Finite Group Databasebeta

← #38753175

Subgroups — #38753175

Order 25216.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#20263358C212yesmaximal, complemented
2C2×#20263353C212yesmaximal, complemented
3#27161630C211yesmaximal, characteristic, complemented
4#124×D197C211yesmaximal, characteristic, complemented
5#27161471C211yesmaximal, characteristic, complemented
6#27161469C211yesmaximal, characteristic, complemented
7#27161468C211yesmaximal, characteristic, complemented
8#27161464C212yesmaximal, complemented
9#27161460C212yesmaximal, complemented
10#27161031C211yesmaximal, characteristic, complemented
11#497×C197C211yesmaximal, characteristic, complemented
12C2×#15691359C2212yes—
13C2×Q8×D197C2211yescharacteristic
14C2×D4×D197C2211yescharacteristic
15#20263358—28no—
16#20263353—28no—
17C2×#15691348C2212yescomplemented
18C2×#15691347C2212yescomplemented
19C2×#15691345C2212yes—
20C2×#15691345C2212yescomplemented
21C2×D1576C2212yes—
22C2×#15691343C2212yes—
23C2×#15691341C2212yes—
24#20263323C2211yescharacteristic, complemented
25#20263315C2211yescharacteristic
26#20263315C2212yescomplemented
27#20263314C2211yescharacteristic
28#20263313C2211yescharacteristic
29#20263312—22no—
30#20263301—22no—
31#20263288—22no—
32C22⋊C4×D197—22no—
33C42×D197C2211yescharacteristic
34#20263237C2211yescharacteristic, complemented
35C8×Dic197C2212yescomplemented
36C2×Q16×C197C2212yescomplemented
37C2×SD16×C197C2212yes—
38C4⋊Q8×C197C2211yescharacteristic, complemented
39#124×C197C2211yescharacteristic
40#97×C197C2211yescharacteristic, complemented
41C23×D197—21no—
42#15691359—24no—
43#15691359—24no—
44C2×Dic394—21no—
45C2×Dic394C2311yescharacteristic
46Q8×D197—24no—
47C2×D788—22no—
48C2×D788—22no—
49C2×D788C2311yescharacteristic
50C2×D788C2312yes—
51C2×C22⋊D197—21no—
52D4×D197—24no—
53C2×C4×D197—21no—
54C2×C4×D197—22no—
55C2×C4×D197D412yes—
56C2×C4×D197C2311yescharacteristic
57#15691348—24no—
58#15691347—24no—
59#15691345—24no—
60#15691345—24no—
61D1576—24no—
62#15691343—24no—
63#15691341D418yescomplemented
64C2×#13096802C2312yes—
65C22⋊Dic197—22no—
66#15691335—22no—
67#15691335—24no—
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—22no—
82C22×D197—44no—
83C22×D197—42no—
84C22×D197C2×D411yescharacteristic
85Dic394—22no—
86Dic394—42no—
87D788—22no—
88D788—22no—
89D788—22no—
90D788—44no—
91D788—44no—
92C22⋊D197—42no—
93C2×Dic197—21no—
94C2×Dic197C2×D412yes—
95C2×Dic197C2×D411yescharacteristic
96C4×D197—42no—
97C4×D197—42no—
98C4×D197—44no—
99C4×D197C2×D414yes—
100#13096802C2×D414yes—
101C23×C197—21no—
102Q8×C197—22no—
103Q8×C197—22no—
104Q8×C197—22no—
105D4×C197—22no—
106C2×C4×C197—21no—
107C2×C4×C197—22no—
108C2×C4×C197C2×D412yes—
109C2×C4×C197C2411yescharacteristic
110C8×C197C2×D414yes—
111C22×C197—41no—
112C22×C197—42no—
113C22×C197C22×D411yescharacteristic
114C2×D197—22no—
115C2×D197—42no—
116C2×D197—41no—
117C2×D197—44no—
118C2×D197—42no—
119C2×D197—84no—
120C2×D197#12712yes—
121C4×C197—22no—
122C4×C197—42no—
123C4×C197—41no—
124C4×C197C22×D412yes—
125Dic197—22no—
126Dic197—41no—
127Dic197#12712yes—
128C2×C197—81no—
129C2×C197C2×#12711yescharacteristic
130C2×C197#57512yes—
131D197—22no—
132D197—82no—
133D197—81no—
134C197#315311yescharacteristic, complemented, Sylow
135#3153—1971nomaximal, Sylow
136C2×#137—1972no—
137C2×AGL(1,ℤ/8)—1972no—
138#532—1971no—
139C2×#124—1971no—
