Finite Group Databasebeta

← #38752643

Subgroups — #38752643

Order 25216.

414 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 133 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#27161590C211yesmaximal, characteristic
2#27161420C214yesmaximal, complemented
3#27161403C214yesmaximal, complemented
4#27161119C212yesmaximal, complemented
5#27161078C212yesmaximal, complemented
6#27161043C211yesmaximal, characteristic, complemented
7C2×#117×C197C211yesmaximal, characteristic, complemented
8C2×#15691358C2211yescharacteristic
9C2×#15691342C2212yes—
10#20263309—21no—
11#20263305C2212yes—
12#20263300C2212yes—
13#20263295—21no—
14C4×C22⋊D197—22no—
15#20263285C2211yescharacteristic
16#20263259C2214yescomplemented
17#20263257C2214yescomplemented
18#20263249C2214yescomplemented
19#20263246C2214yescomplemented
20C22⋊#13096802C2214yes—
21C23×C4×C197C2211yescharacteristic
22#117×C197—22no—
23#117×C197C2214yes—
24C2×C4⋊C4×C197C2212yes—
25C2×C22⋊C4×C197—21no—
26C2×C42×C197—21no—
27#15691358—24no—
28C2×Dic394C2311yescharacteristic
29C2×D788C2311yescharacteristic
30C2×C22⋊D197—22no—
31C2×C22⋊D197—21no—
32C2×C4×D197—21no—
33#15691342—24no—
34C2×#13096802C2314yes—
35C4⋊Dic197C2312yes—
36#15691337—22no—
37C22⋊Dic197—21no—
38C22⋊Dic197—22no—
39#15691335—22no—
40C4×Dic197—21no—
41C24×C197D411yescharacteristic
42C22×C4×C197—24no—
43C22×C4×C197—22no—
44C22×C4×C197D411yescharacteristic
45C22×C4×C197D412yes—
46C22×C4×C197C2312yes—
47C22×C4×C197C2311yescharacteristic
48C4⋊C4×C197—22no—
49C4⋊C4×C197C2314yes—
50C22⋊C4×C197—24no—
51C42×C197—24no—
52C22×D197—21no—
53Dic394—22no—
54D788—22no—
55C22⋊D197—44no—
56C22⋊D197—42no—
57C2×Dic197—22no—
58C2×Dic197—21no—
59C4×D197—42no—
60#13096802—24no—
61C23×C197—22no—
62C23×C197—42no—
63C23×C197C2×D412yes—
64C23×C197C2×D411yescharacteristic
65C2×C4×C197—22no—
66C2×C4×C197—24no—
67C2×C4×C197—24no—
68C2×C4×C197—24no—
69C2×C4×C197—44no—
70C2×C4×C197C2×D412yes—
71C2×C4×C197C2×D411yescharacteristic
72C2×C4×C197#4114yes—
73C2×C4×C197C2411yescharacteristic
74C22×C197—22no—
75C22×C197—22no—
76C22×C197—22no—
77C22×C197—22no—
78C22×C197—44no—
79C22×C197#12312yes—
80C22×C197C22×D411yescharacteristic
81C2×D197—41no—
82C2×D197—42no—
83C4×C197—22no—
84C4×C197—44no—
85C4×C197#12312yes—
86C4×C197C2×#4112yes—
87Dic197—41no—
88Dic197—42no—
89C2×C197—22no—
90C2×C197—42no—
91C2×C197C2×#12311yescharacteristic
92C2×C197#57512yes—
93D197—81no—
94C197#309311yescharacteristic, complemented, Sylow
95#3093—1971nomaximal, Sylow
96#525—1971no—
97C2×#117D19711yescharacteristic, complemented
98#485—1974no—
99#482—1974no—
100#428—1972no—
101#417—1972no—
102#407—1971no—
103C2×#41—1971no—
104C23×C4C2×D19711yescharacteristic
105C2×M16—1972no—
106#123—3941no—
107C22⋊Q8—1972no—
108#121—1972no—
109#120—3941no—
110C4×D4—3942no—
111#117—22no—
112#117—1971no—
113#117C2×D19714yes—
114C2×C4⋊C4C2×D19712yes—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117#107—1974no—
118#106—1974no—
119#103—1974no—
120#102—1974no—
121C22⋊C8—1974no—
122C24C22⋊D19711yescharacteristic
123#41—3944no—
124C2×Q8—1971no—
125C2×D4—3942no—
126C2×D4—3941no—
127C2×D4—1971no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—3941no—
131C22×C4C22⋊D19711yescharacteristic
132C22×C4C22⋊D19712yes—
133C22×C4C22×D19712yes—
134C22×C4C22×D19711yescharacteristic
135M16—3944no—
136C2×C8—1974no—
137C4⋊C4—22no—
138C4⋊C4—3942no—
139C4⋊C4—1972no—
140C4⋊C4C22×D19714yes—
141C22⋊C4—24no—
142C22⋊C4—3941no—
143C22⋊C4—3942no—
144C22⋊C4—3942no—
145C42—24no—
146C42—3941no—
147C23—22no—
148C23—42no—
149C23—3941no—
150C23C2×C22⋊D19712yes—
151C23C2×C22⋊D19711yescharacteristic
152Q8—3942no—
153D4—7884no—
154D4—7882no—
155D4—3942no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—44no—
161C2×C4—7882no—
162C2×C4—3942no—
163C2×C4—3941no—
164C2×C4C2×C22⋊D19712yes—
165C2×C4C2×C22⋊D19711yescharacteristic
166C2×C4#1569135814yes—
167C2×C4C23×D19711yescharacteristic
168C8—3944no—
169C22—22no—
170C22—22no—
171C22—22no—
172C22—22no—
173C22—44no—
174C22—7882no—
175C22—7881no—
176C22#2026330912yes—
177C22C22×C22⋊D19711yescharacteristic
178C4—22no—
179C4—44no—
180C4—7882no—
181C4—7881no—
182C4#2026330912yes—
183C4C2×#1569135812yes—
184C2—22no—
185C2—42no—
186C2—15761no—
187C2C2×#2026330911yescharacteristic
188C2#2716187712yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#117×C197C2
socleC22×C197C22×D4
prime radical#27161590C2
hypercenterC2×#117D197
hypocenterC197#3093
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #27161420C2semidirect
3. #27161403C2semidirect
4. #27161119C2semidirect
5. #27161078C2semidirect
6. #27161043C2semidirect
7. C2×#117×C197C2semidirect
16. #20263259C22semidirect
17. #20263257C22semidirect
18. #20263249C22semidirect
19. #20263246C22semidirect
94. C197#3093semidirect
97. C2×#117D197semidirect

Maximal subgroups

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

subgroupindex
#271615902
#271614202
#271614032
#271611192
#271610782
#271610432
C2×#117×C1972
#3093197