Finite Group Databasebeta

← #497×D197 #38753148

Subgroups — #497×D197 #38753148

Order 25216.

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

normal subgroups only

This group has 564 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×Q16×D197C212yesmaximal, complemented
2C2×SD16×D197C212yesmaximal
3C4⋊Q8×D197C211yesmaximal, characteristic, complemented
4#124×D197C211yesmaximal, characteristic
5#27161474C211yesmaximal, characteristic, complemented
6#27161473C211yesmaximal, characteristic, complemented
7#27161472C212yesmaximal, complemented
8#27161471C211yesmaximal, characteristic, complemented
9#27161470C212yesmaximal, complemented
10#97×D197C211yesmaximal, characteristic, complemented
11#497×C197C211yesmaximal, characteristic, complemented
12C2×Q8×D197C2212yes—
13C2×Q8×D197C2211yescharacteristic
14C2×D4×D197C2211yescharacteristic
15C2×Dic788C2212yescomplemented
16C2×#15691348C2212yescomplemented
17C2×#15691348C2212yescomplemented
18Q16×D197—28no—
19C2×#15691347C2212yes—
20C2×#15691346C2212yes—
21C2×#15691345C2212yes—
22SD16×D197—28no—
23C2×C8×D197C2212yes—
24#20263326C2211yescharacteristic, complemented
25#20263325—22no—
26#20263324C2212yescomplemented
27#20263315C2211yescharacteristic
28#20263314C2211yescharacteristic
29#20263313C2211yescharacteristic
30#20263312—22no—
31C4⋊C4×D197—22no—
32C22⋊C4×D197—22no—
33C42×D197C2211yescharacteristic
34#20263237C2211yescharacteristic, complemented
35#20263236C2212yescomplemented
36C2×Q16×C197C2212yescomplemented
37C2×SD16×C197C2212yes—
38C4⋊Q8×C197C2211yescharacteristic, complemented
39#124×C197C2211yescharacteristic
40#97×C197C2211yescharacteristic, complemented
41C23×D197—21no—
42C2×Dic394—21no—
43C2×Dic394—22no—
44C2×Dic394C2311yescharacteristic
45C2×Dic394C2312yes—
46Q8×D197—24no—
47Q8×D197—24no—
48Q8×D197—24no—
49C2×D788C2311yescharacteristic
50C2×C22⋊D197—21no—
51D4×D197—24no—
52C2×C4×D197—21no—
53C2×C4×D197—22no—
54C2×C4×D197D412yes—
55C2×C4×D197C2311yescharacteristic
56Dic788—24no—
57#15691348—24no—
58#15691348—24no—
59#15691347—24no—
60#15691346—24no—
61#15691345—24no—
62C2×#13096802C2312yes—
63C8×D197D418yes—
64C4⋊Dic197—22no—
65#15691337—24no—
66C22⋊Dic197—22no—
67#15691335—22no—
68#15691335—22no—
69C4×Dic197D412yes—
70C4×Dic197C2311yescharacteristic
71C2×Q8×C197C2311yescharacteristic
72C2×Q8×C197C2312yes—
73C2×D4×C197C2311yescharacteristic
74Q16×C197—24no—
75SD16×C197—24no—
76C2×C8×C197C2312yes—
77C4⋊C4×C197—22no—
78C22⋊C4×C197—22no—
79C42×C197C2311yescharacteristic
80C22×D197—21no—
81C22×D197—42no—
82C22×D197—44no—
83C22×D197C2×D411yescharacteristic
84Dic394—22no—
85Dic394—22no—
86Dic394—22no—
87Dic394—44no—
88Dic394—42no—
89D788—22no—
90C22⋊D197—42no—
91C2×Dic197—21no—
92C2×Dic197—22no—
93C2×Dic197C2×D412yes—
94C2×Dic197C2×D411yescharacteristic
95C4×D197—24no—
96C4×D197—42no—
97C4×D197—44no—
98C4×D197C2×D414yes—
99#13096802C2×D414yes—
100C23×C197—21no—
101Q8×C197—22no—
102Q8×C197—22no—
103Q8×C197—22no—
104D4×C197—22no—
105C2×C4×C197—21no—
106C2×C4×C197—22no—
107C2×C4×C197C2×D412yes—
108C2×C4×C197C2411yescharacteristic
109C8×C197C2×D414yes—
110C22×C197—41no—
111C22×C197—42no—
112C22×C197C22×D411yescharacteristic
113C2×D197—41no—
114C2×D197—42no—
115C2×D197—84no—
116C2×D197#12712yes—
117C2×D197#13714yes—
118C4×C197—22no—
119C4×C197—41no—
120C4×C197—42no—
121C4×C197C22×D412yes—
122Dic197—22no—
123Dic197—41no—
124Dic197—42no—
125Dic197#12712yes—
126C2×C197—81no—
127C2×C197C2×#12711yescharacteristic
128C2×C197C2×#13712yes—
129D197—81no—
130D197#49714yesdirect product factor
