Finite Group Databasebeta

← #38753004

Subgroups — #38753004

Order 25216.

382 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 121 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#27161684C211yesmaximal, characteristic
2#27161556C211yesmaximal, characteristic, complemented
3#27161404C212yesmaximal, complemented
4#27161387C212yesmaximal, complemented
5#27161280C212yesmaximal, complemented
6#27161220C212yesmaximal, complemented
7#27161181C212yesmaximal, complemented
8C4×#15691347C212yesmaximal, complemented
9#433×C197C211yesmaximal, characteristic, complemented
10C22×Dic394C2211yescharacteristic
11C2×#15691347C2212yes—
12C2×#15691347C2212yes—
13#20263323C2211yescharacteristic
14#20263318C2211yescharacteristic
15#20263311—22no—
16#20263300C2212yes—
17#20263299—22no—
18C4×Dic394C2212yes—
19C4×Dic394C2212yes—
20#20263291—22no—
21C4×D788C2212yes—
22#20263285C2211yescharacteristic
23#20263282—22no—
24#20263260C2212yes—
25#20263258C2212yes—
26#20263251C2212yes—
27#20263251C2212yes—
28#20263248C2212yes—
29#20263248C2212yes—
30#117×C197C2211yescharacteristic
31C4⋊C8×C197C2212yes—
32C22⋊C8×C197C2212yes—
33#97×C197C2211yescharacteristic
34C4×C8×C197C2211yescharacteristic
35C2×Dic394—24no—
36C2×Dic394—21no—
37C2×Dic394C2312yes—
38C2×D788—22no—
39C2×D788C2312yes—
40C2×C22⋊D197—22no—
41C22×Dic197—21no—
42C2×C4×D197—22no—
43#15691347—24no—
44#15691347—24no—
45C4⋊Dic197C2312yes—
46C4⋊Dic197C2312yes—
47#15691337—22no—
48#15691337—22no—
49#15691337—22no—
50C22⋊Dic197—21no—
51#15691335—22no—
52#15691335—22no—
53C4×Dic197—22no—
54C4×Dic197—22no—
55C4×Dic197—21no—
56C22×C4×C197C2311yescharacteristic
57C2×C8×C197C2312yes—
58C2×C8×C197C2312yes—
59C4⋊C4×C197D412yes—
60C22⋊C4×C197D412yes—
61C42×C197C2311yescharacteristic
62C42×C197C2311yescharacteristic
63C22×D197—22no—
64Dic394—24no—
65Dic394—22no—
66Dic394#4114yes—
67D788—22no—
68D788—22no—
69D788—82no—
70C22⋊D197—82no—
71C2×Dic197—24no—
72C2×Dic197—22no—
73C2×Dic197—22no—
74C2×Dic197—22no—
75C4×D197—82no—
76C23×C197C2×D411yescharacteristic
77C2×C4×C197—22no—
78C2×C4×C197C2×D412yes—
79C2×C4×C197C2×D412yes—
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2411yescharacteristic
82C8×C197—22no—
83C8×C197—22no—
84C22×C197—21no—
85C22×C197—22no—
86C22×C197C22×D411yescharacteristic
87C2×D197—42no—
88C2×D197—42no—
89C2×D197—42no—
90C4×C197—21no—
91C4×C197—22no—
92C4×C197—22no—
93C4×C197C2×#4112yes—
94Dic197—24no—
95Dic197—42no—
96Dic197—42no—
97C2×C197—41no—
98C2×C197#52511yescharacteristic
99C2×C197#57712yes—
100D197—82no—
101C197#312611yescharacteristic, complemented, Sylow
102#3126—1971nomaximal, Sylow
103#540—1971no—
104#519—1971no—
105#482—1972no—
106#479—1972no—
107#460—1972no—
108#448—1972no—
109#440—1972no—
110C4×SD16—1972no—
111#433D19711yescharacteristic, complemented
112C22×Q8—1971no—
113C2×SD16—1972no—
114C2×SD16—1972no—
115#127—1971no—
116#125—1971no—
117#123—3942no—
118#121—3942no—
119#121—1972no—
120C4×Q8—3942no—
121C4×Q8—1972no—
122C4×Q8—1972no—
123C4×D4—1972no—
124#117—3942no—
125#117—1971no—
126#117C2×D19711yescharacteristic
127#107—1972no—
128#106—1972no—
129C4⋊C8C2×D19712yes—
130#103—1972no—
131#103—1972no—
132#102—1972no—
133#102—1972no—
134C22⋊C8C2×D19712yes—
135#97C2×D19711yescharacteristic
136C4×C8C2×D19711yescharacteristic
137C2×Q8—3944no—
138C2×Q8—3941no—
139C2×Q8—1972no—
140C2×D4—3942no—
141C2×D4—3942no—
142C2×D4—1972no—
143C22×C4—3942no—
144C22×C4—3941no—
145C22×C4C22×D19711yescharacteristic
146SD16—3944no—
147SD16—3944no—
148C2×C8C22×D19712yes—
149C2×C8C22×D19712yes—
150C4⋊C4—3942no—
151C4⋊C4—3942no—
152C4⋊C4—3942no—
153C4⋊C4—1972no—
154C4⋊C4—1972no—
155C4⋊C4D78812yes—
156C22⋊C4—3941no—
157C22⋊C4—3942no—
158C22⋊C4—3942no—
159C22⋊C4D78812yes—
160C42—3942no—
161C42—3942no—
162C42—3941no—
163C42C22×D19711yescharacteristic
164C42C22×D19711yescharacteristic
165C23—3942no—
166C23C2×D78811yescharacteristic
167Q8—3944no—
168Q8—3942no—
169Q8—1974no—
170D4—15762no—
171D4—15762no—
172D4—3942no—
173D4—3942no—
174C2×C4—22no—
175C2×C4—15762no—
176C2×C4—3942no—
177C2×C4—3942no—
178C2×C4—3944no—
179C2×C4—3942no—
180C2×C4C2×D78812yes—
181C2×C4C2×D78812yes—
182C2×C4C2×D78811yescharacteristic
183C2×C4C23×D19711yescharacteristic
184C8—22no—
185C8—22no—
186C22—22no—
187C22—21no—
188C22—7882no—
189C22—7882no—
190C22—7882no—
191C22C22×D78811yescharacteristic
192C4—22no—
193C4—22no—
194C4—21no—
195C4—7882no—
196C4—7882no—
197C4—3944no—
198C4#41×D19712yes—
199C2—41no—
200C2—15762no—
201C2#2716159111yescharacteristic
202C2#2716189112yes—
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#433×C197C2
socleC22×C197C22×D4
prime radical#27161684C2
hypercenter#433D197
hypocenterC197#3126
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #27161556C2semidirect
3. #27161404C2semidirect
4. #27161387C2semidirect
5. #27161280C2semidirect
6. #27161220C2semidirect
7. #27161181C2semidirect
8. C4×#15691347C2semidirect
9. #433×C197C2semidirect
101. C197#3126semidirect
111. #433D197semidirect

Maximal subgroups

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

subgroupindex
#271616842
#271615562
#271614042
#271613872
#271612802
#271612202
#271611812
C4×#156913472
#433×C1972
#3126197