Finite Group Databasebeta

← #38753166

Subgroups — #38753166

Order 25216.

552 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 145 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×#20263355C214yesmaximal, complemented
2#27161627C211yesmaximal, characteristic
3C4⋊Q8×D197C211yesmaximal, characteristic, complemented
4#27161470C214yesmaximal, complemented
5#27161459C211yesmaximal, characteristic, complemented
6#27161458C211yesmaximal, characteristic, complemented
7#27161451C211yesmaximal, characteristic, complemented
8C4×#15691341C211yesmaximal, characteristic, complemented
9#492×C197C211yesmaximal, characteristic, complemented
10C2×#15691357C2212yes—
11C2×Q8×D197C2212yes—
12#20263355—216no—
13C2×Dic788C2214yes—
14C2×#15691348C2214yes—
15C2×#15691346C2214yes—
16C2×#15691341C2212yes—
17#20263326C2211yescharacteristic
18#20263326C2211yescharacteristic
19#20263325—22no—
20#20263324C2212yes—
21#20263313C2212yes—
22#20263297—22no—
23D4×Dic197—22no—
24C4⋊C4×D197—22no—
25C42×D197C2211yescharacteristic
26#20263236C2212yes—
27C4×#13096802C2211yescharacteristic
28C2×SD16×C197C2214yes—
29C4⋊Q8×C197C2211yescharacteristic
30#127×C197C2211yescharacteristic
31C4×C8×C197C2211yescharacteristic
32#15691357—28no—
33C2×Dic394—22no—
34C2×Dic394C2312yes—
35C2×Dic394C2312yes—
36Q8×D197—28no—
37C2×C22⋊D197—22no—
38C22×Dic197—22no—
39C2×C4×D197—22no—
40C2×C4×D197D412yes—
41C2×C4×D197C2311yescharacteristic
42Dic788—28no—
43#15691348—28no—
44#15691346—28no—
45#15691341D418yescomplemented
46C2×#13096802C2312yes—
47C4⋊Dic197—22no—
48C4⋊Dic197—22no—
49#15691337—24no—
50C22⋊Dic197—24no—
51C4×Dic197D412yes—
52C4×Dic197C2311yescharacteristic
53C2×Q8×C197C2312yes—
54C2×D4×C197—22no—
55C2×D4×C197C2312yes—
56SD16×C197—28no—
57C2×C8×C197C2312yes—
58C4⋊C4×C197—22no—
59C42×C197C2311yescharacteristic
60C22×D197C2×D411yescharacteristic
61Dic394—24no—
62Dic394—24no—
63Dic394—44no—
64C22⋊D197—44no—
65C2×Dic197—22no—
66C2×Dic197—22no—
67C2×Dic197—44no—
68C2×Dic197C2×D412yes—
69C2×Dic197C2×D411yescharacteristic
70C4×D197—24no—
71C4×D197—44no—
72C4×D197C2×D414yes—
73#13096802C2×D414yes—
74C23×C197—22no—
75Q8×C197—24no—
76D4×C197—24no—
77D4×C197—44no—
78C2×C4×C197—22no—
79C2×C4×C197C2×D412yes—
80C2×C4×C197C2411yescharacteristic
81C8×C197C2×D414yes—
82C22×C197—42no—
83C22×C197—44no—
84C22×C197C22×D411yescharacteristic
85C2×D197—22no—
86C2×D197#12712yes—
87C4×C197—42no—
88C4×C197#13714yescomplemented
89C4×C197C22×D412yes—
90Dic197—22no—
91Dic197—42no—
92Dic197—42no—
93Dic197#12712yes—
94C2×C197—82no—
95C2×C197C2×#12711yescharacteristic
96C2×C197C2×#13712yes—
97D197—22no—
98C197#315211yescharacteristic, complemented, Sylow
99#3152—1971nomaximal, Sylow
100C2×#137—1974no—
101#532—1971no—
102C2×C4⋊Q8—1971no—
103#497—1974no—
104#494—1971no—
105#494—1971no—
106#492—1971no—
107#492D19711yescharacteristic, complemented
108C4×M16—1971no—
109C2×#41—1972no—
110C22×Q8—1972no—
111#137—39416no—
112C2×Q16—1974no—
113C2×Q16—1974no—
114C2×SD16—1974no—
115C2×SD16C2×D19714yes—
116C2×M16—1972no—
117C4⋊Q8—3942no—
118C4⋊Q8—1972no—
119C4⋊Q8—1971no—
120C4⋊Q8—1971no—
121C4⋊Q8C2×D19711yescharacteristic
122#127C2×D19711yescharacteristic
123#124—1972no—
124#121—3942no—
125C4×D4—3942no—
126C2×C4⋊C4—3942no—
127C2×C42—1971no—
128#97—1972no—
129C4×C8—1971no—
130C4×C8C2×D19711yescharacteristic
131#41—3948no—
132C2×Q8—3942no—
133C2×Q8—3948no—
134C2×Q8—1972no—
135C2×Q8—1972no—
136C2×Q8C22×D19712yes—
137C2×D4—22no—
138C2×D4—3942no—
139C2×D4C22×D19712yes—
140C22×C4—3942no—
141C22×C4—3942no—
142C22×C4—1972no—
143C22×C4—1971no—
144Q16—3948no—
145Q16—3948no—
146SD16—28no—
147SD16—3948no—
148M16—1978no—
149C2×C8—1972no—
150C2×C8C22×D19712yes—
151C4⋊C4—22no—
152C4⋊C4—3944no—
153C4⋊C4—3942no—
154C4⋊C4—3942no—
155C22⋊C4—3944no—
156C42—1972no—
157C42—1971no—
158C42C22×D19711yescharacteristic
159C23—22no—
160C23—1971no—
161Q8—24no—
162Q8—7884no—
163Q8—3944no—
164Q8—3944no—
165D4—24no—
166D4—44no—
167D4—7884no—
168C2×C4—22no—
169C2×C4—7884no—
170C2×C4—7884no—
171C2×C4—3942no—
172C2×C4—3942no—
173C2×C4—3944no—
174C2×C4—1972no—
175C2×C4—1971no—
176C2×C4—1974no—
177C2×C4D4×D19712yes—
178C2×C4C23×D19711yescharacteristic
179C8—1974no—
180C8D4×D19714yes—
181C22—44no—
182C22—42no—
183C22—3942no—
184C22—1972no—
185C22C2×D4×D19711yescharacteristic
186C4—42no—
187C4—7882no—
188C4—7882no—
189C4—3942no—
190C4—1972no—
191C4#2026335514yescomplemented
192C4C2×D4×D19712yes—
193C2—82no—
194C2—3942no—
195C2#127×D19711yescharacteristic
196C2C2×#2026335512yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#492×C197C2
socleC22×C197C22×D4
prime radical#27161627C2
hypercenter#492D197
hypocenterC197#3152
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#20263355C2semidirect
3. C4⋊Q8×D197C2semidirect
4. #27161470C2semidirect
5. #27161459C2semidirect
6. #27161458C2semidirect
7. #27161451C2semidirect
8. C4×#15691341C2semidirect
9. #492×C197C2semidirect
45. #15691341D4semidirect
88. C4×C197#137semidirect
98. C197#3152semidirect
107. #492D197semidirect
191. C4#20263355semidirect

Maximal subgroups

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

subgroupindex
C2×#202633552
#271616272
C4⋊Q8×D1972
#271614702
#271614592
#271614582
#271614512
C4×#156913412
#492×C1972
#3152197