Finite Group Databasebeta

← #38741754

Subgroups — #38741754

Order 25216.

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

normal subgroups only

This group has 202 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#27161412C212yesmaximal
2#27161018C211yesmaximal, characteristic
3#27160816C212yesmaximal
4#27160805C211yesmaximal, characteristic
5#544×C197C211yesmaximal, characteristic
6C2×C4⋊Dic197C2211yescharacteristic
7#20263259—22no—
8#20263250—22no—
9C22⋊#13096802C2212yes—
10#20263233—21no—
11#20263233—21no—
12#20263233C2211yescharacteristic
13C4⋊Q8×C197—22no—
14#125×C197—22no—
15C22⋊Q8×C197—21no—
16C22⋊Q8×C197C2212yes—
17#117×C197—21no—
18C2×C4⋊C4×C197C2211yescharacteristic
19C22×Dic197—21no—
20C22×Dic197—21no—
21C2×#13096802—22no—
22C4⋊Dic197—22no—
23C2×Q8×C197—21no—
24C2×Q8×C197D412yes—
25C22×C4×C197D411yescharacteristic
26C22×C4×C197D411yescharacteristic
27C22×C4×C197C2311yescharacteristic
28C4⋊C4×C197—21no—
29C4⋊C4×C197—21no—
30C4⋊C4×C197—21no—
31C4⋊C4×C197—22no—
32C4⋊C4×C197—22no—
33C4⋊C4×C197—22no—
34C4⋊C4×C197D412yes—
35C22⋊C4×C197—22no—
36C42×C197—22no—
37C2×Dic197—41no—
38C2×Dic197—41no—
39C2×Dic197—42no—
40C2×Dic197—41no—
41C2×Dic197—41no—
42#13096802—42no—
43C23×C197C2×D411yescharacteristic
44Q8×C197—42no—
45Q8×C197—81no—
46C2×C4×C197—21no—
47C2×C4×C197—21no—
48C2×C4×C197—22no—
49C2×C4×C197—22no—
50C2×C4×C197—21no—
51C2×C4×C197—41no—
52C2×C4×C197—41no—
53C2×C4×C197Q1612yes—
54C2×C4×C197C2×D412yes—
55C22×C197—21no—
56C22×C197—21no—
57C22×C197#12011yescharacteristic
58C22×C197C2×Q1611yescharacteristic
59C22×C197#13711yescharacteristic
60C4×C197—22no—
61C4×C197—22no—
62C4×C197—42no—
63C4×C197—41no—
64C4×C197—42no—
65Dic197—81no—
66Dic197—81no—
67C2×C197—21no—
68C2×C197—21no—
69C2×C197#45111yescharacteristic
70C2×C197#45411yescharacteristic
71C2×C197#45811yescharacteristic
72C197#161711yescharacteristic, complemented, Sylow
73#1617—1971nomaximal, Sylow
74#544D19711yescharacteristic
75#484—1972no—
76#400—1971no—
77#328—1972no—
78#324—1971no—
79C4⋊Q8—22no—
80#125—22no—
81C22⋊Q8—21no—
82C22⋊Q8C2×D19712yes—
83#117—21no—
84C2×C4⋊C4—1971no—
85C2×C4⋊C4C2×D19711yescharacteristic
86#107—3942no—
87#103—3942no—
88C22⋊C8—1972no—
89#95—3941no—
90#95—3941no—
91#95—1971no—
92C2×Q8—21no—
93C2×Q8C22⋊D19712yes—
94C22×C4—3941no—
95C22×C4—3941no—
96C22×C4C22⋊D19711yescharacteristic
97C22×C4C22⋊D19711yescharacteristic
98C22×C4C22×D19711yescharacteristic
99C2×C8—3942no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—3942no—
107C4⋊C4C22⋊D19712yes—
108C22⋊C4—22no—
109C42—22no—
110C23C2×C22⋊D19711yescharacteristic
111Q8—42no—
112Q8—81no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—21no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—7881no—
121C2×C4—7881no—
122C2×C4—7882no—
123C2×C4—7881no—
124C2×C4—7881no—
125C2×C4#1569134812yes—
126C2×C4C2×C22⋊D19712yes—
127C8—7882no—
128C22—21no—
129C22—21no—
130C22#2026329511yescharacteristic
131C22C2×#1569134811yescharacteristic
132C22#2026335711yescharacteristic
133C4—22no—
134C4—22no—
135C4—42no—
136C4—41no—
137C4—42no—
138C4—15761no—
139C4—15761no—
140C2—21no—
141C2—21no—
142C2#2716123511yescharacteristic
143C2#2716125211yescharacteristic
144C2#2716126911yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#544×C197C2
socleC22×C197#120
prime radicalC23×C197C2×D4
hypercenter#544D197
hypocenterC197#1617
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
72. C197#1617semidirect

Maximal subgroups

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

subgroupindex
#271614122
#271610182
#271608162
#271608052
#544×C1972
#1617197