
21C2×Dic394—21no—
22C2×C22⋊D197—22no—
23D4×D197—41no—
24C22×Dic197—21no—
25C22×Dic197D412yes—
26#15691346—41no—
27#15691343—41no—
28#15691342—21no—
29C4⋊Dic197—21no—
30C4⋊Dic197—42no—
31#15691337—22no—
32C22⋊Dic197—22no—
33C22⋊Dic197—21no—
34C22⋊Dic197—22no—
35C22⋊Dic197D412yes—
36#15691335—42no—
37C4×Dic197—22no—
38C2×D4×C197—22no—
39C2×D4×C197D411yescharacteristic
40C2×D4×C197C2311yescharacteristic
41C22⋊C4×C197—22no—
42C42×C197D411yescharacteristic
43C22×D197—22no—
44Dic394—21no—
45Dic394—42no—
46D788—21no—
47C22⋊D197—42no—
48C22⋊D197—41no—
49C2×Dic197—22no—
50C2×Dic197—22no—
51C2×Dic197—22no—
52C2×Dic197—42no—
53C2×Dic197—81no—
54C4×D197—41no—
55#13096802—41no—
56C23×C197—21no—
57C23×C197C2×D412yes—
58D4×C197—41no—
59D4×C197—41no—
60D4×C197—44no—
61C2×C4×C197—21no—
62C2×C4×C197—42no—
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—82no—
74Dic197—42no—
75Dic197—41no—
76Dic197—42no—
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#559—1971no—
85#458—1972no—
86#453—1971no—
87#353D19711yescharacteristic, complemented
88#352—1972no—
89H(5,𝔽2)—1971no—
90AGL(1,ℤ/8)—3941no—
91#127C2×D19711yescharacteristic
92#124—3942no—
93#123—3942no—
94#123—3941no—
95#123—1972no—
96C4×Q8—3941no—
97C4×D4—3942no—
98#104—3941no—
99#100—1971no—
100#99—3942no—
101#99C2×D19712yes—
102#41—7881no—
103#41—3941no—
104C2×Q8—3941no—
105C2×D4—22no—
106C2×D4—7881no—
107C2×D4—3942no—
108C2×D4D78811yescharacteristic
109C2×D4C22×D19711yescharacteristic
110C22×C4—3941no—
111C22×C4—1972no—
112SD16—7881no—
113D8—7881no—
114M16—3941no—
115C4⋊C4—7882no—
116C4⋊C4—3941no—
117C4⋊C4—3942no—
118C22⋊C4—22no—
119C22⋊C4—7882no—
120C22⋊C4—3942no—
121C22⋊C4—3941no—
122C22⋊C4—3942no—
123C22⋊C4—1972no—
124C42—3942no—
125C42D78811yescharacteristic
126C23—21no—
127C23—3942no—
128C23D4×D19712yes—
129Q8—7882no—
130Q8—3941no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—7882no—
135D4—7881no—
136D4—3941no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—7881no—
140C2×C4—7882no—
141C2×C4—15761no—
142C2×C4—3942no—
143C2×C4—3942no—
144C2×C4—3942no—
145C2×C4C2×D78811yescharacteristic
146C8—7881no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—7881no—
152C22—15762no—
153C22#2026329611yescharacteristic
154C4—22no—
155C4—21no—
156C4—82no—
157C4—7882no—
158C4—7882no—
159C4—7881no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—15761no—
164C2#2716126711yescharacteristic
subgroupG/H
centerC2#27161267
commutator, cosocleC2×D4×C197C23
FrattiniC2×D4C22×D197
Fitting#353×C197C2
socleC2×C197B(4,𝔽2)
prime radical#27161246C2
hypercenter#353D197
hypocenterC197#2198
LCSG ▷ C2×D4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #27161815C2semidirect
2. #27161270C2semidirect
4. #27160868C2semidirect
5. #353×C197C2semidirect
82. C197#2198semidirect
87. #353D197semidirect

Maximal subgroups

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

subgroupindex
#271618152
#271612702
#271612462
#271608682
#353×C1972
#2198197