Finite Group Databasebeta

← #38755854

Subgroups — #38755854

Order 25216.

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

normal subgroups only

This group has 360 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#27161892C211yesmaximal, characteristic, complemented
2#27161880C211yesmaximal, characteristic, complemented
3#27161520C211yesmaximal, characteristic, complemented
4#27161518C212yesmaximal, complemented
5#27161516C212yesmaximal, complemented
6#27161515C211yesmaximal, characteristic, complemented
7#27161511C211yesmaximal, characteristic, complemented
8#27161510C212yesmaximal, complemented
9SD32×D197C212yesmaximal, complemented
10#27161495C211yesmaximal, characteristic, complemented
11C2×SD32×C197C211yesmaximal, characteristic, complemented
12#20263374—21no—
13#20263371—21no—
14#20263358—22no—
15#20263357—21no—
16#20263352—21no—
17#20263350—22no—
18#20263349C2212yescomplemented
19#20263348C2211yescharacteristic
20#20263348C2211yescharacteristic, complemented
21#20263345C2212yes—
22Q16×D197C2212yescomplemented
23D8×D197C2212yes—
24#20263332C2211yescharacteristic
25Dic1576C2211yescharacteristic, complemented
26#20263272C2212yescomplemented
27#20263271C2211yescharacteristic, complemented
28#20263271C2211yescharacteristic, complemented
29#20263270C2212yescomplemented
30#20263269C2212yescomplemented
31D3152C2211yescharacteristic, complemented
32#20263267C2212yescomplemented
33#20263266C2211yescharacteristic, complemented
34#20263265C2212yescomplemented
35C16×D197C2212yescomplemented
36C2×Q16×C197C2211yescharacteristic, complemented
37C2×D8×C197C2211yescharacteristic
38SD32×C197C2212yescomplemented
39SD32×C197C2212yescomplemented
40C2×C16×C197C2211yescharacteristic, complemented
41#15691359—22no—
42#15691359—41no—
43#15691358—21no—
44#15691358—21no—
45#15691358—42no—
46#15691358D411yescharacteristic
47#15691357—22no—
48#15691357—41no—
49Q8×D197—22no—
50Q8×D197—41no—
51C2×C22⋊D197—42no—
52D4×D197—22no—
53D4×D197—41no—
54Dic788C2311yescharacteristic
55Dic788C2311yescharacteristic
56#15691348—22no—
57#15691347—21no—
58#15691347—21no—
59#15691346—22no—
60#15691345—22no—
61D1576C2311yescharacteristic
62D1576C2311yescharacteristic
63#15691343—22no—
64#15691342D411yescharacteristic
65#15691341D412yes—
66C8×D197C2312yes—
67#15691333C2312yes—
68C2×Q8×C197—21no—
69C2×D4×C197—21no—
70Q16×C197—21no—
71Q16×C197C2312yes—
72D8×C197—21no—
73D8×C197C2312yes—
74C2×C8×C197C2311yescharacteristic
75C16×C197C2311yescharacteristic
76C16×C197C2311yescharacteristic
77C22×D197—42no—
78Dic394—21no—
79Dic394—21no—
80Dic394—42no—
81Dic394D811yescharacteristic
82D788—21no—
83D788—21no—
84D788—42no—
85D788D811yescharacteristic
86C22⋊D197—41no—
87C22⋊D197—41no—
88C22⋊D197—42no—
89C22⋊D197—42no—
90C22⋊D197D812yes—
91C2×Dic197—42no—
92C4×D197—42no—
93C4×D197—41no—
94C4×D197—41no—
95C4×D197—42no—
96C4×D197C2×D412yes—
97#13096802C2×D412yes—
98C23×C197—41no—
99Q8×C197—22no—
100Q8×C197—41no—
101D4×C197—22no—
102D4×C197—41no—
103C2×C4×C197—41no—
104C2×C4×C197C2×D411yescharacteristic
105C8×C197C2×D411yescharacteristic
106C8×C197C2411yescharacteristic
107C22×C197—42no—
108C22×C197—161no—
109C22×C197C2×D811yescharacteristic
110C2×D197—41no—
111C2×D197—41no—
112C2×D197—162no—
113C2×D197C2×D812yes—
114C4×C197—42no—
115C4×C197C2×D811yescharacteristic
116C4×C197C22×D411yescharacteristic
117Dic197—41no—
118Dic197—41no—
119Dic197C2×D812yes—
120C2×C197—21no—
121C2×C197—82no—
122C2×C197C22×D811yescharacteristic
123D197—22no—
124D197—81no—
125D197—81no—
126C197#341611yescharacteristic, complemented, Sylow
127#3416—1971nomaximal, Sylow
128#578—1971no—
129#576—1971no—
130#510—1971no—
131#510—1972no—
132#509—1971no—
133#509—1972no—
134#508—1972no—
135#508—1971no—
136C2×SD32—1972no—
137C2×SD32D19711yescharacteristic, complemented
138#504—1971no—
1392−1+4—3941no—
140H(5,𝔽2)—3941no—
141#137—3942no—
142#137—3941no—
143AGL(1,ℤ/8)—3942no—
144AGL(1,ℤ/8)—3941no—
145#135—1972no—
146#135—1972no—
147#135—1971no—
148#135—1971no—
149C2×Q16—1972no—
150C2×Q16C2×D19711yescharacteristic, complemented
151C2×D8—1972no—
152C2×D8C2×D19711yescharacteristic
153#131—1971no—
154Q32—1972no—
155Q32—1971no—
156SD32—1972no—
157SD32—1972no—
158SD32—1971no—
159SD32—1971no—
160SD32C2×D19712yescomplemented
161SD32C2×D19712yescomplemented
162D16—1972no—
163D16—1971no—
164M32—1971no—
165M32—1972no—
166C2×C16—1972no—
167C2×C16C2×D19711yescharacteristic, complemented
168#41—7882no—
169#41—7881no—
170#41—7881no—
171#41—3942no—
172#41—3942no—
173#41—3941no—
174#41—3941no—
175#41—1971no—
176C2×Q8—21no—
177C2×Q8—7881no—
178C2×Q8—3942no—
179C2×D4—21no—
180C2×D4—7881no—
181C2×D4—7882no—
182C2×D4—3942no—
183Q16—21no—
184Q16—3942no—
185Q16—1971no—
186Q16—1971no—
187Q16C22×D19712yes—
188SD16—3942no—
189SD16—3942no—
190SD16—3941no—
191SD16—3941no—
192D8—21no—
193D8—3942no—
194D8—1971no—
195D8—1971no—
196D8C22×D19712yes—
197M16—1971no—
198M16—1972no—
199C2×C8—1972no—
200C2×C8C22×D19711yescharacteristic
201C16—1972no—
202C16C22×D19711yescharacteristic
203C16C22×D19711yescharacteristic
204C23—41no—
205C23—7882no—
206Q8—22no—
207Q8—41no—
208Q8—7882no—
209Q8—3941no—
210Q8—3941no—
211Q8—1971no—
212D4—22no—
213D4—41no—
214D4—7882no—
215D4—7882no—
216D4—7881no—
217D4—7881no—
218D4—7882no—
219D4—3941no—
220D4—3941no—
221D4—1972no—
222D4—1971no—
223C2×C4—41no—
224C2×C4—7882no—
225C2×C4—7882no—
226C2×C4—7882no—
227C2×C4—7881no—
228C2×C4—7881no—
229C2×C4—1972no—
230C2×C4D4×D19711yescharacteristic
231C8—1972no—
232C8D4×D19711yescharacteristic
233C8C23×D19711yescharacteristic
234C22—42no—
235C22—161no—
236C22—7881no—
237C22—7881no—
238C22—31522no—
239C22—1972no—
240C22D8×D19711yescharacteristic
241C4—42no—
242C4—7881no—
243C4—7881no—
244C4—1972no—
245C4D8×D19711yescharacteristic
246C4C2×D4×D19711yescharacteristic
247C2—21no—
248C2—82no—
249C2—15761no—
250C2—15761no—
251C2—3942no—
252C2C2×D8×D19711yescharacteristic
subgroupG/H
centerC2C2×D8×D197
commutator, cosocleC8×C197C24
FrattiniC8C23×D197
FittingC2×SD32×C197C2
socleC2×C197C22×D8
hypercenterC2×SD32D197
hypocenterC197#3416
LCSG ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

