Finite Group Databasebeta

← #38755855

Subgroups — #38755855

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#27161893C211yesmaximal, characteristic, complemented
2#27161881C211yesmaximal, characteristic, complemented
3#27161521C212yesmaximal, complemented
4#27161519C211yesmaximal, characteristic, complemented
5#27161514C211yesmaximal, characteristic, complemented
6#27161513C212yesmaximal, complemented
7#27161510C212yesmaximal, complemented
8C2×#20263271C211yesmaximal, characteristic, complemented
9SD32×D197C212yesmaximal, complemented
10#27161495C211yesmaximal, characteristic, complemented
11#508×C197C211yesmaximal, characteristic, complemented
12#20263373—21no—
13#20263372—21no—
14#20263356—21no—
15#20263355—22no—
16#20263353—22no—
17#20263351—21no—
18#20263349C2212yes—
19#20263345C2212yescomplemented
20C2×Dic788C2211yescharacteristic, complemented
21Q16×D197C2212yescomplemented
22C2×D1576C2211yescharacteristic
23D8×D197C2212yes—
24#20263332C2211yescharacteristic
25#20263272C2212yescomplemented
26#20263271C2212yescomplemented
27#20263271C2212yescomplemented
28#20263270C2212yescomplemented
29#20263269C2212yescomplemented
30#20263267C2212yescomplemented
31#20263266C2211yescharacteristic, complemented
32#20263265C2212yescomplemented
33C16×D197C2212yescomplemented
34#135×C197C2211yescharacteristic
35#135×C197C2211yescharacteristic, complemented
36Q32×C197C2211yescharacteristic, complemented
37SD32×C197C2211yescharacteristic, complemented
38SD32×C197C2211yescharacteristic, complemented
39D16×C197C2211yescharacteristic, complemented
40C2×C16×C197C2211yescharacteristic, complemented
41#15691359—22no—
42#15691358—41no—
43#15691358—41no—
44#15691358D411yescharacteristic
45#15691357—22no—
46#15691357—42no—
47C2×Dic394—21no—
48C2×Dic394—41no—
49Q8×D197—22no—
50C2×D788—21no—
51C2×D788—41no—
52D4×D197—22no—
53D4×D197—42no—
54Dic788—21no—
55Dic788C2312yes—
56#15691348—22no—
57#15691346—22no—
58#15691345—22no—
59D1576—21no—
60D1576C2312yes—
61#15691343—22no—
62#15691342D411yescharacteristic
63#15691341D412yes—
64C8×D197C2312yes—
65#15691333C2312yes—
66#41×C197—21no—
67#41×C197—21no—
68Q16×C197C2311yescharacteristic
69Q16×C197C2311yescharacteristic
70SD16×C197—21no—
71SD16×C197—21no—
72D8×C197C2311yescharacteristic
73D8×C197C2311yescharacteristic
74C2×C8×C197C2311yescharacteristic
75C16×C197C2311yescharacteristic
76C16×C197C2311yescharacteristic
77C22×D197—41no—
78C22×D197—42no—
79Dic394—22no—
80Dic394—41no—
81Dic394—42no—
82Dic394D811yescharacteristic
83D788—22no—
84D788—42no—
85D788—41no—
86D788D811yescharacteristic
87C22⋊D197—42no—
88C22⋊D197—42no—
89C22⋊D197D812yes—
90C2×Dic197—41no—
91C2×Dic197—42no—
92C4×D197—42no—
93C4×D197—42no—
94C4×D197C2×D412yes—
95#13096802C2×D412yes—
96Q8×C197—21no—
97Q8×C197—21no—
98D4×C197—21no—
99D4×C197—21no—
100D4×C197—41no—
101D4×C197—41no—
102C2×C4×C197—41no—
103C2×C4×C197—41no—
104C2×C4×C197C2×D411yescharacteristic
105C8×C197C2×D411yescharacteristic
106C8×C197C2411yescharacteristic
107C22×C197—41no—
108C22×C197—41no—
109C22×C197C2×D811yescharacteristic
110C2×D197—42no—
111C2×D197—162no—
112C2×D197—161no—
113C2×D197C2×D812yes—
114C4×C197—41no—
115C4×C197—41no—
116C4×C197C2×D811yescharacteristic
117C4×C197C22×D411yescharacteristic
118Dic197—42no—
119Dic197C2×D812yes—
120C2×C197—21no—
121C2×C197—81no—
122C2×C197—81no—
123C2×C197C22×D811yescharacteristic
124D197—22no—
125D197—82no—
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#508D19711yescharacteristic, complemented
