Finite Group Databasebeta

← #38755852

Subgroups — #38755852

Order 25216.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#27161881C211yesmaximal, characteristic, complemented
2#27161880C211yesmaximal, characteristic, complemented
3#27161517C211yesmaximal, characteristic, complemented
4#27161516C212yesmaximal, complemented
5#27161513C212yesmaximal, complemented
6#27161510C212yesmaximal, complemented
7#27161509C211yesmaximal, characteristic, complemented
8C2×#20263267C211yesmaximal, characteristic, complemented
9D16×D197C212yesmaximal, complemented
10#27161497C211yesmaximal, characteristic, complemented
11#509×C197C211yesmaximal, characteristic, complemented
12#20263372—21no—
13#20263371—21no—
14#20263353—22no—
15#20263352—21no—
16#20263351—21no—
17#20263350—22no—
18#20263349C2212yescomplemented
19#20263348C2211yescharacteristic, complemented
20#20263345C2212yescomplemented
21C2×D1576C2211yescharacteristic, complemented
22D8×D197C2212yescomplemented
23D8×D197C2212yescomplemented
24#20263332C2211yescharacteristic
25#20263272C2211yescharacteristic, complemented
26#20263271C2212yescomplemented
27#20263270C2211yescharacteristic, complemented
28#20263269C2211yescharacteristic, complemented
29D3152C2212yescomplemented
30#20263267C2211yescharacteristic, complemented
31#20263267C2212yescomplemented
32#20263267C2212yescomplemented
33#20263265C2212yescomplemented
34C2×#15691333C2211yescharacteristic, complemented
35C16×D197C2212yescomplemented
36#135×C197C2211yescharacteristic, complemented
37C2×D8×C197C2211yescharacteristic, complemented
38SD32×C197C2212yescomplemented
39D16×C197C2212yescomplemented
40M32×C197C2211yescharacteristic, complemented
41#15691359—22no—
42#15691358—21no—
43#15691358—41no—
44#15691358D411yescharacteristic
45#15691357—22no—
46#15691357—41no—
47C2×D788—21no—
48C2×D788—41no—
49C2×C22⋊D197—42no—
50D4×D197—22no—
51D4×D197—22no—
52D4×D197—41no—
53D4×D197—42no—
54Dic788C2311yescharacteristic
55#15691347—21no—
56#15691346—22no—
57#15691345—22no—
58D1576—21no—
59D1576C2312yes—
60D1576C2311yescharacteristic
61#15691343—22no—
62#15691343—22no—
63#15691342D411yescharacteristic
64#15691341D412yes—
65C8×D197C2312yes—
66#15691333C2311yescharacteristic
67#15691333C2311yescharacteristic
68#41×C197—21no—
69C2×D4×C197—21no—
70Q16×C197C2311yescharacteristic
71SD16×C197—21no—
72D8×C197—21no—
73D8×C197C2311yescharacteristic
74D8×C197C2312yes—
75C2×C8×C197C2311yescharacteristic
76C16×C197C2312yes—
77C22×D197—41no—
78C22×D197—42no—
79C22×D197—42no—
80Dic394—21no—
81Dic394D811yescharacteristic
82D788—21no—
83D788—22no—
84D788—42no—
85D788—41no—
86D788D811yescharacteristic
87C22⋊D197—41no—
88C22⋊D197—42no—
89C22⋊D197—42no—
90C22⋊D197—42no—
91C22⋊D197D812yes—
92C2×Dic197—42no—
93C4×D197—41no—
94C4×D197—42no—
95C4×D197C2×D412yes—
96#13096802C2×D412yes—
97C23×C197—41no—
98Q8×C197—21no—
99D4×C197—22no—
100D4×C197—21no—
101D4×C197—41no—
102D4×C197—41no—
103C2×C4×C197—41no—
104C2×C4×C197C2×D411yescharacteristic
105C8×C197C2×D411yescharacteristic
106C8×C197C2411yescharacteristic
107C22×C197—41no—
108C22×C197—42no—
109C22×C197—161no—
110C22×C197C2×D811yescharacteristic
111C2×D197—41no—
112C2×D197—42no—
113C2×D197—162no—
114C2×D197—162no—
115C2×D197—161no—
116C2×D197C2×D812yes—
117C4×C197—41no—
118C4×C197C2×D811yescharacteristic
119C4×C197C22×D411yescharacteristic
120Dic197—41no—
121Dic197C2×D812yes—
122C2×C197—21no—
123C2×C197—81no—
124C2×C197—82no—
125C2×C197C22×D811yescharacteristic
126D197—22no—
127D197—81no—
128D197—82no—
129C197#341511yescharacteristic, complemented, Sylow
130#3415—1971nomaximal, Sylow
131#576—1971no—
132#576—1971no—
133#509—1972no—
134#509—1972no—
135#509—1971no—
