Finite Group Databasebeta

← #38751239

Subgroups — #38751239

Order 25216.

778 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 253 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×#20263372C211yesmaximal, characteristic
2#27161562C211yesmaximal, characteristic, complemented
3#27161085C213yesmaximal, complemented
4#27161081C213yesmaximal, complemented
5#27161076C213yesmaximal, complemented
6C2×#20263248C213yesmaximal, complemented
7C2×#131×C197C211yesmaximal, characteristic, complemented
8#20263372—24no—
9#20263372C418yescomplemented
10C2×#15691359—21no—
11C2×#15691358C2213yes—
12C22×D788C2213yes—
13C2×D4×D197—23no—
14Q8×Dic197—21no—
15D4×Dic197—23no—
16#20263285C2213yescomplemented
17C2×C4⋊Dic197C2213yescomplemented
18#20263251C2211yescharacteristic, complemented
19#20263251C2213yescomplemented
20#20263248C2213yescomplemented
21#20263248C2216yescomplemented
22#20263248C2213yescomplemented
23C2×#41×C197C2211yescharacteristic
24#131×C197—24no—
25C2×M16×C197C2213yescomplemented
26C22×C8×C197C2213yescomplemented
27C23×D197—23no—
28#15691359—28no—
29#15691358—26no—
30#15691358C2×C4112yescomplemented
31C2×Dic394C2311yescharacteristic
32C2×D788—23no—
33C2×D788—26no—
34C2×D788C2×C4112yescomplemented
35C2×D788C2313yes—
36C2×C22⋊D197—23no—
37D4×D197—224no—
38C22×Dic197—23no—
39C2×C4×D197—23no—
40C4⋊Dic197—23no—
41C4⋊Dic197C2313yes—
42C4⋊Dic197C2311yescharacteristic
43C22⋊Dic197—23no—
44C4×Dic197—23no—
45#41×C197C2×C414yes—
46#41×C197D414yes—
47C2×Q8×C197D411yescharacteristic
48C2×D4×C197D413yes—
49C22×C4×C197C2313yes—
50M16×C197—26no—
51C2×C8×C197—26no—
52C2×C8×C197C2311yescharacteristic
53C2×C8×C197C2313yes—
54C22×D197—23no—
55C22×D197—212no—
56C22×D197—83no—
57C22×D197—83no—
58Dic394—21no—
59Dic394C22×C412yes—
60D788—212no—
61D788—23no—
62D788C22×C416yes—
63C22⋊D197—212no—
64C2×Dic197—21no—
65C2×Dic197—21no—
66C2×Dic197—23no—
67C2×Dic197—83no—
68C4×D197—212no—
69C23×C197C2×D413yes—
70Q8×C197C22⋊C414yes—
71D4×C197C22⋊C4112yes—
72C2×C4×C197C22×C416yes—
73C2×C4×C197C2×D416yes—
74C2×C4×C197C2×D413yes—
75C2×C4×C197C2411yescharacteristic
76C8×C197—21no—
77C8×C197—23no—
78C22×C197—23no—
79C22×C197—23no—
80C22×C197C2×C22⋊C416yes—
81C22×C197C22×D411yescharacteristic
82C2×D197—26no—
83C2×D197—43no—
84C2×D197—43no—
85C2×D197—812no—
86C4×C197C2×C22⋊C416yes—
87C4×C197C23×C411yescharacteristic
88C4×C197C22×D411yescharacteristic
89Dic197—22no—
90Dic197—43no—
91Dic197—41no—
92C2×C197—26no—
93C2×C197C22×C22⋊C411yescharacteristic
94C2×C197#57611yescharacteristic
95C2×C197#57711yescharacteristic
96D197—46no—
97C197#289711yescharacteristic, complemented, Sylow
98#2897—1971nomaximal, Sylow
99C2×H(5,𝔽2)—1971no—
100C2×#131D19711yescharacteristic, complemented
101#520—1971no—
102#418—1973no—
103#417—1973no—
