Finite Group Databasebeta

← #38753193

Subgroups — #38753193

Order 25216.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#20263352C212yesmaximal, complemented
2C2×#20263348C211yesmaximal, characteristic, complemented
3#27161646C212yesmaximal, complemented
4#27161467C214yesmaximal, complemented
5#27161462C212yesmaximal, complemented
6#27161454C212yesmaximal, complemented
7#27161034C211yesmaximal, characteristic, complemented
8C22×D8×C197C211yesmaximal, characteristic, complemented
9C2×#15691358C2212yes—
10#20263352—28no—
11#20263348—24no—
12C2×Dic788C2211yescharacteristic, complemented
13C2×#15691347C2212yescomplemented
14C2×#15691346C2214yescomplemented
15C2×D1576C2211yescharacteristic, complemented
16C2×#15691343C2214yescomplemented
17C2×#15691342C2211yescharacteristic
18#20263322C2214yescomplemented
19#20263313C2214yescomplemented
20#20263297—24no—
21#20263295—24no—
22#20263285C2211yescharacteristic
23#20263236C2212yescomplemented
24C8×Dic197C2212yescomplemented
25C22×D4×C197C2212yes—
26C2×D8×C197—24no—
27C2×D8×C197C2212yes—
28C2×D8×C197C2212yes—
29C22×C8×C197C2211yescharacteristic
30#15691358—24no—
31#15691358—24no—
32C2×Dic394C2312yes—
33C2×D788C2312yes—
34C2×C22⋊D197—28no—
35C2×C22⋊D197—22no—
36C2×C4×D197—22no—
37Dic788—22no—
38#15691347—24no—
39#15691346—28no—
40D1576—22no—
41#15691343—28no—
42#15691342D414yescomplemented
43C2×#13096802C2312yes—
44C4⋊Dic197D412yes—
45C22⋊Dic197—28no—
46C22⋊Dic197D412yes—
47C4×Dic197C2312yes—
48C24×C197—22no—
49C2×D4×C197—24no—
50C2×D4×C197—24no—
51C2×D4×C197—22no—
52C2×D4×C197C2314yes—
53C22×C4×C197C2311yescharacteristic
54D8×C197—24no—
55D8×C197—24no—
56C2×C8×C197D414yescomplemented
57C2×C8×C197C2311yescharacteristic
58C2×C8×C197C2311yescharacteristic
59C22×D197—22no—
60Dic394—22no—
61Dic394—22no—
62D788—22no—
63D788—22no—
64C22⋊D197—44no—
65C22⋊D197—88no—
66C2×Dic197—22no—
67C2×Dic197C2×D414yes—
68C4×D197—44no—
69#13096802C2×D414yes—
70C23×C197—24no—
71C23×C197—44no—
72C23×C197—84no—
73C23×C197C2×D411yescharacteristic
74D4×C197—24no—
75D4×C197—24no—
76D4×C197—44no—
77C2×C4×C197C2×D411yescharacteristic
78C2×C4×C197C2×D412yes—
79C2×C4×C197C2×D412yes—
80C2×C4×C197C2411yescharacteristic
81C8×C197C2×D412yes—
82C8×C197C2×D412yes—
83C22×C197—22no—
84C22×C197—44no—
85C22×C197—48no—
86C22×C197—88no—
87C22×C197#12712yes—
88C22×C197C22×D411yescharacteristic
89C2×D197—42no—
90C2×D197—44no—
91C4×C197#12712yes—
92C4×C197C22×D411yescharacteristic
93C4×C197C22×D411yescharacteristic
94Dic197—24no—
95Dic197—42no—
96C2×C197—22no—
97C2×C197—84no—
98C2×C197C2×#12711yescharacteristic
99C2×C197#57612yes—
100D197—82no—
101C197#315511yescharacteristic, complemented, Sylow
102#3155—1971nomaximal, Sylow
103C2×AGL(1,ℤ/8)—1972no—
104C2×#135—1971no—