136#509D19711yescharacteristic, complemented
137#508—1971no—
138#508—1972no—
139C2×D16—1971no—
140C2×D16—1972no—
141#504—1971no—
142H(5,𝔽2)—3941no—
143H(5,𝔽2)—3941no—
144AGL(1,ℤ/8)—3942no—
145AGL(1,ℤ/8)—3942no—
146AGL(1,ℤ/8)—3941no—
147AGL(1,ℤ/8)—3941no—
148#135—1972no—
149#135—1972no—
150#135—1971no—
151#135C2×D19711yescharacteristic, complemented
152C2×D8—1972no—
153C2×D8—1972no—
154C2×D8—1971no—
155C2×D8C2×D19711yescharacteristic, complemented
156#131—1971no—
157Q32—1971no—
158SD32—1971no—
159SD32—1971no—
160SD32—1972no—
161SD32C2×D19712yescomplemented
162D16—1972no—
163D16—1971no—
164D16—1972no—
165D16—1972no—
166D16C2×D19712yescomplemented
167M32—1972no—
168M32C2×D19711yescharacteristic, complemented
169C2×C16—1971no—
170C2×C16—1972no—
171#41—21no—
172#41—7881no—
173#41—7881no—
174#41—3942no—
175#41—3942no—
176#41—3941no—
177#41—1971no—
178C2×D4—21no—
179C2×D4—7881no—
180C2×D4—7882no—
181C2×D4—7881no—
182C2×D4—7882no—
183C2×D4—3942no—
184C2×D4—3942no—
185C2×D4—3941no—
186Q16—1971no—
187Q16C22×D19711yescharacteristic
188SD16—21no—
189SD16—3942no—
190SD16—3942no—
191SD16—3941no—
192D8—21no—
193D8—3942no—
194D8—3942no—
195D8—3941no—
196D8—1972no—
197D8—1971no—
198D8C22×D19712yes—
199D8C22×D19711yescharacteristic
200M16—1971no—
201M16—1972no—
202C2×C8—1972no—
203C2×C8C22×D19711yescharacteristic
204C16—1971no—
205C16—1971no—
206C16C22×D19712yes—
207C23—41no—
208C23—7882no—
209C23—7882no—
210C23—7881no—
211Q8—21no—
212Q8—3941no—
213Q8—1971no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—7882no—
219D4—7882no—
220D4—7881no—
221D4—7882no—
222D4—7882no—
223D4—7881no—
224D4—3941no—
225D4—3942no—
226D4—1972no—
227D4—1971no—
228C2×C4—41no—
229C2×C4—7882no—
230C2×C4—7882no—
231C2×C4—7881no—
232C2×C4—1972no—
233C2×C4D4×D19711yescharacteristic
234C8—1972no—
235C8D4×D19711yescharacteristic
236C8C23×D19711yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—7882no—
241C22—7881no—
242C22—31521no—
243C22—31522no—
244C22—31522no—
245C22—1972no—
246C22D8×D19711yescharacteristic
247C4—41no—
248C4—7881no—
249C4—1972no—
250C4D8×D19711yescharacteristic
251C4C2×D4×D19711yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—15762no—
256C2—15761no—
257C2—3942no—
258C2C2×D8×D19711yescharacteristic
subgroupG/H
centerC2C2×D8×D197
commutator, cosocleC8×C197C24
FrattiniC8C23×D197
Fitting#509×C197C2
socleC2×C197C22×D8
hypercenter#509D197
hypocenterC197#3415
LCSG ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #27161881C2semidirect
2. #27161880C2semidirect
3. #27161517C2semidirect
4. #27161516C2semidirect
5. #27161513C2semidirect
6. #27161510C2semidirect
7. #27161509C2semidirect
8. C2×#20263267C2semidirect
9. D16×D197C2semidirect
10. #27161497C2semidirect
11. #509×C197C2semidirect
18. #20263349C22semidirect
19. #20263348C22semidirect
20. #20263345C22semidirect
21. C2×D1576C22semidirect
22. D8×D197C22semidirect
23. D8×D197C22semidirect
25. #20263272C22semidirect
26. #20263271C22semidirect
27. #20263270C22semidirect
28. #20263269C22semidirect
29. D3152C22semidirect
30. #20263267C22semidirect
31. #20263267C22semidirect
32. #20263267C22semidirect
33. #20263265C22semidirect
34. C2×#15691333C22semidirect
35. C16×D197C22semidirect
36. #135×C197C22semidirect
37. C2×D8×C197C22semidirect
38. SD32×C197C22semidirect
39. D16×C197C22semidirect
40. M32×C197C22semidirect
129. C197#3415semidirect
136. #509D197semidirect
151. #135C2×D197semidirect
155. C2×D8C2×D197semidirect
161. SD32C2×D197semidirect
166. D16C2×D197semidirect
168. M32C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618812
#271618802
#271615172
#271615162
#271615132
#271615102
#271615092
C2×#202632672
D16×D1972
#271614972
#509×C1972
#3415197