136C2×SD32—1972no—
137C2×SD32—1971no—
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#135C2×D19711yescharacteristic
148#135C2×D19711yescharacteristic, complemented
149C2×Q16—1972no—
150C2×Q16—1971no—
151C2×D8—1972no—
152C2×D8—1971no—
153#131—1971no—
154Q32—1972no—
155Q32C2×D19711yescharacteristic, complemented
156SD32—1972no—
157SD32—1972no—
158SD32—1972no—
159SD32—1972no—
160SD32C2×D19711yescharacteristic, complemented
161SD32C2×D19711yescharacteristic, complemented
162D16—1972no—
163D16C2×D19711yescharacteristic, complemented
164M32—1971no—
165M32—1972no—
166C2×C16—1972no—
167C2×C16C2×D19711yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—7881no—
171#41—7881no—
172#41—7882no—
173#41—3942no—
174#41—3942no—
175#41—1971no—
176C2×Q8—7881no—
177C2×Q8—3942no—
178C2×Q8—3941no—
179C2×D4—7881no—
180C2×D4—7882no—
181C2×D4—3942no—
182C2×D4—3941no—
183Q16—3942no—
184Q16—3941no—
185Q16—1972no—
186Q16C22×D19711yescharacteristic
187Q16C22×D19711yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—3942no—
191SD16—3942no—
192D8—3942no—
193D8—3941no—
194D8—1972no—
195D8C22×D19711yescharacteristic
196D8C22×D19711yescharacteristic
197M16—1971no—
198M16—1972no—
199C2×C8—1972no—
200C2×C8C22×D19711yescharacteristic
201C16—1972no—
202C16C22×D19711yescharacteristic
203C16C22×D19711yescharacteristic
204C23—7882no—
205C23—7881no—
206Q8—21no—
207Q8—21no—
208Q8—7882no—
209Q8—7881no—
210Q8—3942no—
211Q8—1971no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—7882no—
217D4—7882no—
218D4—7882no—
219D4—7881no—
220D4—3942no—
221D4—1972no—
222D4—1971no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—7882no—
226C2×C4—7882no—
227C2×C4—7881no—
228C2×C4—7882no—
229C2×C4—1972no—
230C2×C4D4×D19711yescharacteristic
231C8—1972no—
232C8D4×D19711yescharacteristic
233C8C23×D19711yescharacteristic
234C22—41no—
235C22—41no—
236C22—7882no—
237C22—31521no—
238C22—31522no—
239C22—1972no—
240C22D8×D19711yescharacteristic
241C4—41no—
242C4—41no—
243C4—7882no—
244C4—1972no—
245C4D8×D19711yescharacteristic
246C4C2×D4×D19711yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—15762no—
251C2—3942no—
252C2C2×D8×D19711yescharacteristic
subgroupG/H
centerC2C2×D8×D197
commutator, cosocleC8×C197C24
FrattiniC8C23×D197
Fitting#508×C197C2
socleC2×C197C22×D8
hypercenter#508D197
hypocenterC197#3416
LCSG ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #27161893C2semidirect
2. #27161881C2semidirect
3. #27161521C2semidirect
4. #27161519C2semidirect
5. #27161514C2semidirect
6. #27161513C2semidirect
7. #27161510C2semidirect
8. C2×#20263271C2semidirect
9. SD32×D197C2semidirect
10. #27161495C2semidirect
11. #508×C197C2semidirect
19. #20263345C22semidirect
20. C2×Dic788C22semidirect
21. Q16×D197C22semidirect
25. #20263272C22semidirect
26. #20263271C22semidirect
27. #20263271C22semidirect
28. #20263270C22semidirect
29. #20263269C22semidirect
30. #20263267C22semidirect
31. #20263266C22semidirect
32. #20263265C22semidirect
33. C16×D197C22semidirect
35. #135×C197C22semidirect
36. Q32×C197C22semidirect
37. SD32×C197C22semidirect
38. SD32×C197C22semidirect
39. D16×C197C22semidirect
40. C2×C16×C197C22semidirect
126. C197#3416semidirect
135. #508D197semidirect
148. #135C2×D197semidirect
155. Q32C2×D197semidirect
160. SD32C2×D197semidirect
161. SD32C2×D197semidirect
163. D16C2×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
#271618932
#271618812
#271615212
#271615192
#271615142
#271615132
#271615102
C2×#202632712
SD32×D1972
#271614952
#508×C1972
#3416197