105C22×D8D19711yescharacteristic, complemented
106#535—1972no—
107#496—1974no—
108#495—1972no—
109#493—1972no—
110#405—1971no—
111C2×#41—1972no—
112C22×D4C2×D19712yes—
113AGL(1,ℤ/8)—3948no—
114#135—3944no—
115C2×Q16—1971no—
116C2×SD16—1974no—
117C2×SD16—1972no—
118C2×D8—24no—
119C2×D8—1974no—
120C2×D8—1971no—
121C2×D8C2×D19712yes—
122C2×D8C2×D19712yes—
123C2×M16—1971no—
124C22×C8C2×D19711yescharacteristic
125#127—1974no—
126#124—1974no—
127#121—3944no—
128#120—3944no—
129#117—1971no—
130#97—1972no—
131C4×C8—1972no—
132C24—22no—
133#41—3944no—
134#41—3944no—
135C2×Q8—1972no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—24no—
139C2×D4—3942no—
140C2×D4—3948no—
141C2×D4—1972no—
142C2×D4C22×D19714yes—
143C22×C4—3942no—
144C22×C4C22×D19711yescharacteristic
145Q16—3942no—
146SD16—3948no—
147SD16—3944no—
148D8—24no—
149D8—24no—
150D8—3948no—
151D8—3942no—
152M16—1974no—
153C2×C8—1972no—
154C2×C8C22⋊D19714yescomplemented
155C2×C8C22×D19711yescharacteristic
156C2×C8C22×D19711yescharacteristic
157C4⋊C4—1972no—
158C22⋊C4—3948no—
159C22⋊C4—1972no—
160C42—1972no—
161C23—24no—
162C23—44no—
163C23—84no—
164C23—3942no—
165C23D4×D19711yescharacteristic
166Q8—3942no—
167Q8—3942no—
168D4—24no—
169D4—24no—
170D4—44no—
171D4—7884no—
172D4—15768no—
173D4—3942no—
174D4—3942no—
175C2×C4—7884no—
176C2×C4—3942no—
177C2×C4—1974no—
178C2×C4D4×D19711yescharacteristic
179C2×C4D4×D19712yes—
180C2×C4C2×C22⋊D19712yes—
181C2×C4C23×D19711yescharacteristic
182C8—1974no—
183C8C2×C22⋊D19712yes—
184C8C2×C22⋊D19712yes—
185C22—22no—
186C22—44no—
187C22—48no—
188C22—88no—
189C22—7882no—
190C22—7884no—
191C22#2026332212yes—
192C22C2×D4×D19711yescharacteristic
193C4—7882no—
194C4—3944no—
195C4#2026332212yes—
196C4C2×D4×D19711yescharacteristic
197C4C22×C22⋊D19711yescharacteristic
198C2—22no—
199C2—84no—
200C2—15762no—
201C2C2×#2026332211yescharacteristic
202C2#2716188012yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×D8×C197C2
socleC22×C197C22×D4
hypercenterC22×D8D197
hypocenterC197#3155
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20263352C2semidirect
2. C2×#20263348C2semidirect
3. #27161646C2semidirect
4. #27161467C2semidirect
5. #27161462C2semidirect
6. #27161454C2semidirect
7. #27161034C2semidirect
8. C22×D8×C197C2semidirect
12. C2×Dic788C22semidirect
13. C2×#15691347C22semidirect
14. C2×#15691346C22semidirect
15. C2×D1576C22semidirect
16. C2×#15691343C22semidirect
18. #20263322C22semidirect
19. #20263313C22semidirect
23. #20263236C22semidirect
24. C8×Dic197C22semidirect
42. #15691342D4semidirect
56. C2×C8×C197D4semidirect
101. C197#3155semidirect
105. C22×D8D197semidirect
154. C2×C8C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
C2×#202633482
#271616462
#271614672
#271614622
#271614542
#271610342
C22×D8×C1972
#3155197