Finite Group Databasebeta

← #38744660

Subgroups — #38744660

Order 25216.

226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161879C211yesmaximal, characteristic, complemented
2#27161517C212yesmaximal, complemented
3#27161332C211yesmaximal, characteristic, complemented
4#27160890C212yesmaximal, complemented
5#349×C197C211yesmaximal, characteristic, complemented
6C2×#15691358—21no—
7#20263359—22no—
8#20263354—22no—
9#20263348—21no—
10#20263348C2212yes—
11C2×D1576C2211yescharacteristic, complemented
12#20263300—21no—
13#20263271—22no—
14D3152—22no—
15#20263258C2211yescharacteristic, complemented
16#20263248—21no—
17C2×M16×C197C2211yescharacteristic
18M32×C197C2212yescomplemented
19#15691358—22no—
20#15691358—22no—
21#15691358—41no—
22#15691358—41no—
23C2×Dic394—21no—
24C2×D788—21no—
25C2×D788—21no—
26C2×C22⋊D197—41no—
27C2×C22⋊D197—41no—
28C2×C4×D197—41no—
29Dic788—21no—
30Dic788D412yescomplemented
31#15691347—22no—
32#15691347—22no—
33D1576—21no—
34D1576—22no—
35D1576D412yescomplemented
36C4⋊Dic197—21no—
37#15691335—41no—
38C22×C4×C197D411yescharacteristic
39M16×C197—22no—
40C2×C8×C197D411yescharacteristic
41C2×C8×C197C2311yescharacteristic
42C16×C197—22no—
43C22×D197—41no—
44C22×D197—41no—
45Dic394—22no—
46Dic394—41no—
47D788—22no—
48D788—41no—
49D788—41no—
50D788—41no—
51C22⋊D197—42no—
52C22⋊D197—42no—
53C22⋊D197—82no—
54C2×Dic197—41no—
55C2×Dic197—41no—
56C4×D197—42no—
57C4×D197—42no—
58C23×C197D811yescharacteristic
59C2×C4×C197—21no—
60C2×C4×C197—21no—
61C2×C4×C197D811yescharacteristic
62C2×C4×C197C2×D411yescharacteristic
63C8×C197—21no—
64C8×C197C2×D412yes—
65C22×C197—21no—
66C22×C197—41no—
67C22×C197C2×D811yescharacteristic
68C2×D197—42no—
69C2×D197—82no—
70C2×D197—81no—
71C2×D197—81no—
72C2×D197—81no—
73C4×C197—21no—
74C4×C197#12011yescharacteristic
75C4×C197AGL(1,ℤ/8)11yescharacteristic
76Dic197—42no—
77Dic197—81no—
78C2×C197—21no—
79C2×C197—41no—
80C2×C197#44711yescharacteristic
81D197—82no—
82D197—161no—
83C197#219011yescharacteristic, complemented, Sylow
84#2190—1971nomaximal, Sylow
85#575—1971no—
86#509—1972no—
87#469—1971no—
88#360—1972no—
89#349D19711yescharacteristic, complemented
90C2×#41—3941no—
91#137—3942no—
92AGL(1,ℤ/8)—3942no—
93#135—3941no—
94#135—1972no—
95C2×D8—1971no—
96C2×M16C2×D19711yescharacteristic
97#121—3941no—
98SD32—3942no—
99D16—3942no—
100M32C2×D19712yescomplemented
101#106—1971no—
102#102—3941no—
103#41—7881no—
104#41—7881no—
105#41—3942no—
106#41—3942no—
107C2×Q8—3941no—
108C2×D4—7881no—
109C2×D4—7881no—
110C2×D4—3941no—
111C2×D4—3941no—
112C22×C4—7881no—
113C22×C4D78811yescharacteristic
114Q16—3941no—
115Q16—1972no—
116SD16—3942no—
117SD16—3942no—
118D8—3941no—
119D8—3942no—
120D8—1972no—
121M16—22no—
122C2×C8D78811yescharacteristic
123C2×C8C22×D19711yescharacteristic
124C4⋊C4—3941no—
125C22⋊C4—7881no—
126C16—22no—
127C23—7881no—
128C23—7881no—
129C23D157611yescharacteristic
130Q8—7881no—
131Q8—3942no—
132D4—7881no—
133D4—7882no—
134D4—7882no—
135D4—7881no—
136D4—7881no—
137D4—15762no—
138D4—3942no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—7882no—
142C2×C4—7881no—
143C2×C4—7882no—
144C2×C4—7881no—
145C2×C4D157611yescharacteristic
146C2×C4C2×D78811yescharacteristic
147C8—21no—
148C8D4×D19712yes—
149C22—21no—
150C22—41no—
151C22—7882no—
152C22—15761no—
153C22—15761no—
154C22—15762no—
155C22—15761no—
156C22C2×D157611yescharacteristic
157C4—21no—
158C4—7882no—
159C4—15761no—
160C4#2026329611yescharacteristic
161C4#2026335411yescharacteristic
162C2—21no—
163C2—41no—
164C2—15762no—
165C2—31521no—
166C2#2716121511yescharacteristic
subgroupG/H
centerC2#27161215
commutator, cosocleC2×C8×C197C23
FrattiniC2×C8C22×D197
Fitting#349×C197C2
socleC2×C197#447
hypercenter#349D197
hypocenterC197#2190
LCSG ▷ C2×C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
1. #27161879C2semidirect
2. #27161517C2semidirect
3. #27161332C2semidirect
4. #27160890C2semidirect
5. #349×C197C2semidirect
11. C2×D1576C22semidirect
15. #20263258C22semidirect
18. M32×C197C22semidirect
30. Dic788D4semidirect
35. D1576D4semidirect
83. C197#2190semidirect
89. #349D197semidirect
100. M32C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618792
#271615172
#271613322
#271608902
#349×C1972
#2190197