Finite Group Databasebeta

← #38743242

Subgroups — #38743242

Order 25216.

362 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 85 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#27161646C211yesmaximal, characteristic
2#27161554C211yesmaximal, characteristic, complemented
3#27161095C212yesmaximal, complemented
4#27160836C212yesmaximal, complemented
5C2×#100×C197C211yesmaximal, characteristic, complemented
6C2×#15691358C2211yescharacteristic
7#20263322C412yescomplemented
8#20263313C412yescomplemented
9#20263297—22no—
10#20263295—22no—
11D4×Dic197—24no—
12#20263285C2211yescharacteristic
13#20263285C2212yes—
14C2×C22⋊Dic197—22no—
15#20263252—24no—
16C22×D4×C197C2211yescharacteristic
17C2×M16×C197C2212yes—
18#100×C197—22no—
19#15691358—22no—
20#15691358—22no—
21C2×Dic394C2×C411yescharacteristic, complemented
22C2×D788C2×C411yescharacteristic, complemented
23C2×C22⋊D197—24no—
24C2×C22⋊D197—21no—
25C22×Dic197—24no—
26C2×C4×D197—21no—
27C4⋊Dic197—22no—
28C4⋊Dic197—21no—
29C22⋊Dic197—24no—
30C22⋊Dic197—24no—
31C22⋊Dic197—22no—
32C22⋊Dic197—21no—
33C4×Dic197C2×C412yescomplemented
34C4×Dic197D414yescomplemented
35C24×C197D412yes—
36C2×D4×C197—22no—
37C2×D4×C197—22no—
38C2×D4×C197C2×C412yes—
39C2×D4×C197D412yes—
40C22×C4×C197C2311yescharacteristic
41M16×C197—22no—
42M16×C197—22no—
43C2×C8×C197—22no—
44C22×D197—21no—
45Dic394—22no—
46D788—22no—
47C22⋊D197—41no—
48C22⋊D197—41no—
49C22⋊D197—84no—
50C2×Dic197—22no—
51C2×Dic197—21no—
52C2×Dic197—24no—
53C2×Dic197—84no—
54C4×D197—42no—
55C23×C197—22no—
56C23×C197—22no—
57C23×C197—82no—
58C23×C197C22⋊C414yes—
59C23×C197C2×D411yescharacteristic
60D4×C197—24no—
61D4×C197—24no—
62C2×C4×C197C22×C411yescharacteristic
63C2×C4×C197C2×D412yes—
64C2×C4×C197C2×D411yescharacteristic
65C2×C4×C197#4112yes—
66C8×C197—42no—
67C22×C197—24no—
68C22×C197—41no—
69C22×C197—44no—
70C22×C197—44no—
71C22×C197—82no—
72C22×C197C2×C22⋊C411yescharacteristic
73C22×C197#12011yescharacteristic
74C22×C197#12311yescharacteristic
75C2×D197—42no—
76C2×D197—41no—
77C4×C197C4×D412yes—
78C4×C197#12112yes—
79Dic197—44no—
80Dic197—42no—
81Dic197—41no—
82C2×C197—21no—
83C2×C197—21no—
84C2×C197—21no—
85C2×C197—44no—
86C2×C197#38611yescharacteristic
87D197—81no—
88C197#191111yescharacteristic, complemented, Sylow
89#1911—1971nomaximal, Sylow
90#535—1971no—
91#518—1971no—
92#421—1972no—
93C2×#100D19711yescharacteristic, complemented
94H(3,ℤ/4)—1972no—
95C2×#41—1971no—
96C22×D4C2×D19711yescharacteristic
97C2×M16C2×D19712yes—
98#127—1972no—
99#124—1972no—
100#121—3942no—
101#120—3942no—
102C4×D4—3944no—
103#117—1972no—
104#117—1971no—
105C2×C22⋊C4—3942no—
106#104—3944no—
107#100—22no—
108C24D78812yes—
109#41—3942no—
110#41—3942no—
111C2×Q8—1971no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—3941no—
115C2×D4—3944no—
116C2×D4—1971no—
117C2×D4C4×D19712yes—
118C2×D4C22⋊D19712yes—
119C22×C4—3941no—
120C22×C4—3944no—
121C22×C4C22×D19711yescharacteristic
122M16—22no—
123M16—22no—
124C2×C8—22no—
125C4⋊C4—3942no—
126C4⋊C4—3941no—
127C22⋊C4—3944no—
128C22⋊C4—3944no—
129C22⋊C4—3942no—
130C22⋊C4—3941no—
131C42—1972no—
132C42—1974no—
133C23—22no—
134C23—22no—
135C23—82no—
136C23—3941no—
137C23#1569133514yes—
138C23C2×D78811yescharacteristic
139Q8—3942no—
140D4—24no—
141D4—24no—
142D4—7881no—
143D4—7881no—
144D4—15764no—
145D4—3942no—
146C2×C4—7882no—
147C2×C4—15764no—
148C2×C4—3942no—
149C2×C4—3941no—
150C2×C4—3944no—
151C2×C4C2×C4×D19711yescharacteristic
152C2×C4D4×D19712yes—
153C2×C4C2×C22⋊D19711yescharacteristic
154C2×C4#1569135712yes—
155C8—42no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—7882no—
162C22—7881no—
163C22C2×#1569133511yescharacteristic
164C22#2026329611yescharacteristic
165C22#2026331011yescharacteristic
166C4—7884no—
167C4—7882no—
168C4—7881no—
169C4#2026328612yes—
170C4#2026329812yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—15761no—
176C2#2716095611yescharacteristic
subgroupG/H
centerC2#27160956
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
FittingC2×#100×C197C2
socleC2×C197#386
prime radical#27161646C2
cosocleC22×C4×C197C23
hypercenterC2×#100D197
hypocenterC197#1911
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #27161554C2semidirect
3. #27161095C2semidirect
4. #27160836C2semidirect
5. C2×#100×C197C2semidirect
7. #20263322C4semidirect
8. #20263313C4semidirect
21. C2×Dic394C2×C4semidirect
22. C2×D788C2×C4semidirect
33. C4×Dic197C2×C4semidirect
34. C4×Dic197D4semidirect
88. C197#1911semidirect
93. C2×#100D197semidirect

Maximal subgroups

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

subgroupindex
#271616462
#271615542
#271610952
#271608362
C2×#100×C1972
#1911197