104#416—1973no—
105C2×#102—1973no—
106H(5,𝔽2)—3944no—
107H(5,𝔽2)—1978no—
108C2×#41—3941no—
109C2×#41—1973no—
110C2×#41C2×D19711yescharacteristic
111C22×D4—3943no—
112C22×D4—1973no—
113#131—24no—
114C2×M16C2×D19713yescomplemented
115C22×C8C2×D19713yescomplemented
116C4×Q8—3941no—
117C4×D4—3943no—
118#117—1973no—
119C2×C4⋊C4—1973no—
120#103—1971no—
121#103—1973no—
122#102—1973no—
123#102—1976no—
124#102—1973no—
125C24—3943no—
126#41—3948no—
127#41—3946no—
128#41—19712no—
129#41C4×D19714yes—
130#41C22⋊D19714yes—
131C2×Q8—1971no—
132C2×Q8D78811yescharacteristic
133C2×D4—3943no—
134C2×D4—39424no—
135C2×D4—3946no—
136C2×D4—3943no—
137C2×D4—19712no—
138C2×D4—1973no—
139C2×D4D78813yes—
140C22×C4—3943no—
141C22×C4—3943no—
142C22×C4C22×D19713yes—
143M16—26no—
144C2×C8—26no—
145C2×C8C22×D19711yescharacteristic
146C2×C8C22×D19713yes—
147C4⋊C4—3943no—
148C4⋊C4—1973no—
149C4⋊C4—1971no—
150C22⋊C4—3943no—
151C42—3943no—
152C23—15763no—
153C23—15763no—
154C23—3943no—
155C23—39412no—
156C23C2×D78813yes—
157Q8—3941no—
158Q8—1972no—
159Q8#1569133514yes—
160D4—39412no—
161D4—39412no—
162D4—3943no—
163D4—1976no—
164D4#15691335112yes—
165C2×C4—15763no—
166C2×C4—3941no—
167C2×C4—3943no—
168C2×C4—3941no—
169C2×C4—39412no—
170C2×C4C2×C4×D19716yes—
171C2×C4C2×C22⋊D19716yes—
172C2×C4C2×D78813yes—
173C2×C4C23×D19711yescharacteristic
174C8—21no—
175C8—23no—
176C22—23no—
177C22—23no—
178C22—7883no—
179C22—7883no—
180C22—157612no—
181C22—3946no—
182C22C2×#1569133516yes—
183C22C22×D78811yescharacteristic
184C4—7883no—
185C4—7881no—
186C4—3942no—
187C4C2×#1569133516yes—
188C4C22×C4×D19711yescharacteristic
189C4C22×C22⋊D19711yescharacteristic
190C2—26no—
191C2—7886no—
192C2C22×#1569133511yescharacteristic
193C2#2716188411yescharacteristic
194C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutatorC4×C197C23×C4
FrattiniC2×C4C23×D197
FittingC2×#131×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263372C2
cosocleC2×C4×C197C24
hypercenterC2×#131D197
hypocenterC197#2897
LCSG ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
2. #27161562C2semidirect
3. #27161085C2semidirect
4. #27161081C2semidirect
5. #27161076C2semidirect
6. C2×#20263248C2semidirect
7. C2×#131×C197C2semidirect
9. #20263372C4semidirect
16. #20263285C22semidirect
17. C2×C4⋊Dic197C22semidirect
18. #20263251C22semidirect
19. #20263251C22semidirect
20. #20263248C22semidirect
21. #20263248C22semidirect
22. #20263248C22semidirect
25. C2×M16×C197C22semidirect
26. C22×C8×C197C22semidirect
30. #15691358C2×C4semidirect
34. C2×D788C2×C4semidirect
97. C197#2897semidirect
100. C2×#131D197semidirect
114. C2×M16C2×D197semidirect
115. C22×C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633722
#271615622
#271610852
#271610812
#271610762
C2×#202632482
C2×#131×C1972
#2897197