Finite Group Databasebeta

← #38744712

Subgroups — #38744712

Order 25216.

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

normal subgroups only

This group has 270 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#27161821C211yesmaximal, characteristic, complemented
2#27161266C212yesmaximal, complemented
3#27160874C211yesmaximal, characteristic, complemented
4#27160867C212yesmaximal, complemented
5#455×C197C211yesmaximal, characteristic, complemented
6#20263315—21no—
7#20263314—21no—
8#20263312—22no—
9#20263296—22no—
10#20263294—22no—
11#20263294C2212yes—
12#20263241—22no—
13#20263241C2212yes—
14H(5,𝔽2)×C197C2211yescharacteristic
15#137×C197—21no—
16#124×C197C2211yescharacteristic
17#104×C197—21no—
18#100×C197C2211yescharacteristic
19C23×D197D412yescomplemented
20C2×Dic394—21no—
21C2×D788—21no—
22C2×D788—21no—
23C2×C22⋊D197—22no—
24C2×C22⋊D197—42no—
25C22⋊Dic197—22no—
26C22⋊Dic197D412yescomplemented
27#15691335—22no—
28#15691335—22no—
29#15691335—22no—
30#15691335—42no—
31C4×Dic197—21no—
32#41×C197—21no—
33#41×C197—41no—
34C2×Q8×C197D411yescharacteristic, complemented
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, complemented
43C22×D197—22no—
44C22×D197—42no—
45C22×D197—82no—
46Dic394—81no—
47D788—81no—
48D788—81no—
49C22⋊D197—82no—
50C22⋊D197—84no—
51C2×Dic197—22no—
52C2×Dic197—42no—
53C23×C197—22no—
54C23×C197C2×D412yes—
55Q8×C197—21no—
56Q8×C197—41no—
57D4×C197—21no—
58D4×C197—41no—
59D4×C197—42no—
60D4×C197—41no—
61C2×C4×C197—22no—
62C2×C4×C197—21no—
63C2×C4×C197—21no—
64C2×C4×C197—41no—
65C2×C4×C197C2×D411yescharacteristic
66C8×C197—41no—
67C22×C197—22no—
68C22×C197—42no—
69C22×C197—41no—
70C22×C197—82no—
71C22×C197#12011yescharacteristic
72C2×D197—42no—
73C2×D197—42no—
74C2×D197—44no—
75C2×D197—42no—
76C2×D197—82no—
77C4×C197—21no—
78C4×C197—41no—
79C4×C197—41no—
80C4×C197—41no—
81Dic197—42no—
82Dic197—82no—
83C2×C197—21no—
84C2×C197—42no—
85C2×C197—81no—
86C2×C197B(4,𝔽2)11yescharacteristic
87D197—82no—
88C197#219911yescharacteristic, complemented, Sylow
89#2199—1971nomaximal, Sylow
90H(3,𝔽4)—1971no—
91B(4,𝔽2)—1972no—
92#455D19711yescharacteristic, complemented
93#354—1971no—
94AΣL(1,𝔽16)—1972no—
95H(5,𝔽2)C2×D19711yescharacteristic
96#137—21no—
97#124—3942no—
98#124—3941no—
99#124—3941no—
100#124C2×D19711yescharacteristic
101#120—3942no—
102#120—3942no—
103#120—1972no—
104#104—21no—
105#100C2×D19711yescharacteristic
106#99—3942no—
107#99—1972no—
108C24—1972no—
109#41—21no—
110#41—41no—
111C2×Q8—3941no—
112C2×Q8C22⋊D19711yescharacteristic, complemented
113C2×D4—22no—
114C2×D4—41no—
115C2×D4—7882no—
116C2×D4—3941no—
117C2×D4—3941no—
118C2×D4—3942no—
119C2×D4C22×D19711yescharacteristic
120Q16—41no—
121SD16—41no—
122M16—21no—
123C22⋊C4—22no—
124C22⋊C4—7882no—
125C22⋊C4—3942no—
126C22⋊C4—3942no—
127C22⋊C4—3942no—
128C22⋊C4—3942no—
129C22⋊C4—1972no—
130C42—3941no—
131C42C22⋊D19711yescharacteristic, complemented
132C23—22no—
133C23—7882no—
134C23—15762no—
135C23—3942no—
136C23D4×D19712yes—
137Q8—21no—
138Q8—41no—
139Q8—15761no—
140D4—21no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—15761no—
145D4—15761no—
146D4—15762no—
147D4—15764no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—7882no—
153C2×C4—3942no—
154C2×C4C2×C22⋊D19711yescharacteristic
155C8—41no—
156C22—22no—
157C22—42no—
158C22—41no—
159C22—82no—
160C22—7882no—
161C22—7882no—
162C22—7884no—
163C22—7882no—
164C22—15762no—
165C22#2026329411yescharacteristic
166C4—21no—
167C4—41no—
168C4—41no—
169C4—41no—
170C4—7882no—
171C4—15762no—
172C2—21no—
173C2—42no—
174C2—81no—
175C2—15762no—
176C2#2716126511yescharacteristic
subgroupG/H
centerC2#27161265
commutator, cosocleC2×D4×C197C23
FrattiniC2×D4C22×D197
Fitting#455×C197C2
socleC2×C197B(4,𝔽2)
hypercenter#455D197
hypocenterC197#2199
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. #27161821C2semidirect
2. #27161266C2semidirect
3. #27160874C2semidirect
4. #27160867C2semidirect
5. #455×C197C2semidirect
19. C23×D197D4semidirect
26. C22⋊Dic197D4semidirect
34. C2×Q8×C197D4semidirect
42. C42×C197D4semidirect
88. C197#2199semidirect
92. #455D197semidirect
112. C2×Q8C22⋊D197semidirect
131. C42C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271618212
#271612662
#271608742
#271608672
#455×C1972
#2199197