Finite Group Databasebeta

← #38741688

Subgroups — #38741688

Order 25216.

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

normal subgroups only

This group has 238 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#27161237C212yesmaximal
3#27160999C211yesmaximal, characteristic
4#27160817C212yesmaximal
5#324×C197C211yesmaximal, characteristic
6C22×Dic394C2211yescharacteristic
7C2×Dic788—22no—
8#20263325—22no—
9#20263317—22no—
10#20263305—21no—
11#20263305C2212yes—
12#20263285—21no—
13C2×#15691337—21no—
14#20263251—22no—
15#20263233—21no—
16#20263233C2211yescharacteristic
17C2×C4⋊C4×C197C2211yescharacteristic
18C22⋊C8×C197C2212yes—
19C2×Dic394—21no—
20C2×Dic394—22no—
21C2×Dic394—42no—
22C2×Dic394D412yes—
23C22×Dic197—21no—
24C22×Dic197—21no—
25Dic788—44no—
26C4⋊Dic197—21no—
27C4⋊Dic197—21no—
28C4⋊Dic197—21no—
29C4⋊Dic197D412yes—
30#15691337—22no—
31#15691337—22no—
32#15691337—42no—
33C22⋊Dic197—22no—
34C4×Dic197—22no—
35C22×C4×C197D411yescharacteristic
36C22×C4×C197D411yescharacteristic
37C22×C4×C197C2311yescharacteristic
38C2×C8×C197—22no—
39C4⋊C4×C197—22no—
40Dic394—24no—
41Dic394—42no—
42Dic394—42no—
43Dic394—81no—
44C2×Dic197—22no—
45C2×Dic197—22no—
46C2×Dic197—22no—
47C2×Dic197—22no—
48C2×Dic197—41no—
49C2×Dic197—41no—
50C2×Dic197—41no—
51C2×Dic197—41no—
52C23×C197C2×D411yescharacteristic
53C2×C4×C197—21no—
54C2×C4×C197—21no—
55C2×C4×C197—21no—
56C2×C4×C197—41no—
57C2×C4×C197—41no—
58C2×C4×C197Q1612yescomplemented
59C2×C4×C197C2×D412yes—
60C8×C197—42no—
61C22×C197—21no—
62C22×C197—21no—
63C22×C197#12011yescharacteristic
64C22×C197C2×Q1611yescharacteristic
65C22×C197#13711yescharacteristic
66C4×C197—22no—
67C4×C197—22no—
68C4×C197—41no—
69Dic197—42no—
70Dic197—42no—
71Dic197—42no—
72Dic197—42no—
73C2×C197—21no—
74C2×C197—21no—
75C2×C197#45111yescharacteristic
76C2×C197#45611yescharacteristic
77C2×C197B(4,𝔽2)11yescharacteristic
78C197#160511yescharacteristic, complemented, Sylow
79#1605—1971nomaximal, Sylow
80#544—1971no—
81#451—1972no—
82#395—1971no—
83#328—1972no—
84#324D19711yescharacteristic
85C22×Q8—1971no—
86C2×Q16—3942no—
87C4⋊Q8—3942no—
88#125—3942no—
89C22⋊Q8—3941no—
90C22⋊Q8—1972no—
91#117—3941no—
92C2×C4⋊C4—3941no—
93C2×C4⋊C4C2×D19711yescharacteristic
94#103—3942no—
95C22⋊C8C2×D19712yes—
96#95—3941no—
97#95—1971no—
98C2×Q8—7882no—
99C2×Q8—3941no—
100C2×Q8—3942no—
101C2×Q8—1972no—
102C22×C4—3941no—
103C22×C4—3941no—
104C22×C4D78811yescharacteristic
105C22×C4D78811yescharacteristic
106C22×C4C22×D19711yescharacteristic
107Q16—7884no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—7882no—
111C4⋊C4—3942no—
112C4⋊C4—3941no—
113C4⋊C4—3941no—
114C4⋊C4—3941no—
115C4⋊C4—3942no—
116C4⋊C4—1972no—
117C22⋊C4—3942no—
118C42—3942no—
119C23C2×D78811yescharacteristic
120Q8—7882no—
121Q8—7882no—
122Q8—15761no—
123Q8—3944no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—7881no—
130C2×C4—7881no—
131C2×C4—7881no—
132C2×C4—7881no—
133C2×C4—3942no—
134C2×C4—3942no—
135C2×C4—3942no—
136C2×C4—3942no—
137C2×C4Dic78812yescomplemented
138C2×C4D4×D19712yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#2026329611yescharacteristic
143C22C2×Dic78811yescharacteristic
144C22#2026335911yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—7882no—
149C4—7882no—
150C4—7882no—
151C4—7882no—
152C2—21no—
153C2—21no—
154C2#2716123711yescharacteristic
155C2#2716126411yescharacteristic
156C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#324×C197C2
socleC22×C197#120
prime radicalC23×C197C2×D4
hypercenter#324D197
hypocenterC197#1605
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
58. C2×C4×C197Q16semidirect
78. C197#1605semidirect
137. C2×C4Dic788semidirect

Maximal subgroups

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

subgroupindex
#271617102
#271612372
#271609992
#271608172
#324×C1972
#1605197