Finite Group Databasebeta

← #38741751

Subgroups — #38741751

Order 25216.

246 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 53 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#27161660C211yesmaximal, characteristic
2#27161398C212yesmaximal, complemented
3#27161018C211yesmaximal, characteristic, complemented
4#27160814C212yesmaximal, complemented
5#324×C197C211yesmaximal, characteristic, complemented
6C2×#15691358—21no—
7#20263306—22no—
8#20263301—22no—
9#20263300—21no—
10#20263300C2212yes—
11C2×C4⋊Dic197C2211yescharacteristic
12#20263258—22no—
13#20263248—22no—
14#20263233—21no—
15#20263233—21no—
16#20263233C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19#15691358—42no—
20#15691358—42no—
21C2×Dic394—21no—
22C2×D788—21no—
23C2×D788—21no—
24C2×D788D412yes—
25C2×C22⋊D197—22no—
26C22×Dic197—21no—
27C22×Dic197—21no—
28C2×C4×D197—22no—
29C4⋊Dic197—21no—
30C4⋊Dic197—22no—
31C4⋊Dic197D412yes—
32#15691335—22no—
33#15691335—22no—
34C22×C4×C197D411yescharacteristic
35C22×C4×C197D411yescharacteristic
36C22×C4×C197C2311yescharacteristic
37C2×C8×C197—22no—
38C4⋊C4×C197—22no—
39C22×D197—22no—
40Dic394—42no—
41D788—42no—
42D788—42no—
43D788—81no—
44C22⋊D197—42no—
45C22⋊D197—42no—
46C2×Dic197—22no—
47C2×Dic197—41no—
48C2×Dic197—41no—
49C2×Dic197—42no—
50C2×Dic197—41no—
51C2×Dic197—41no—
52C4×D197—44no—
53C23×C197C2×D411yescharacteristic
54C2×C4×C197—21no—
55C2×C4×C197—21no—
56C2×C4×C197—21no—
57C2×C4×C197—41no—
58C2×C4×C197—41no—
59C2×C4×C197SD1612yes—
60C2×C4×C197C2×D412yes—
61C8×C197—42no—
62C22×C197—21no—
63C22×C197—21no—
64C22×C197#12011yescharacteristic
65C22×C197C2×SD1611yescharacteristic
66C22×C197AGL(1,ℤ/8)11yescharacteristic
67C2×D197—42no—
68C2×D197—42no—
69C2×D197—42no—
70C4×C197—22no—
71C4×C197—22no—
72C4×C197—41no—
73Dic197—42no—
74Dic197—81no—
75Dic197—81no—
76C2×C197—21no—
77C2×C197—21no—
78C2×C197#45011yescharacteristic
79C2×C197#45411yescharacteristic
80C2×C197#45811yescharacteristic
81D197—82no—
82C197#161611yescharacteristic, complemented, Sylow
83#1616—1971nomaximal, Sylow
84#537—1971no—
85#481—1972no—
86#400—1971no—
87#327—1972no—
88#324D19711yescharacteristic, complemented
89C2×#41—3941no—
90C22⋊Q8—3942no—
91#121—3942no—
92#121—3941no—
93#121—1972no—
94C2×C4⋊C4—1971no—
95C2×C4⋊C4C2×D19711yescharacteristic
96#106—3942no—
97#102—3942no—
98C22⋊C8C2×D19712yes—
99#95—3941no—
100#95—3941no—
101#95—1971no—
102#41—7882no—
103#41—7882no—
104C2×Q8—3941no—
105C2×D4—3941no—
106C2×D4—3941no—
107C2×D4—3942no—
108C2×D4—1972no—
109C22×C4—3942no—
110C22×C4—3941no—
111C22×C4—3941no—
112C22×C4D78811yescharacteristic
113C22×C4D78811yescharacteristic
114C22×C4C22×D19711yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—3941no—
118C4⋊C4—3942no—
119C4⋊C4—1972no—
120C22⋊C4—3942no—
121C22⋊C4—3942no—
122C23—3942no—
123C23C2×D78811yescharacteristic
124Q8—7882no—
125D4—7882no—
126D4—7882no—
127D4—7882no—
128D4—7882no—
129D4—15761no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—7884no—
136C2×C4—7881no—
137C2×C4—7881no—
138C2×C4—7882no—
139C2×C4—7881no—
140C2×C4—7881no—
141C2×C4—3942no—
142C2×C4#1569134712yes—
143C2×C4D4×D19712yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—7882no—
148C22—7882no—
149C22—7882no—
150C22#2026329611yescharacteristic
151C22C2×#1569134711yescharacteristic
152C22#2026335411yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—7882no—
157C4—15761no—
158C4—15761no—
159C2—21no—
160C2—21no—
161C2—15762no—
162C2#2716123211yescharacteristic
163C2#2716125411yescharacteristic
164C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#324×C197C2
socleC22×C197#120
prime radical#27161660C2
hypercenter#324D197
hypocenterC197#1616
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27161398C2semidirect
3. #27161018C2semidirect
4. #27160814C2semidirect
5. #324×C197C2semidirect
82. C197#1616semidirect
88. #324D197semidirect

Maximal subgroups

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

subgroupindex
#271616602
#271613982
#271610182
#271608142
#324×C1972
#1616197