Finite Group Databasebeta

← #38752072

Subgroups — #38752072

Order 25216.

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

normal subgroups only

This group has 762 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×#20263371C211yesmaximal, characteristic, complemented
2#27161879C212yesmaximal, complemented
3#27161646C211yesmaximal, characteristic, complemented
4#27161259C214yesmaximal, complemented
5#27161249C214yesmaximal, complemented
6#27161095C212yesmaximal, complemented
7C2×#100×C197C211yesmaximal, characteristic, complemented
8#20263371—24no—
9#20263371—22no—
10#20263371C2214yes—
11C2×#15691358C2212yes—
12C2×#15691358C2211yescharacteristic, complemented
13C2×#15691357—22no—
14C22×C22⋊D197—22no—
15C2×D4×D197—22no—
16#20263359—22no—
17#20263359C2214yescomplemented
18#20263354—22no—
19#20263354C2214yescomplemented
20#20263348—24no—
21#20263322C2212yescomplemented
22#20263313C2212yescomplemented
23#20263297—22no—
24#20263295—22no—
25#20263285C2211yescharacteristic, complemented
26#20263252C2218yescomplemented
27C22×D4×C197C2211yescharacteristic
28C2×M16×C197C2212yescomplemented
29#100×C197C2214yescomplemented
30C23×D197—22no—
31#15691358—22no—
32#15691358—22no—
33#15691358—22no—
34#15691358—22no—
35#15691358D414yescomplemented
36#15691358C2314yes—
37#15691357—28no—
38#15691357—28no—
39C2×Dic394D412yescomplemented
40C2×Dic394C2311yescharacteristic, complemented
41C2×D788D412yescomplemented
42C2×D788C2311yescharacteristic, complemented
43C2×C22⋊D197—24no—
44C2×C22⋊D197—28no—
45C2×C22⋊D197—22no—
46C2×C22⋊D197—28no—
47C2×C22⋊D197—21no—
48C2×C22⋊D197—24no—
49D4×D197—28no—
50D4×D197—28no—
51C22×Dic197—22no—
52C2×C4×D197—22no—
53C2×C4×D197—21no—
54Dic788—24no—
55#15691347—24no—
56#15691347—24no—
57D1576—24no—
58C4⋊Dic197—21no—
59C22⋊Dic197—21no—
60C22⋊Dic197—24no—
61C4×Dic197C2312yescomplemented
62C24×C197D412yes—
63C2×D4×C197—22no—
64C2×D4×C197—22no—
65C2×D4×C197D412yes—
66C2×D4×C197C2312yes—
67C22×C4×C197C2311yescharacteristic
68M16×C197—22no—
69M16×C197C2314yescomplemented
70C2×C8×C197—22no—
71C22×D197—22no—
72C22×D197—28no—
73C22×D197—21no—
74C22×D197—84no—
75Dic394—22no—
76Dic394—22no—
77Dic394C2×D414yes—
78D788—22no—
79D788—22no—
80D788C2×D414yes—
81C22⋊D197—216no—
82C22⋊D197—24no—
83C22⋊D197—24no—
84C22⋊D197—41no—
85C22⋊D197—41no—
86C22⋊D197—84no—
87C2×Dic197—28no—
88C2×Dic197—22no—
89C2×Dic197—22no—
90C2×Dic197—21no—
91C4×D197—24no—
92C4×D197—24no—
93C4×D197—42no—
94C23×C197—22no—
95C23×C197—22no—
96C23×C197—82no—
97C23×C197C2×D414yes—
98C23×C197C2×D411yescharacteristic
99D4×C197—24no—
100D4×C197—24no—
101C2×C4×C197C2×D412yes—
102C2×C4×C197C2×D412yes—
103C2×C4×C197C2×D411yescharacteristic
104C2×C4×C197C2411yescharacteristic
105C8×C197—24no—
106C22×C197—24no—
107C22×C197—41no—
108C22×C197—44no—
109C22×C197—44no—
110C22×C197—82no—
111C22×C197#12011yescharacteristic
112C22×C197#12011yescharacteristic
113C22×C197C22×D411yescharacteristic
114C2×D197—24no—
115C2×D197—44no—
116C2×D197—42no—
117C2×D197—41no—
118C2×D197—88no—
119C4×C197#12012yes—
120C4×C197C22×D412yes—
121Dic197—24no—
122Dic197—42no—
123Dic197—41no—
124C2×C197—21no—
125C2×C197—21no—
126C2×C197—21no—
127C2×C197—44no—
128C2×C197C2×#12011yescharacteristic
129D197—44no—
130D197—81no—
131C197#301911yescharacteristic, complemented, Sylow
132#3019—1971nomaximal, Sylow
133C2×H(5,𝔽2)—1971no—
134#575—1972no—
