Finite Group Databasebeta

← #38741719

Subgroups — #38741719

Order 25216.

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

normal subgroups only

This group has 210 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#27161409C212yesmaximal
3#27160815C212yesmaximal
4#27160805C211yesmaximal, characteristic
5#398×C197C211yesmaximal, characteristic
6#20263325—22no—
7#20263317—22no—
8#20263305—21no—
9#20263305C2212yes—
10#20263285—21no—
11#20263257—22no—
12#20263249—22no—
13C22⋊#13096802C2212yes—
14C2×C4⋊C4×C197—21no—
15C2×C4⋊C4×C197C2211yescharacteristic
16C2×C4⋊C4×C197C2211yescharacteristic
17#95×C197—21no—
18#95×C197C2211yescharacteristic
19C2×Dic394—21no—
20C2×Dic394D412yes—
21C2×#13096802—22no—
22C4⋊Dic197—21no—
23C4⋊Dic197—21no—
24C4⋊Dic197—21no—
25C4⋊Dic197D412yes—
26#15691337—22no—
27#15691337—22no—
28C22⋊Dic197—22no—
29C4×Dic197—22no—
30C22×C4×C197—21no—
31C22×C4×C197—21no—
32C22×C4×C197D411yescharacteristic
33C22×C4×C197D411yescharacteristic
34C22×C4×C197C2311yescharacteristic
35C4⋊C4×C197—22no—
36C4⋊C4×C197—22no—
37C4⋊C4×C197—42no—
38Dic394—42no—
39Dic394—81no—
40C2×Dic197—22no—
41C2×Dic197—22no—
42#13096802—42no—
43C23×C197C2×D411yescharacteristic
44C2×C4×C197—21no—
45C2×C4×C197—21no—
46C2×C4×C197—22no—
47C2×C4×C197—21no—
48C2×C4×C197—41no—
49C2×C4×C197—41no—
50C2×C4×C197—41no—
51C2×C4×C197—41no—
52C2×C4×C197—41no—
53C2×C4×C197—41no—
54C2×C4×C197—41no—
55C2×C4×C197SD1612yes—
56C2×C4×C197C2×D412yes—
57C22×C197—21no—
58C22×C197—21no—
59C22×C197#12011yescharacteristic
60C22×C197C2×SD1611yescharacteristic
61C22×C197#13711yescharacteristic
62C4×C197—22no—
63C4×C197—22no—
64C4×C197—42no—
65C4×C197—41no—
66C4×C197—81no—
67Dic197—42no—
68Dic197—42no—
69C2×C197—21no—
70C2×C197—21no—
71C2×C197#44811yescharacteristic
72C2×C197#45611yescharacteristic
73C2×C197#45811yescharacteristic
74C197#161111yescharacteristic, complemented, Sylow
75#1611—1971nomaximal, Sylow
76#544—1971no—
77#483—1972no—
78#398D19711yescharacteristic
79#328—1972no—
80#324—1971no—
81C4⋊Q8—3942no—
82#125—3942no—
83C22⋊Q8—3941no—
84C22⋊Q8—1972no—
85#117—3941no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2×D19711yescharacteristic
88C2×C4⋊C4C2×D19711yescharacteristic
89#106—3942no—
90#103—3942no—
91C22⋊C8—1972no—
92#95—21no—
93#95C2×D19711yescharacteristic
94C2×Q8—3941no—
95C2×Q8—1972no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D19711yescharacteristic
99C22×C4C22⋊D19711yescharacteristic
100C22×C4C22×D19711yescharacteristic
101C2×C8—3942no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4—3942no—
106C4⋊C4—3941no—
107C4⋊C4—3941no—
108C4⋊C4—3941no—
109C4⋊C4—3942no—
110C4⋊C4—1972no—
111C22⋊C4—3942no—
112C42—3942no—
113C23C2×C22⋊D19711yescharacteristic
114Q8—7882no—
115Q8—15761no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—3942no—
128C2×C4—3942no—
129C2×C4#1569134612yes—
130C2×C4D4×D19712yes—
131C8—7882no—
132C22—21no—
133C22—21no—
134C22#2026329411yescharacteristic
135C22C2×#1569134611yescharacteristic
136C22#2026335611yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—81no—
142C4—7882no—
143C4—7882no—
144C2—21no—
145C2—21no—
146C2#2716121711yescharacteristic
147C2#2716126111yescharacteristic
148C2#2716126811yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#398×C197C2
socleC22×C197#120
prime radicalC23×C197C2×D4
hypercenter#398D197
hypocenterC197#1611
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
74. C197#1611semidirect

Maximal subgroups

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

subgroupindex
#271617102
#271614092
#271608152
#271608052
#398×C1972
#1611197