Finite Group Databasebeta

← #38742813

Subgroups — #38742813

Order 25216.

262 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 81 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#41×Dic197C211yesmaximal, characteristic
2#27161063C211yesmaximal, characteristic
3#27160838C212yesmaximal, complemented
4#27160835C212yesmaximal, complemented
5#410×C197C211yesmaximal, characteristic
6C2×#15691342C2212yes—
7Q8×Dic197—21no—
8D4×Dic197—22no—
9D4×Dic197—21no—
10#20263285C412yescomplemented
11#20263285C2211yescharacteristic
12C2×C4×Dic197C412yescomplemented
13C2×C4×Dic197C2211yescharacteristic
14#20263245—21no—
15#20263243—21no—
16C2×#41×C197C2211yescharacteristic
17#117×C197C2212yes—
18#99×C197—22no—
19C22×Dic197—22no—
20C22×Dic197—21no—
21#15691342—22no—
22#15691342—22no—
23C2×#13096802—22no—
24C4⋊Dic197—22no—
25C4⋊Dic197—21no—
26C22⋊Dic197—22no—
27C22⋊Dic197—21no—
28C4×Dic197—22no—
29C4×Dic197—21no—
30C4×Dic197C2×C411yescharacteristic
31C4×Dic197C2×C412yes—
32C4×Dic197C2×C411yescharacteristic
33#41×C197—22no—
34#41×C197—22no—
35C2×Q8×C197D411yescharacteristic
36C2×D4×C197D412yes—
37C2×D4×C197D411yescharacteristic
38C22×C4×C197C2×C412yes—
39C22×C4×C197C2311yescharacteristic
40C4⋊C4×C197—22no—
41C22⋊C4×C197—22no—
42C42×C197—22no—
43C2×Dic197—21no—
44C2×Dic197—22no—
45C2×Dic197—21no—
46C2×Dic197—42no—
47C2×Dic197—44no—
48#13096802—42no—
49C23×C197C22⋊C412yes—
50C23×C197C2×D411yescharacteristic
51Q8×C197—22no—
52D4×C197—24no—
53D4×C197—22no—
54C2×C4×C197—22no—
55C2×C4×C197—22no—
56C2×C4×C197—42no—
57C2×C4×C197C22⋊C412yes—
58C2×C4×C197C22×C411yescharacteristic
59C2×C4×C197C2×D411yescharacteristic
60C2×C4×C197#4112yes—
61C2×C4×C197#4112yes—
62C22×C197—22no—
63C22×C197—41no—
64C22×C197—42no—
65C22×C197C2×C22⋊C411yescharacteristic
66C22×C197#12312yes—
67C4×C197—22no—
68C4×C197—82no—
69C4×C197#11711yescharacteristic
70C4×C197#11711yescharacteristic
71C4×C197#12312yes—
72Dic197—22no—
73Dic197—42no—
74Dic197—41no—
75C2×C197—21no—
76C2×C197—22no—
77C2×C197—42no—
78C2×C197#38111yescharacteristic
79C197#182711yescharacteristic, complemented, Sylow
80#1827—1971nomaximal, Sylow
81C4×#41—1971no—
82#413—1971no—
83#410D19711yescharacteristic
84#339—1972no—
85H(3,ℤ/4)—1972no—
86C2×#41C2×D19711yescharacteristic
87C2×M16—1972no—
88C4×Q8—3941no—
89C4×D4—3942no—
90C4×D4—3941no—
91#117—1972no—
92#117—1971no—
93#117C2×D19712yes—
94C2×C42—1972no—
95C2×C42—1971no—
96#101—3941no—
97#100—3941no—
98#99—22no—
99#41—22no—
100#41—22no—
101C2×Q8D78811yescharacteristic
102C2×D4C22⋊D19712yes—
103C2×D4D78811yescharacteristic
104C22×C4—3942no—
105C22×C4—3941no—
106C22×C4C4×D19712yes—
107C22×C4C22×D19711yescharacteristic
108M16—3942no—
109M16—3942no—
110C2×C8—3942no—
111C4⋊C4—22no—
112C4⋊C4—3942no—
113C4⋊C4—3941no—
114C22⋊C4—22no—
115C22⋊C4—3942no—
116C22⋊C4—3941no—
117C42—22no—
118C42—3942no—
119C42—3941no—
120C42—1971no—
121C42—1972no—
122C42—1971no—
123C23#1569133512yes—
124C23C2×C22⋊D19711yescharacteristic
125Q8—22no—
126D4—24no—
127D4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—7882no—
132C2×C4—7884no—
133C2×C4—3942no—
134C2×C4—3941no—
135C2×C4—3941no—
136C2×C4#1569133512yes—
137C2×C4C2×C4×D19711yescharacteristic
138C2×C4C2×D78811yescharacteristic
139C2×C4#1569135712yes—
140C2×C4#1569135712yes—
141C8—7882no—
142C22—22no—
143C22—41no—
144C22—42no—
145C22C2×#1569133511yescharacteristic
146C22#2026330712yes—
147C4—22no—
148C4—82no—
149C4—7881no—
150C4—7882no—
151C4—3942no—
152C4#2026328211yescharacteristic
153C4#2026328211yescharacteristic
154C4#2026331012yes—
155C2—22no—
156C2—21no—
157C2—42no—
158C2#2716094011yescharacteristic
subgroupG/H
centerC4#20263282
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#410×C197C2
socleC2×C197#381
prime radicalC2×#41×C197C22
cosocleC22×C4×C197C23
hypercenter#410D197
hypocenterC197#1827
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #27160838C2semidirect
4. #27160835C2semidirect
10. #20263285C4semidirect
12. C2×C4×Dic197C4semidirect
79. C197#1827semidirect

Maximal subgroups

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

subgroupindex
#41×Dic1972
#271610632
#271608382
#271608352
#410×C1972
#1827197