135#535—1971no—
136#455—1974no—
137#453—1974no—
138#421—1972no—
139C2×#100D19711yescharacteristic, complemented
140H(5,𝔽2)—3944no—
141H(5,𝔽2)—3942no—
142H(5,𝔽2)—1974no—
143C2×#41—3942no—
144C2×#41—1971no—
145C2×#41—1972no—
146C22×D4—3942no—
147C22×D4—3942no—
148C22×D4C2×D19711yescharacteristic
149#137—3942no—
150#137—1974no—
151AGL(1,ℤ/8)—3942no—
152AGL(1,ℤ/8)—1974no—
153#135—3944no—
154C2×M16C2×D19712yescomplemented
155#127—1972no—
156#124—1972no—
157#121—3942no—
158#120—3942no—
159#117—1971no—
160#104—1978no—
161#100C2×D19714yescomplemented
162C24—3942no—
163C24D78812yes—
164#41—3942no—
165#41—3942no—
166#41—3948no—
167#41—3948no—
168#41—3942no—
169#41—3942no—
170#41—1974no—
171#41—1974no—
172C2×Q8—1972no—
173C2×Q8—1971no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—3948no—
177C2×D4—3948no—
178C2×D4—3941no—
179C2×D4—3944no—
180C2×D4—3944no—
181C2×D4—3948no—
182C2×D4—3942no—
183C2×D4—3948no—
184C2×D4—1972no—
185C2×D4—1971no—
186C2×D4D78812yes—
187C2×D4C22×D19712yes—
188C22×C4—3941no—
189C22×C4—3942no—
190C22×C4—3942no—
191C22×C4C22×D19711yescharacteristic
192Q16—3944no—
193SD16—3944no—
194SD16—3944no—
195D8—3944no—
196M16—22no—
197M16C22×D19714yescomplemented
198C2×C8—22no—
199C4⋊C4—3941no—
200C22⋊C4—3941no—
201C22⋊C4—3944no—
202C42—1972no—
203C23—22no—
204C23—22no—
205C23—82no—
206C23—15764no—
207C23—3948no—
208C23—3941no—
209C23—3942no—
210C23C2×D78814yes—
211C23C2×D78811yescharacteristic
212Q8—3942no—
213Q8—3942no—
214Q8—1974no—
215D4—24no—
216D4—24no—
217D4—7881no—
218D4—7881no—
219D4—15764no—
220D4—39416no—
221D4—3944no—
222D4—3944no—
223D4—3942no—
224D4—3942no—
225D4—1974no—
226C2×C4—7882no—
227C2×C4—3942no—
228C2×C4—3941no—
229C2×C4—3948no—
230C2×C4—3942no—
231C2×C4—3944no—
232C2×C4—3944no—
233C2×C4D4×D19712yes—
234C2×C4D4×D19712yes—
235C2×C4C2×D78811yescharacteristic
236C2×C4C23×D19711yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—82no—
243C22—7884no—
244C22—7882no—
245C22—7881no—
246C22—15768no—
247C22—3944no—
248C22#2026329611yescharacteristic
249C22#2026329611yescharacteristic
250C22C22×D78811yescharacteristic
251C4—7882no—
252C4—7881no—
253C4—3944no—
254C4#2026329612yes—
255C4C2×D4×D19712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—7884no—
261C2—15761no—
262C2C2×#2026329611yescharacteristic
subgroupG/H
centerC2C2×#20263296
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC2×#100×C197C2
socleC2×C197C2×#120
hypercenterC2×#100D197
hypocenterC197#3019
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#20263371C2semidirect
2. #27161879C2semidirect
3. #27161646C2semidirect
4. #27161259C2semidirect
5. #27161249C2semidirect
6. #27161095C2semidirect
7. C2×#100×C197C2semidirect
12. C2×#15691358C22semidirect
17. #20263359C22semidirect
19. #20263354C22semidirect
21. #20263322C22semidirect
22. #20263313C22semidirect
25. #20263285C22semidirect
26. #20263252C22semidirect
28. C2×M16×C197C22semidirect
29. #100×C197C22semidirect
35. #15691358D4semidirect
39. C2×Dic394D4semidirect
40. C2×Dic394C23semidirect
41. C2×D788D4semidirect
42. C2×D788C23semidirect
61. C4×Dic197C23semidirect
69. M16×C197C23semidirect
131. C197#3019semidirect
139. C2×#100D197semidirect
154. C2×M16C2×D197semidirect
161. #100C2×D197semidirect
197. M16C22×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633712
#271618792
#271616462
#271612592
#271612492
#271610952
C2×#100×C1972
#3019197