Finite Group Databasebeta

← #38741749

Subgroups — #38741749

Order 25216.

246 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 57 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#27161395C212yesmaximal, complemented
2#27161018C211yesmaximal, characteristic, complemented
3#27160813C212yesmaximal, complemented
4#27160805C211yesmaximal, characteristic, complemented
5#537×C197C211yesmaximal, characteristic
6C2×C4⋊Dic197C2211yescharacteristic
7#20263257—22no—
8#20263247—22no—
9C22⋊#13096802C2212yes—
10#20263233—21no—
11#20263233—21no—
12#20263233C2211yescharacteristic
13C2×#41×C197—21no—
14C22⋊Q8×C197—22no—
15#121×C197—22no—
16#121×C197—21no—
17#121×C197C2212yes—
18C2×C4⋊C4×C197C2211yescharacteristic
19C22×Dic197—21no—
20C22×Dic197—21no—
21C2×#13096802—22no—
22C4⋊Dic197—22no—
23#41×C197—42no—
24#41×C197—42no—
25C2×Q8×C197—21no—
26C2×D4×C197—21no—
27C2×D4×C197—22no—
28C2×D4×C197—21no—
29C2×D4×C197D412yes—
30C22×C4×C197—22no—
31C22×C4×C197D411yescharacteristic
32C22×C4×C197D411yescharacteristic
33C22×C4×C197C2311yescharacteristic
34C4⋊C4×C197—21no—
35C4⋊C4×C197—22no—
36C4⋊C4×C197D412yes—
37C22⋊C4×C197—22no—
38C22⋊C4×C197—22no—
39C2×Dic197—41no—
40C2×Dic197—41no—
41C2×Dic197—42no—
42C2×Dic197—41no—
43C2×Dic197—41no—
44#13096802—42no—
45C23×C197—22no—
46C23×C197C2×D411yescharacteristic
47Q8×C197—42no—
48D4×C197—42no—
49D4×C197—42no—
50D4×C197—42no—
51D4×C197—42no—
52D4×C197—81no—
53C2×C4×C197—21no—
54C2×C4×C197—21no—
55C2×C4×C197—21no—
56C2×C4×C197—22no—
57C2×C4×C197—41no—
58C2×C4×C197—44no—
59C2×C4×C197—41no—
60C2×C4×C197SD1612yes—
61C2×C4×C197C2×D412yes—
62C22×C197—21no—
63C22×C197—21no—
64C22×C197—42no—
65C22×C197—42no—
66C22×C197—42no—
67C22×C197#12011yescharacteristic
68C22×C197C2×SD1611yescharacteristic
69C22×C197AGL(1,ℤ/8)11yescharacteristic
70C4×C197—22no—
71C4×C197—22no—
72C4×C197—42no—
73C4×C197—41no—
74Dic197—81no—
75Dic197—81no—
76C2×C197—21no—
77C2×C197—21no—
78C2×C197—82no—
79C2×C197#45011yescharacteristic
80C2×C197#45411yescharacteristic
81C2×C197#45811yescharacteristic
82C197#161611yescharacteristic, complemented, Sylow
83#1616—1971nomaximal, Sylow
84#537D19711yescharacteristic
85#481—1972no—
86#400—1971no—
87#327—1972no—
88#324—1971no—
89C2×#41—21no—
90C22⋊Q8—22no—
91#121—22no—
92#121—21no—
93#121C2×D19712yes—
94C2×C4⋊C4—1971no—
95C2×C4⋊C4C2×D19711yescharacteristic
96#106—3942no—
97#102—3942no—
98C22⋊C8—1972no—
99#95—3941no—
100#95—3941no—
101#95—1971no—
102#41—42no—
103#41—42no—
104C2×Q8—21no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4C22⋊D19712yes—
109C22×C4—22no—
110C22×C4—3941no—
111C22×C4—3941no—
112C22×C4C22⋊D19711yescharacteristic
113C22×C4C22⋊D19711yescharacteristic
114C22×C4C22×D19711yescharacteristic
115C2×C8—3942no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—3942no—
119C4⋊C4C22⋊D19712yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C23—22no—
123C23C2×C22⋊D19711yescharacteristic
124Q8—42no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—81no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—44no—
136C2×C4—41no—
137C2×C4—7881no—
138C2×C4—7881no—
139C2×C4—7882no—
140C2×C4—7881no—
141C2×C4—7881no—
142C2×C4#1569134612yes—
143C2×C4C2×C22⋊D19712yes—
144C8—7882no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22#2026329511yescharacteristic
151C22C2×#1569134611yescharacteristic
152C22#2026335211yescharacteristic
153C4—22no—
154C4—22no—
155C4—42no—
156C4—41no—
157C4—15761no—
158C4—15761no—
159C2—21no—
160C2—21no—
161C2—82no—
162C2#2716123011yescharacteristic
163C2#2716125211yescharacteristic
164C2#2716126911yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting, prime radical#537×C197C2
socleC22×C197#120
hypercenter#537D197
hypocenterC197#1616
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #27161395C2semidirect
2. #27161018C2semidirect
3. #27160813C2semidirect
4. #27160805C2semidirect
82. C197#1616semidirect

Maximal subgroups

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

subgroupindex
#271613952
#271610182
#271608132
#271608052
#537×C1972
#1616197