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

NH ≅ G/Ntype
1. #27161892C2semidirect
2. #27161880C2semidirect
3. #27161520C2semidirect
4. #27161518C2semidirect
5. #27161516C2semidirect
6. #27161515C2semidirect
7. #27161511C2semidirect
8. #27161510C2semidirect
9. SD32×D197C2semidirect
10. #27161495C2semidirect
11. C2×SD32×C197C2semidirect
18. #20263349C22semidirect
20. #20263348C22semidirect
22. Q16×D197C22semidirect
25. Dic1576C22semidirect
26. #20263272C22semidirect
27. #20263271C22semidirect
28. #20263271C22semidirect
29. #20263270C22semidirect
30. #20263269C22semidirect
31. D3152C22semidirect
32. #20263267C22semidirect
33. #20263266C22semidirect
34. #20263265C22semidirect
35. C16×D197C22semidirect
36. C2×Q16×C197C22semidirect
38. SD32×C197C22semidirect
39. SD32×C197C22semidirect
40. C2×C16×C197C22semidirect
126. C197#3416semidirect
137. C2×SD32D197semidirect
150. C2×Q16C2×D197semidirect
160. SD32C2×D197semidirect
161. SD32C2×D197semidirect
167. C2×C16C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618922
#271618802
#271615202
#271615182
#271615162
#271615152
#271615112
#271615102
SD32×D1972
#271614952
C2×SD32×C1972
#3416197