Finite Group Databasebeta

← #38744724

Subgroups — #38744724

Order 25216.

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

normal subgroups only

This group has 226 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#27161829C211yesmaximal, characteristic, complemented
2#27161269C212yesmaximal
3#27160874C211yesmaximal, characteristic, complemented
4#27160870C212yesmaximal, complemented
5#455×C197C211yesmaximal, characteristic
6#20263324—21no—
7#20263319—22no—
8#20263318—21no—
9#20263307—22no—
10#20263307C2212yes—
11#20263303—22no—
12#20263241—22no—
13#20263241C2212yes—
14H(5,𝔽2)×C197C2211yescharacteristic
15#137×C197—21no—
16#124×C197C2211yescharacteristic
17#104×C197—21no—
18#100×C197C2211yescharacteristic
19C2×Dic394—21no—
20C22×Dic197D412yes—
21C4⋊Dic197—21no—
22C4⋊Dic197—21no—
23#15691337—22no—
24#15691337—22no—
25#15691337—22no—
26#15691337—42no—
27C22⋊Dic197—22no—
28C22⋊Dic197—22no—
29C22⋊Dic197—42no—
30C22⋊Dic197D412yes—
31C4×Dic197—21no—
32#41×C197—21no—
33#41×C197—41no—
34C2×Q8×C197D411yescharacteristic
35C2×D4×C197—22no—
36C2×D4×C197—41no—
37C2×D4×C197C2311yescharacteristic
38Q16×C197—41no—
39SD16×C197—41no—
40M16×C197—21no—
41C22⋊C4×C197—22no—
42C42×C197D411yescharacteristic
43Dic394—81no—
44C2×Dic197—22no—
45C2×Dic197—22no—
46C2×Dic197—42no—
47C2×Dic197—42no—
48C23×C197—22no—
49C23×C197C2×D412yes—
50Q8×C197—21no—
51Q8×C197—41no—
52D4×C197—21no—
53D4×C197—41no—
54D4×C197—42no—
55D4×C197—41no—
56C2×C4×C197—22no—
57C2×C4×C197—21no—
58C2×C4×C197—21no—
59C2×C4×C197—41no—
60C2×C4×C197C2×D411yescharacteristic
61C8×C197—41no—
62C22×C197—22no—
63C22×C197—42no—
64C22×C197—41no—
65C22×C197—82no—
66C22×C197#12011yescharacteristic
67C4×C197—21no—
68C4×C197—41no—
69C4×C197—41no—
70C4×C197—41no—
71Dic197—42no—
72Dic197—42no—
73Dic197—82no—
74C2×C197—21no—
75C2×C197—42no—
76C2×C197—81no—
77C2×C197B(4,𝔽2)11yescharacteristic
78C197#220111yescharacteristic, complemented, Sylow
79#2201—1971nomaximal, Sylow
80#563—1971no—
81#458—1972no—
82#455D19711yescharacteristic
83#354—1971no—
84#352—1972no—
85H(5,𝔽2)C2×D19711yescharacteristic
86#137—21no—
87C4⋊Q8—3941no—
88#126—3942no—
89#125—3941no—
90#124C2×D19711yescharacteristic
91#123—3942no—
92#123—1972no—
93C22⋊Q8—3942no—
94#104—21no—
95#100C2×D19711yescharacteristic
96#99—3942no—
97#99—1972no—
98#41—21no—
99#41—41no—
100C2×Q8—3941no—
101C2×Q8C22⋊D19711yescharacteristic
102C2×D4—22no—
103C2×D4—41no—
104C2×D4C22×D19711yescharacteristic
105C22×C4—1972no—
106Q16—41no—
107SD16—41no—
108M16—21no—
109C4⋊C4—7882no—
110C4⋊C4—3942no—
111C4⋊C4—3942no—
112C4⋊C4—3941no—
113C4⋊C4—3941no—
114C4⋊C4—3942no—
115C22⋊C4—22no—
116C22⋊C4—7882no—
117C22⋊C4—3942no—
118C22⋊C4—3942no—
119C22⋊C4—1972no—
120C42—3941no—
121C42C22⋊D19711yescharacteristic
122C23—22no—
123C23D4×D19712yes—
124Q8—21no—
125Q8—41no—
126Q8—15761no—
127D4—21no—
128D4—41no—
129D4—42no—
130D4—41no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—7882no—
136C2×C4—7882no—
137C2×C4—3942no—
138C2×C4—3942no—
139C2×C4C2×C22⋊D19711yescharacteristic
140C8—41no—
141C22—22no—
142C22—42no—
143C22—41no—
144C22—82no—
145C22#2026329411yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—41no—
150C4—7882no—
151C4—7882no—
152C4—15762no—
153C2—21no—
154C2—42no—
155C2—81no—
156C2#2716126511yescharacteristic
subgroupG/H
centerC2#27161265
commutator, cosocleC2×D4×C197C23
FrattiniC2×D4C22×D197
Fitting#455×C197C2
socleC2×C197B(4,𝔽2)
prime radicalH(5,𝔽2)×C197C22
hypercenter#455D197
hypocenterC197#2201
LCSG ▷ C2×D4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #27161829C2semidirect
3. #27160874C2semidirect
4. #27160870C2semidirect
78. C197#2201semidirect

Maximal subgroups

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

subgroupindex
#271618292
#271612692
#271608742
#271608702
#455×C1972
#2201197