Finite Group Databasebeta

← #38754508

Subgroups — #38754508

Order 25216.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#27161883C212yesmaximal, complemented
2#27161882C212yesmaximal, complemented
3#27161455C211yesmaximal, characteristic, complemented
4#27161348C212yesmaximal, complemented
5#27161347C212yesmaximal, complemented
6#27161248C212yesmaximal, complemented
7#27161246C211yesmaximal, characteristic, complemented
8#27161193C212yesmaximal, complemented
9#453×C197C211yesmaximal, characteristic, complemented
10#20263372C2212yescomplemented
11#20263371C2211yescharacteristic, complemented
12#20263354C2212yescomplemented
13#20263353—22no—
14#20263352—22no—
15#20263352C2212yescomplemented
16#20263351—22no—
17#20263351C2212yescomplemented
18#20263350—22no—
19#20263347—22no—
20#20263346—22no—
21#20263346C2212yes—
22C2×#15691343—22no—
23C2×#15691343C2212yescomplemented
24C2×#15691343C2212yescomplemented
25D8×D197—22no—
26#20263334C2212yescomplemented
27#20263333C2212yescomplemented
28#20263323C2211yescharacteristic, complemented
29#20263261C2212yescomplemented
30#20263254C2212yescomplemented
31#20263243C2211yescharacteristic, complemented
32#20263242C2212yescomplemented
33C4×#13096802C2211yescharacteristic, complemented
34H(5,𝔽2)×C197C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C197C2212yescomplemented
36#127×C197C2211yescharacteristic, complemented
37#104×C197C2212yescomplemented
38#100×C197C2211yescharacteristic, complemented
39#15691359—22no—
40#15691358—22no—
41#15691358C2312yes—
42#15691357—22no—
43C2×D788—22no—
44C2×D788—41no—
45C2×D788C2312yes—
46C2×C22⋊D197—41no—
47D4×D197—22no—
48D4×D197—22no—
49D4×D197—42no—
50#15691348D412yescomplemented
51#15691347—22no—
52#15691346—22no—
53#15691346D412yescomplemented
54#15691345—22no—
55#15691345D412yescomplemented
56D1576—22no—
57#15691343—22no—
58#15691343—22no—
59#15691343—22no—
60#15691343—22no—
61#15691343—22no—
62#15691343—22no—
63#15691343D412yescomplemented
64#15691342—22no—
65#15691342C2312yes—
66#15691341—22no—
67C2×#13096802—22no—
68C2×#13096802C2312yes—
69C8×D197—22no—
70#41×C197—22no—
71#41×C197C2312yes—
72C2×D4×C197—22no—
73C2×D4×C197—41no—
74C2×D4×C197—41no—
75C2×D4×C197C2311yescharacteristic
76C2×D4×C197C2311yescharacteristic
77SD16×C197—22no—
78D8×C197—22no—
79M16×C197C2312yes—
80C42×C197C2311yescharacteristic
81C22×D197—22no—
82C22×D197—41no—
83C22×D197—42no—
84Dic394C2×D412yes—
85D788—22no—
86D788—22no—
87D788—42no—
88D788—82no—
89D788C2×D412yes—
90C22⋊D197—22no—
91C22⋊D197—42no—
92C22⋊D197—42no—
93C2×Dic197—41no—
94C4×D197—22no—
95C4×D197—42no—
96#13096802—22no—
97#13096802C2×D412yes—
98#13096802C2×D412yes—
99C23×C197—21no—
100C23×C197—21no—
101C23×C197—41no—
102Q8×C197C2×D412yes—
103D4×C197—22no—
104D4×C197—22no—
105D4×C197—22no—
106D4×C197—42no—
107D4×C197—82no—
108D4×C197C2×D412yes—
109C2×C4×C197—21no—
110C2×C4×C197—22no—
111C2×C4×C197—41no—
112C2×C4×C197C2411yescharacteristic
113C8×C197—22no—
114C22×C197—22no—
115C22×C197—41no—
116C22×C197—41no—
117C22×C197—81no—
118C22×C197—81no—
119C22×C197—161no—
120C22×C197H(5,𝔽2)11yescharacteristic
121C2×D197—22no—
122C2×D197—42no—
123C2×D197—82no—
124C2×D197—161no—
125C2×D197—162no—
126C4×C197—22no—
127C4×C197—22no—
128C4×C197C22×D412yes—
129Dic197—22no—
130C2×C197—21no—
131C2×C197—42no—
132C2×C197—81no—
133C2×C197—81no—
134C2×C197D4211yescharacteristic
135D197—42no—
136D197—82no—
137C197#329211yescharacteristic, complemented, Sylow
138#3292—1971nomaximal, Sylow
139#576—1972no—
140#576—1972no—
141#493—1971no—
142#472—1972no—
143#472—1972no—
144#453—1971no—
145#453—1972no—
146#453D19711yescharacteristic, complemented