131C197C2×#49711yescharacteristic, complemented, Sylow
132C2×#497—1971nomaximal, Sylow
133C22×Q16—1972no—
134C22×SD16—1972no—
135C2×C4⋊Q8—1971no—
136C2×#124—1971no—
137#497—1972no—
138#497—1972no—
139#497—1971no—
140#497—1971no—
141#497—1971no—
142#497D19711yescharacteristic, direct product factor
143C2×#97—1971no—
144C22×Q8—1971no—
145C22×Q8—1972no—
146C22×D4—1971no—
147C2×Q16—3948no—
148C2×Q16—1972no—
149C2×Q16—1972no—
150C2×Q16—1972no—
151C2×Q16C2×D19712yescomplemented
152C2×SD16—3948no—
153C2×SD16—1972no—
154C2×SD16—1972no—
155C2×SD16—1972no—
156C2×SD16C2×D19712yes—
157C22×C8—1972no—
158C4⋊Q8—3942no—
159C4⋊Q8—1972no—
160C4⋊Q8—1971no—
161C4⋊Q8C2×D19711yescharacteristic, complemented
162#124—3942no—
163#124—1971no—
164#124—1971no—
165#124—1971no—
166#124C2×D19711yescharacteristic
167C2×C4⋊C4—3942no—
168C2×C22⋊C4—3942no—
169C2×C42—1971no—
170#97—1972no—
171#97—1971no—
172#97C2×D19711yescharacteristic, complemented
173C24—3941no—
174C2×Q8—3941no—
175C2×Q8—3942no—
176C2×Q8—3944no—
177C2×Q8—3944no—
178C2×Q8—3944no—
179C2×Q8—1971no—
180C2×Q8—1972no—
181C2×Q8C22×D19711yescharacteristic
182C2×Q8C22×D19712yes—
183C2×D4—3944no—
184C2×D4—3941no—
185C2×D4—1971no—
186C2×D4C22×D19711yescharacteristic
187C22×C4—3941no—
188C22×C4—3942no—
189C22×C4—1972no—
190C22×C4—1971no—
191Q16—24no—
192Q16—3944no—
193Q16—3944no—
194Q16—3944no—
195SD16—24no—
196SD16—3944no—
197SD16—3944no—
198SD16—3944no—
199C2×C8—1972no—
200C2×C8—1978no—
201C2×C8C22×D19712yes—
202C4⋊C4—22no—
203C4⋊C4—3944no—
204C4⋊C4—3942no—
205C22⋊C4—22no—
206C22⋊C4—3942no—
207C22⋊C4—3942no—
208C22⋊C4—3942no—
209C42—1972no—
210C42—1971no—
211C42C22×D19711yescharacteristic
212C23—21no—
213C23—7882no—
214C23—7884no—
215C23—3941no—
216C23—1971no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—7882no—
221Q8—7884no—
222Q8—3942no—
223Q8—3942no—
224Q8—3942no—
225D4—22no—
226D4—7882no—
227D4—3942no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—7882no—
231C2×C4—7884no—
232C2×C4—3944no—
233C2×C4—3941no—
234C2×C4—3942no—
235C2×C4—1972no—
236C2×C4—1971no—
237C2×C4—1974no—
238C2×C4D4×D19712yes—
239C2×C4C23×D19711yescharacteristic
240C8—1974no—
241C8D4×D19714yes—
242C22—42no—
243C22—41no—
244C22—7882no—
245C22—7881no—
246C22—15764no—
247C22—1974no—
248C22—1972no—
249C22C2×D4×D19711yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—7881no—
254C4—7882no—
255C4—3942no—
256C4—1972no—
257C4C2×D4×D19712yes—
258C2—81no—
259C2—15761no—
260C2—1974no—
261C2#127×D19711yescharacteristic
262C2#137×D19712yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#497×C197C2
socleC22×C197C22×D4
prime radicalC2×D4×D197C22
hypercenter#497D197
hypocenterC197C2×#497
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D197C2semidirect
3. C4⋊Q8×D197C2semidirect
5. #27161474C2semidirect
6. #27161473C2semidirect
7. #27161472C2semidirect
8. #27161471C2semidirect
9. #27161470C2semidirect
10. #97×D197C2semidirect
11. #497×C197C2semidirect
15. C2×Dic788C22semidirect
16. C2×#15691348C22semidirect
17. C2×#15691348C22semidirect
24. #20263326C22semidirect
26. #20263324C22semidirect
34. #20263237C22semidirect
35. #20263236C22semidirect
36. C2×Q16×C197C22semidirect
38. C4⋊Q8×C197C22semidirect
40. #97×C197C22semidirect
130. D197#497direct (N × H)
131. C197C2×#497semidirect
142. #497D197direct (N × H)
151. C2×Q16C2×D197semidirect
161. C4⋊Q8C2×D197semidirect
172. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1972
C2×SD16×D1972
C4⋊Q8×D1972
#124×D1972
#271614742
#271614732
#271614722
#271614712
#271614702
#97×D1972
#497×C1972
C2×#497197