Finite Group Databasebeta

← #38743239

Subgroups — #38743239

Order 25216.

344 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 89 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#27161643C211yesmaximal, characteristic
2#27161085C212yesmaximal, complemented
3#27161053C211yesmaximal, characteristic, complemented
4#27160836C212yesmaximal, complemented
5#568×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7#20263322C412yescomplemented
8#20263315C412yescomplemented
9#20263300—22no—
10#20263294—22no—
11#20263285C2212yes—
12#20263285C2211yescharacteristic
13#20263248—24no—
14#20263241—22no—
15C2×#41×C197C2211yescharacteristic
16#131×C197—24no—
17C2×M16×C197—22no—
18C2×M16×C197C2212yes—
19C23×D197—21no—
20C2×D788—22no—
21C2×D788—21no—
22C2×D788—22no—
23C2×D788C2×C412yes—
24C2×C22⋊D197—24no—
25C4⋊Dic197—21no—
26C4⋊Dic197—22no—
27C22⋊Dic197—21no—
28C22⋊Dic197—22no—
29#15691335—24no—
30C4×Dic197—22no—
31C4×Dic197C2×C412yes—
32#41×C197—22no—
33#41×C197—22no—
34C2×Q8×C197C2×C411yescharacteristic
35C2×D4×C197C2×C411yescharacteristic
36C2×D4×C197D412yes—
37C22×C4×C197D412yes—
38C22×C4×C197C2311yescharacteristic
39M16×C197—24no—
40M16×C197—22no—
41M16×C197—22no—
42C2×C8×C197—24no—
43C2×C8×C197D414yescomplemented
44C22×D197—22no—
45C22×D197—42no—
46C22×D197—82no—
47D788—24no—
48D788—41no—
49D788—41no—
50C22⋊D197—84no—
51C2×Dic197—22no—
52C2×Dic197—42no—
53C23×C197C22⋊C412yes—
54C23×C197C2×D411yescharacteristic
55Q8×C197—22no—
56D4×C197—24no—
57D4×C197—22no—
58C2×C4×C197—22no—
59C2×C4×C197—22no—
60C2×C4×C197C22⋊C412yes—
61C2×C4×C197C22×C411yescharacteristic
62C2×C4×C197C2×D412yes—
63C2×C4×C197C2×D411yescharacteristic
64C2×C4×C197#4112yes—
65C8×C197—24no—
66C22×C197—22no—
67C22×C197—41no—
68C22×C197—42no—
69C22×C197C2×C22⋊C411yescharacteristic
70C22×C197#12112yes—
71C2×D197—44no—
72C2×D197—42no—
73C2×D197—84no—
74C4×C197—22no—
75C4×C197C4×D412yes—
76C4×C197#12011yescharacteristic
77C4×C197#12311yescharacteristic
78Dic197—42no—
79Dic197—82no—
80C2×C197—21no—
81C2×C197—22no—
82C2×C197—42no—
83C2×C197#38611yescharacteristic
84D197—82no—
85C197#191011yescharacteristic, complemented, Sylow
86#1910—1971nomaximal, Sylow
87#568D19711yescharacteristic, complemented
88#535—1971no—
89#418—1972no—
90#410—1971no—
91H(3,ℤ/4)—1972no—
92C2×#41C2×D19711yescharacteristic
93C22×D4—1971no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D19712yes—
97#127—1972no—
98#124—1972no—
99#121—3942no—
100#120—3942no—
101#117—1972no—
102#117—1971no—
103#102—3944no—
104#99—3942no—
105C24—3941no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D19711yescharacteristic
109C2×D4—3944no—
110C2×D4—3941no—
111C2×D4—3942no—
112C2×D4—3942no—
113C2×D4—1972no—
114C2×D4C4×D19711yescharacteristic
115C2×D4D78812yes—
116C22×C4C22⋊D19712yes—
117C22×C4C22×D19711yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D19714yescomplemented
123C4⋊C4—3942no—
124C4⋊C4—3941no—
125C22⋊C4—3942no—
126C22⋊C4—3941no—
127C22⋊C4—3944no—
128C42—3942no—
129C42—1972no—
130C23—7882no—
131C23—15762no—
132C23—3942no—
133C23#1569133512yes—
134C23C2×D78811yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—7881no—
139D4—7881no—
140D4—15764no—
141D4—3944no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—7882no—
145C2×C4—3942no—
146C2×C4#1569133512yes—
147C2×C4C2×C4×D19711yescharacteristic
148C2×C4C2×C22⋊D19712yes—
149C2×C4C2×C22⋊D19711yescharacteristic
150C2×C4#1569135812yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—7884no—
156C22—7882no—
157C22—15764no—
158C22C2×#1569133511yescharacteristic
159C22#2026330012yes—
160C4—22no—
161C4—7882no—
162C4—15762no—
163C4C4×C22⋊D19712yes—
164C4#2026329511yescharacteristic
165C4#2026330911yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—15762no—
170C2#2716095811yescharacteristic
subgroupG/H
centerC2#27160958
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#568×C197C2
socleC2×C197#386
prime radical#27161643C2
cosocleC22×C4×C197C23
hypercenter#568D197
hypocenterC197#1910
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27161085C2semidirect
3. #27161053C2semidirect
4. #27160836C2semidirect
5. #568×C197C2semidirect
7. #20263322C4semidirect
8. #20263315C4semidirect
43. C2×C8×C197D4semidirect
85. C197#1910semidirect
87. #568D197semidirect
122. C2×C8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271616432
#271610852
#271610532
#271608362
#568×C1972
#1910197