Finite Group Databasebeta

← #38743259

Subgroups — #38743259

Order 25216.

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

normal subgroups only

This group has 340 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#27161063C211yesmaximal, characteristic, complemented
3C2×#20263242C212yesmaximal, complemented
4#27160851C212yesmaximal, complemented
5#568×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7C2×#15691342C2212yes—
8#20263322—21no—
9#20263315—21no—
10#20263300—22no—
11#20263294C414yes—
12#20263285C2211yescharacteristic
13#20263245—21no—
14#20263243—21no—
15#20263242—24no—
16C2×#41×C197C2211yescharacteristic
17#131×C197—24no—
18C2×M16×C197—22no—
19C2×M16×C197C2212yes—
20C23×D197C2×C412yes—
21C2×D788—22no—
22C2×D788—21no—
23C2×D788—21no—
24C2×D788—22no—
25C2×C22⋊D197—24no—
26#15691342—22no—
27#15691342—22no—
28C2×#13096802—22no—
29C4⋊Dic197—21no—
30C22⋊Dic197C2×C412yes—
31#15691335—24no—
32C4×Dic197—21no—
33#41×C197—22no—
34#41×C197—22no—
35C2×Q8×C197D411yescharacteristic
36C2×D4×C197C2×C412yes—
37C2×D4×C197D411yescharacteristic
38C22×C4×C197D412yes—
39C22×C4×C197C2311yescharacteristic
40M16×C197—24no—
41M16×C197—22no—
42M16×C197D414yescomplemented
43C2×C8×C197—24no—
44C2×C8×C197—22no—
45C22×D197—22no—
46C22×D197—24no—
47C22×D197—82no—
48D788—41no—
49D788—42no—
50D788—41no—
51C22⋊D197—84no—
52C2×Dic197—22no—
53#13096802—42no—
54C23×C197C22⋊C412yes—
55C23×C197C22×C411yescharacteristic
56Q8×C197—22no—
57D4×C197—24no—
58D4×C197—22no—
59C2×C4×C197—22no—
60C2×C4×C197—22no—
61C2×C4×C197C22⋊C412yes—
62C2×C4×C197C2×D412yes—
63C2×C4×C197C2×D411yescharacteristic
64C2×C4×C197C2×D411yescharacteristic
65C2×C4×C197#4112yes—
66C8×C197—24no—
67C22×C197—22no—
68C22×C197—41no—
69C22×C197—42no—
70C22×C197C2×C22⋊C411yescharacteristic
71C22×C197C4×D412yes—
72C2×D197—44no—
73C2×D197—42no—
74C2×D197—84no—
75C4×C197—22no—
76C4×C197#12011yescharacteristic
77C4×C197#12112yes—
78C4×C197#12311yescharacteristic
79Dic197—42no—
80C2×C197—21no—
81C2×C197—22no—
82C2×C197—42no—
83C2×C197#38611yescharacteristic
84D197—82no—
85C197#191411yescharacteristic, complemented, Sylow
86#1914—1971nomaximal, Sylow
87#568D19711yescharacteristic, complemented
88#535—1971no—
89#413—1971no—
90C2×#100—1972no—
91#344—1972no—
92C2×#41C2×D19711yescharacteristic
93C22×D4—1971no—
94#131—24no—
95C2×M16—22no—
96C2×M16—1972no—
97C2×M16C2×D19712yes—
98#127—3941no—
99#124—3941no—
100#121—3942no—
101#120—1974no—
102#117—1971no—
103#101—3941no—
104#100—3941no—
105#100—3944no—
106C24—1972no—
107#41—22no—
108#41—22no—
109C2×Q8D78811yescharacteristic
110C2×D4—3944no—
111C2×D4—3942no—
112C2×D4—3941no—
113C2×D4—3941no—
114C2×D4—3942no—
115C2×D4C4×D19712yes—
116C2×D4D78811yescharacteristic
117C22×C4C22⋊D19712yes—
118C22×C4C22×D19711yescharacteristic
119M16—24no—
120M16—22no—
121M16—3942no—
122M16—3942no—
123M16C22⋊D19714yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—3942no—
127C4⋊C4—3941no—
128C22⋊C4—3944no—
129C22⋊C4—1972no—
130C42—3941no—
131C23—15762no—
132C23—3942no—
133C23—3944no—
134C23#1569133512yes—
135C23C2×C4×D19711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—7881no—
140D4—7882no—
141D4—7881no—
142D4—15764no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—3942no—
146C2×C4#1569133512yes—
147C2×C4C2×C22⋊D19712yes—
148C2×C4C2×C22⋊D19711yescharacteristic
149C2×C4C2×D78811yescharacteristic
150C2×C4#1569135812yes—
151C8—24no—
152C8—7882no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—7884no—
157C22—7882no—
158C22—15764no—
159C22C2×#1569133511yescharacteristic
160C22C4×C22⋊D19712yes—
161C4—22no—
162C4—7882no—
163C4#2026329511yescharacteristic
164C4#2026330012yes—
165C4#2026330911yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—15762no—
170C2#2716095811yescharacteristic
subgroupG/H
centerC2#27160958
commutatorC23×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#568×C197C2
socleC2×C197#386
prime radical#27161643C2
cosocleC22×C4×C197C23
hypercenter#568D197
hypocenterC197#1914
LCSG ▷ C23×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #27161063C2semidirect
3. C2×#20263242C2semidirect
4. #27160851C2semidirect
5. #568×C197C2semidirect
42. M16×C197D4semidirect
85. C197#1914semidirect
87. #568D197semidirect
123. M16C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
#271616432
#271610632
C2×#202632422
#271608512
#568×C1972
#1914197