147#443—1972no—
148H(5,𝔽2)—1972no—
149H(5,𝔽2)—1971no—
150H(5,𝔽2)C2×D19711yescharacteristic, complemented
151AGL(1,ℤ/8)—3942no—
152AGL(1,ℤ/8)—3942no—
153AGL(1,ℤ/8)—3942no—
154AGL(1,ℤ/8)—3942no—
155AGL(1,ℤ/8)—1972no—
156AGL(1,ℤ/8)—1972no—
157AGL(1,ℤ/8)—1972no—
158AGL(1,ℤ/8)C2×D19712yescomplemented
159#135—3942no—
160#135—3942no—
161#135—1972no—
162C2×D8—3942no—
163C2×D8—3942no—
164C2×D8—1972no—
165C2×D8—1972no—
166#131—1972no—
167#131—1972no—
168#127—1971no—
169#127C2×D19711yescharacteristic, complemented
170#108—1972no—
171#104—1972no—
172#104C2×D19712yescomplemented
173#100—1971no—
174#100—1972no—
175#100C2×D19711yescharacteristic, complemented
176C4×C8—1971no—
177#41—22no—
178#41—3942no—
179#41—3942no—
180#41—3942no—
181#41—1972no—
182#41C22×D19712yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—7881no—
187C2×D4—7882no—
188C2×D4—7881no—
189C2×D4—3942no—
190C2×D4—3942no—
191C2×D4—3942no—
192C2×D4—1972no—
193C2×D4C22×D19711yescharacteristic
194C2×D4C22×D19711yescharacteristic
195Q16—1972no—
196SD16—22no—
197SD16—3942no—
198SD16—3942no—
199SD16—3942no—
200SD16—1972no—
201SD16—1972no—
202D8—22no—
203D8—3942no—
204D8—3942no—
205D8—3942no—
206D8—3942no—
207D8—3942no—
208D8—3942no—
209D8—3942no—
210D8—1972no—
211M16—3942no—
212M16—3942no—
213M16—1972no—
214M16C22×D19712yes—
215C2×C8—3942no—
216C2×C8—3942no—
217C2×C8—1972no—
218C42C22×D19711yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—7882no—
223C23—7881no—
224C23—3942no—
225Q8—1972no—
226Q8D4×D19712yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—7882no—
233D4—7882no—
234D4—7882no—
235D4—15762no—
236D4—3942no—
237D4—3942no—
238D4—3942no—
239D4—1972no—
240D4D4×D19712yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—7882no—
245C2×C4—7881no—
246C2×C4—3942no—
247C2×C4C23×D19711yescharacteristic
248C8—22no—
249C8—3942no—
250C8—1972no—
251C8—1972no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—7882no—
259C22—15762no—
260C22—31522no—
261C22—31521no—
262C22—3942no—
263C22#2026337111yescharacteristic
264C4—22no—
265C4—22no—
266C4—3942no—
267C4C2×D4×D19712yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—7882no—
273C2—15762no—
274C2#2716171111yescharacteristic
subgroupG/H
centerC2#27161711
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#453×C197C2
socleC2×C197D42
hypercenter#453D197
hypocenterC197#3292
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #27161883C2semidirect
2. #27161882C2semidirect
3. #27161455C2semidirect
4. #27161348C2semidirect
5. #27161347C2semidirect
6. #27161248C2semidirect
7. #27161246C2semidirect
8. #27161193C2semidirect
9. #453×C197C2semidirect
10. #20263372C22semidirect
11. #20263371C22semidirect
12. #20263354C22semidirect
15. #20263352C22semidirect
17. #20263351C22semidirect
23. C2×#15691343C22semidirect
24. C2×#15691343C22semidirect
26. #20263334C22semidirect
27. #20263333C22semidirect
28. #20263323C22semidirect
29. #20263261C22semidirect
30. #20263254C22semidirect
31. #20263243C22semidirect
32. #20263242C22semidirect
33. C4×#13096802C22semidirect
34. H(5,𝔽2)×C197C22semidirect
35. AGL(1,ℤ/8)×C197C22semidirect
36. #127×C197C22semidirect
37. #104×C197C22semidirect
38. #100×C197C22semidirect
50. #15691348D4semidirect
53. #15691346D4semidirect
55. #15691345D4semidirect
63. #15691343D4semidirect
137. C197#3292semidirect
146. #453D197semidirect
150. H(5,𝔽2)C2×D197semidirect
158. AGL(1,ℤ/8)C2×D197semidirect
169. #127C2×D197semidirect
172. #104C2×D197semidirect
175. #100C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618832
#271618822
#271614552
#271613482
#271613472
#271612482
#271612462
#271611932
#453×C1972
#3292197