Finite Group Databasebeta

← #38741756

Subgroups — #38741756

Order 25216.

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

normal subgroups only

This group has 202 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#27161710C211yesmaximal, characteristic
2#27161415C212yesmaximal
3#27161018C211yesmaximal, characteristic
4#27160817C212yesmaximal
5#324×C197C211yesmaximal, characteristic
6#20263325—22no—
7#20263317—22no—
8#20263305—21no—
9#20263305C2212yes—
10#20263285—21no—
11C2×C4⋊Dic197C2211yescharacteristic
12#20263260—22no—
13#20263251—22no—
14#20263233—21no—
15#20263233—21no—
16#20263233C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C2×Dic394—21no—
20C2×Dic394D412yes—
21C22×Dic197—21no—
22C22×Dic197—21no—
23C4⋊Dic197—21no—
24C4⋊Dic197—21no—
25C4⋊Dic197—21no—
26C4⋊Dic197—22no—
27C4⋊Dic197D412yes—
28#15691337—22no—
29#15691337—22no—
30C22⋊Dic197—22no—
31C4×Dic197—22no—
32C22×C4×C197D411yescharacteristic
33C22×C4×C197D411yescharacteristic
34C22×C4×C197C2311yescharacteristic
35C2×C8×C197—22no—
36C4⋊C4×C197—22no—
37Dic394—42no—
38Dic394—81no—
39C2×Dic197—22no—
40C2×Dic197—22no—
41C2×Dic197—41no—
42C2×Dic197—41no—
43C2×Dic197—42no—
44C2×Dic197—41no—
45C2×Dic197—41no—
46C23×C197C2×D411yescharacteristic
47C2×C4×C197—21no—
48C2×C4×C197—21no—
49C2×C4×C197—21no—
50C2×C4×C197—41no—
51C2×C4×C197—41no—
52C2×C4×C197Q1612yes—
53C2×C4×C197C2×D412yes—
54C8×C197—42no—
55C22×C197—21no—
56C22×C197—21no—
57C22×C197#12011yescharacteristic
58C22×C197C2×Q1611yescharacteristic
59C22×C197#13711yescharacteristic
60C4×C197—22no—
61C4×C197—22no—
62C4×C197—41no—
63Dic197—42no—
64Dic197—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#544—1971no—
75#484—1972no—
76#400—1971no—
77#328—1972no—
78#324D19711yescharacteristic
79C4⋊Q8—3942no—
80#125—3942no—
81C22⋊Q8—3941no—
82C22⋊Q8—1972no—
83#117—3941no—
84C2×C4⋊C4—1971no—
85C2×C4⋊C4C2×D19711yescharacteristic
86#107—3942no—
87#103—3942no—
88C22⋊C8C2×D19712yes—
89#95—3941no—
90#95—3941no—
91#95—1971no—
92C2×Q8—3941no—
93C2×Q8—1972no—
94C22×C4—3941no—
95C22×C4—3941no—
96C22×C4D78811yescharacteristic
97C22×C4D78811yescharacteristic
98C22×C4C22×D19711yescharacteristic
99C2×C8—22no—
100C4⋊C4—22no—
101C4⋊C4—3942no—
102C4⋊C4—3941no—
103C4⋊C4—3941no—
104C4⋊C4—3941no—
105C4⋊C4—3942no—
106C4⋊C4—3942no—
107C4⋊C4—1972no—
108C22⋊C4—3942no—
109C42—3942no—
110C23C2×D78811yescharacteristic
111Q8—7882no—
112Q8—15761no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—7881no—
119C2×C4—7881no—
120C2×C4—7882no—
121C2×C4—7881no—
122C2×C4—7881no—
123C2×C4—3942no—
124C2×C4—3942no—
125C2×C4Dic78812yes—
126C2×C4D4×D19712yes—
127C8—42no—
128C22—21no—
129C22—21no—
130C22#2026329611yescharacteristic
131C22C2×Dic78811yescharacteristic
132C22#2026335911yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—7882no—
137C4—7882no—
138C4—15761no—
139C4—15761no—
140C2—21no—
141C2—21no—
142C2#2716123711yescharacteristic
143C2#2716125411yescharacteristic
144C2#2716127011yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#324×C197C2
socleC22×C197#120
prime radicalC23×C197C2×D4
hypercenter#324D197
hypocenterC197#1617
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
72. C197#1617semidirect

Maximal subgroups

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

subgroupindex
#271617102
#271614152
#271610182
#271608172
#324×C1972
#1617197