Finite Group Databasebeta

← #38744707

Subgroups — #38744707

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#27161269C212yesmaximal, complemented
3#27160872C211yesmaximal, characteristic, complemented
4#27160870C212yesmaximal, complemented
5#453×C197C211yesmaximal, characteristic
6#20263313—22no—
7#20263307—22no—
8#20263307—21no—
9#20263307C2212yes—
10C4×Dic394—21no—
11D4×Dic197—22no—
12#20263241—22no—
13#20263241C2212yes—
14H(5,𝔽2)×C197C2211yescharacteristic
15AGL(1,ℤ/8)×C197—21no—
16#127×C197C2211yescharacteristic
17#104×C197—21no—
18#100×C197C2211yescharacteristic
19C2×Dic394—21no—
20C22×Dic197—21no—
21C22×Dic197D412yes—
22C4⋊Dic197—21no—
23#15691337—22no—
24#15691337—42no—
25C22⋊Dic197—22no—
26C22⋊Dic197—21no—
27C22⋊Dic197—22no—
28C22⋊Dic197—22no—
29C22⋊Dic197—42no—
30C22⋊Dic197D412yes—
31C4×Dic197—22no—
32#41×C197—21no—
33#41×C197—41no—
34C2×D4×C197—22no—
35C2×D4×C197—22no—
36C2×D4×C197—41no—
37C2×D4×C197D411yescharacteristic
38C2×D4×C197C2311yescharacteristic
39SD16×C197—41no—
40D8×C197—41no—
41M16×C197—21no—
42C42×C197D411yescharacteristic
43Dic394—42no—
44C2×Dic197—22no—
45C2×Dic197—22no—
46C2×Dic197—42no—
47C2×Dic197—42no—
48C2×Dic197—81no—
49C23×C197—22no—
50C23×C197—21no—
51C23×C197C2×D412yes—
52Q8×C197—21no—
53D4×C197—21no—
54D4×C197—41no—
55D4×C197—41no—
56D4×C197—44no—
57D4×C197—42no—
58D4×C197—41no—
59C2×C4×C197—21no—
60C2×C4×C197—22no—
61C2×C4×C197—41no—
62C2×C4×C197C2×D411yescharacteristic
63C8×C197—41no—
64C22×C197—22no—
65C22×C197—42no—
66C22×C197—42no—
67C22×C197—41no—
68C22×C197—41no—
69C22×C197—82no—
70C22×C197#12011yescharacteristic
71C4×C197—22no—
72C4×C197—21no—
73C4×C197—41no—
74Dic197—42no—
75Dic197—42no—
76Dic197—82no—
77C2×C197—21no—
78C2×C197—42no—
79C2×C197—81no—
80C2×C197—81no—
81C2×C197B(4,𝔽2)11yescharacteristic
82C197#219811yescharacteristic, complemented, Sylow
83#2198—1971nomaximal, Sylow
84#559—1971no—
85#458—1972no—
86#453D19711yescharacteristic
87#353—1971no—
88#352—1972no—
89H(5,𝔽2)C2×D19711yescharacteristic
90AGL(1,ℤ/8)—21no—
91#127C2×D19711yescharacteristic
92#124—3942no—
93#123—3942no—
94#123—3941no—
95#123—1972no—
96C4×Q8—3941no—
97C4×D4—3942no—
98#104—21no—
99#100C2×D19711yescharacteristic
100#99—3942no—
101#99—1972no—
102#41—21no—
103#41—41no—
104C2×Q8—3941no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—41no—
108C2×D4C22⋊D19711yescharacteristic
109C2×D4C22×D19711yescharacteristic
110C22×C4—3941no—
111C22×C4—1972no—
112SD16—41no—
113D8—41no—
114M16—21no—
115C4⋊C4—7882no—
116C4⋊C4—3941no—
117C4⋊C4—3942no—
118C22⋊C4—7882no—
119C22⋊C4—3942no—
120C22⋊C4—3942no—
121C22⋊C4—3941no—
122C22⋊C4—3942no—
123C22⋊C4—1972no—
124C42—3942no—
125C42C22⋊D19711yescharacteristic
126C23—22no—
127C23—21no—
128C23D4×D19712yes—
129Q8—21no—
130Q8—7882no—
131D4—21no—
132D4—41no—
133D4—41no—
134D4—44no—
135D4—42no—
136D4—41no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—7882no—
141C2×C4—7882no—
142C2×C4—15761no—
143C2×C4—3942no—
144C2×C4—3942no—
145C2×C4C2×C22⋊D19711yescharacteristic
146C8—41no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—82no—
153C22#2026329411yescharacteristic
154C4—22no—
155C4—21no—
156C4—41no—
157C4—7882no—
158C4—7882no—
159C4—15762no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—81no—
164C2#2716126511yescharacteristic
subgroupG/H
centerC2#27161265
commutator, cosocleC2×D4×C197C23
FrattiniC2×D4C22×D197
Fitting, prime radical#453×C197C2
socleC2×C197B(4,𝔽2)
hypercenter#453D197
hypocenterC197#2198
LCSG ▷ C2×D4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #27161815C2semidirect
2. #27161269C2semidirect
3. #27160872C2semidirect
4. #27160870C2semidirect
82. C197#2198semidirect

Maximal subgroups

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

subgroupindex
#271618152
#271612692
#271608722
#271608702
#453×C1972
#2198197