Finite Group Databasebeta

← #38742678

Subgroups — #38742678

Order 25216.

548 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 97 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#27161915C211yesmaximal, characteristic
2#27161097C212yesmaximal, complemented
3#27161073C212yesmaximal, complemented
4#27161046C211yesmaximal, characteristic, complemented
5#410×C197C211yesmaximal, characteristic, complemented
6#20263374C412yescomplemented
7#20263373—22no—
8#20263372—22no—
9#20263371—22no—
10#20263371C412yescomplemented
11C2×#15691358—22no—
12C2×#15691358C2212yes—
13#41×D197—24no—
14C2×#15691342C2211yescharacteristic, complemented
15C22×#13096802C2211yescharacteristic, complemented
16#20263254—24no—
17#20263249—22no—
18#20263246—22no—
19C22⋊#13096802—22no—
20C2×#41×C197C2211yescharacteristic
21#117×C197C2212yescomplemented
22#99×C197—22no—
23#15691359—24no—
24#15691358—24no—
25#15691358—24no—
26#15691358—24no—
27#15691358—22no—
28#15691358—22no—
29#15691358C2×C414yes—
30#15691358D414yes—
31#15691357—28no—
32#15691357—24no—
33C2×Dic394—22no—
34C2×Dic394D412yes—
35Q8×D197—24no—
36C2×D788—22no—
37C2×D788D412yes—
38C2×C22⋊D197—24no—
39C2×C22⋊D197—22no—
40D4×D197—28no—
41D4×D197—24no—
42C2×C4×D197—24no—
43C2×C4×D197—22no—
44#15691342—22no—
45C2×#13096802—21no—
46C2×#13096802—21no—
47C2×#13096802—22no—
48#41×C197—22no—
49#41×C197—22no—
50C2×Q8×C197C2×C411yescharacteristic
51C2×D4×C197C2×C411yescharacteristic
52C2×D4×C197D412yes—
53C22×C4×C197D412yes—
54C22×C4×C197C2311yescharacteristic
55C4⋊C4×C197—22no—
56C22⋊C4×C197—22no—
57C42×C197—22no—
58C22×D197—24no—
59C22×D197—22no—
60Dic394—24no—
61Dic394—22no—
62Dic394C22⋊C414yes—
63D788—24no—
64D788—22no—
65D788C22⋊C414yes—
66C22⋊D197—28no—
67C22⋊D197—24no—
68C22⋊D197—24no—
69C2×Dic197—24no—
70C2×Dic197—22no—
71C4×D197—28no—
72C4×D197—24no—
73C4×D197—24no—
74#13096802—41no—
75#13096802—41no—
76C23×C197C22⋊C412yes—
77C23×C197C2×D411yescharacteristic
78Q8×C197—22no—
79D4×C197—24no—
80D4×C197—22no—
81C2×C4×C197—22no—
82C2×C4×C197—22no—
83C2×C4×C197—42no—
84C2×C4×C197C22⋊C412yes—
85C2×C4×C197C22×C411yescharacteristic
86C2×C4×C197C2×D412yes—
87C2×C4×C197C2×D412yes—
88C2×C4×C197C2×D411yescharacteristic
89C22×C197—22no—
90C22×C197—41no—
91C22×C197—42no—
92C22×C197C2×C22⋊C411yescharacteristic
93C22×C197#12012yes—
94C2×D197—24no—
95C2×D197—82no—
96C2×D197—84no—
97C4×C197—22no—
98C4×C197—82no—
99C4×C197C2×C22⋊C412yes—
100C4×C197#12011yescharacteristic
101C4×C197#12011yescharacteristic
102Dic197—24no—
103C2×C197—22no—
104C2×C197—21no—
105C2×C197—42no—
106C2×C197#37911yescharacteristic
107D197—44no—
108C197#179411yescharacteristic, complemented, Sylow
109#1794—1971nomaximal, Sylow
110#585—1971no—
111#421—1972no—
112#416—1972no—
113#410D19711yescharacteristic, complemented
114#408—1971no—
1152−1+4—3942no—
1162−1+4—1972no—
117H(5,𝔽2)—3942no—
118H(5,𝔽2)—3942no—
119H(5,𝔽2)—1972no—
120C2×#41—3944no—
121C2×#41—3942no—
122C2×#41—1972no—
123C2×#41C2×D19711yescharacteristic
124C2×M16—1971no—
125C22×C8—1971no—
126#117C2×D19712yescomplemented
127#104—3944no—
128#103—3942no—
129#102—3942no—
130#99—22no—
131C22⋊C8—3942no—
132#41—22no—
133#41—22no—
134#41—3944no—
135#41—3944no—
136#41—3948no—
137#41—3944no—
138#41—3944no—
139#41—3944no—
140#41—3942no—
141#41—3942no—
142#41—1974no—
143#41—1974no—
144C2×Q8—3942no—
145C2×Q8—3944no—
146C2×Q8—1972no—
147C2×Q8C4×D19711yescharacteristic
148C2×D4—3942no—
149C2×D4—3948no—
150C2×D4—3944no—
151C2×D4—3944no—
152C2×D4—3942no—
153C2×D4—1972no—
154C2×D4C4×D19711yescharacteristic
155C2×D4C22⋊D19712yes—
156C22×C4—3944no—
157C22×C4—3942no—
158C22×C4D78812yes—
159C22×C4C22×D19711yescharacteristic
160M16—3942no—
161C2×C8—3942no—
162C2×C8—3941no—
163C2×C8—3941no—
164C4⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C23—3944no—
168C23—3942no—
169C23#1569133512yes—
170C23C2×C22⋊D19711yescharacteristic
171Q8—22no—
172Q8—3944no—
173Q8—3942no—
174Q8—1974no—
175D4—24no—
176D4—22no—
177D4—3944no—
178D4—3948no—
179D4—3944no—
180D4—3944no—
181D4—3942no—
182D4—1974no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—3948no—
187C2×C4—3944no—
188C2×C4—3942no—
189C2×C4—3944no—
190C2×C4—3944no—
191C2×C4#1569133512yes—
192C2×C4C2×C4×D19711yescharacteristic
193C2×C4D4×D19712yes—
194C2×C4D4×D19712yes—
195C2×C4C2×D78811yescharacteristic
196C8—7881no—
197C8—7881no—
198C22—22no—
199C22—41no—
200C22—42no—
201C22—15762no—
202C22—15764no—
203C22—3944no—
204C22C2×#1569133511yescharacteristic
205C22#2026329412yes—
206C4—22no—
207C4—82no—
208C4—3944no—
209C4C22⋊C4×D19712yes—
210C4#2026329611yescharacteristic
211C4#2026329611yescharacteristic
212C2—22no—
213C2—21no—
214C2—42no—
215C2—7884no—
216C2#2716093411yescharacteristic
subgroupG/H
centerC4#20263296
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#410×C197C2
socleC2×C197#379
prime radical#27161915C2
cosocleC22×C4×C197C23
hypercenter#410D197
hypocenterC197#1794
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #27161097C2semidirect
3. #27161073C2semidirect
4. #27161046C2semidirect
5. #410×C197C2semidirect
6. #20263374C4semidirect
10. #20263371C4semidirect
14. C2×#15691342C22semidirect
15. C22×#13096802C22semidirect
21. #117×C197C22semidirect
108. C197#1794semidirect
113. #410D197semidirect
126. #117C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271619152
#271610972
#271610732
#271610462
#410×C1972
#1794197