140#497—1971no—
141#497D19711yescharacteristic, complemented
142#496—1971no—
143#496—1971no—
144#495—1972no—
145#495—1972no—
146C4×M16—1971no—
147C2×#41—1972no—
148C22×Q8—1971no—
149C22×D4—1971no—
150#137—3948no—
151AGL(1,ℤ/8)—3948no—
152C2×Q16—1972no—
153C2×Q16C2×D19712yescomplemented
154C2×SD16—1972no—
155C2×SD16—1972no—
156C2×SD16—1972no—
157C2×SD16C2×D19712yes—
158C2×D8—1972no—
159C2×D8—1972no—
160C2×M16—1972no—
161C4⋊Q8C2×D19711yescharacteristic, complemented
162#127—1971no—
163#124—3942no—
164#124—1971no—
165#124—1972no—
166#124—1971no—
167#124—1971no—
168#124C2×D19711yescharacteristic
169#121—3942no—
170C4×D4—3942no—
171C2×C22⋊C4—3942no—
172C2×C42—1971no—
173#97—1971no—
174#97C2×D19711yescharacteristic, complemented
175C4×C8—1972no—
176C24—3941no—
177#41—3944no—
178#41—3944no—
179C2×Q8—3941no—
180C2×Q8—3944no—
181C2×Q8—1971no—
182C2×Q8C22×D19711yescharacteristic
183C2×Q8C22×D19712yes—
184C2×D4—3942no—
185C2×D4—3944no—
186C2×D4—3941no—
187C2×D4—3942no—
188C2×D4—1971no—
189C2×D4—1972no—
190C2×D4C22×D19711yescharacteristic
191C22×C4—3941no—
192C22×C4—3942no—
193C22×C4—1972no—
194C22×C4—1971no—
195Q16—24no—
196Q16—3944no—
197SD16—24no—
198SD16—3944no—
199SD16—3944no—
200SD16—3944no—
201D8—3944no—
202D8—3944no—
203M16—1978no—
204C2×C8—1972no—
205C2×C8C22×D19712yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—3942no—
209C22⋊C4—3944no—
210C22⋊C4—3942no—
211C22⋊C4—3942no—
212C42—1972no—
213C42—1971no—
214C42C22×D19711yescharacteristic
215C23—21no—
216C23—7884no—
217C23—7882no—
218C23—3941no—
219C23—3942no—
220C23—1971no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—7882no—
225Q8—3942no—
226D4—22no—
227D4—7884no—
228D4—7884no—
229D4—7882no—
230D4—3942no—
231D4—3942no—
232D4—3942no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—7882no—
236C2×C4—7882no—
237C2×C4—7884no—
238C2×C4—3941no—
239C2×C4—1972no—
240C2×C4—1971no—
241C2×C4—1974no—
242C2×C4D4×D19712yes—
243C2×C4C23×D19711yescharacteristic
244C8—1974no—
245C8D4×D19714yes—
246C22—41no—
247C22—42no—
248C22—7882no—
249C22—7881no—
250C22—7884no—
251C22—7882no—
252C22—15764no—
253C22—3942no—
254C22—1972no—
255C22C2×D4×D19711yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—7881no—
260C4—3942no—
261C4—1972no—
262C4C2×D4×D19712yes—
263C2—81no—
264C2—15761no—
265C2—15762no—
266C2—3942no—
267C2#127×D19711yescharacteristic
268C2#2716187812yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#497×C197C2
socleC22×C197C22×D4
hypercenter#497D197
hypocenterC197#3153
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×#20263358C2semidirect
2. C2×#20263353C2semidirect
3. #27161630C2semidirect
4. #124×D197C2semidirect
5. #27161471C2semidirect
6. #27161469C2semidirect
7. #27161468C2semidirect
8. #27161464C2semidirect
9. #27161460C2semidirect
10. #27161031C2semidirect
11. #497×C197C2semidirect
17. C2×#15691348C22semidirect
18. C2×#15691347C22semidirect
20. C2×#15691345C22semidirect
24. #20263323C22semidirect
26. #20263315C22semidirect
34. #20263237C22semidirect
35. C8×Dic197C22semidirect
36. C2×Q16×C197C22semidirect
38. C4⋊Q8×C197C22semidirect
40. #97×C197C22semidirect
63. #15691341D4semidirect
134. C197#3153semidirect
141. #497D197semidirect
153. C2×Q16C2×D197semidirect
161. C4⋊Q8C2×D197semidirect
174. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633582
C2×#202633532
#271616302
#124×D1972
#271614712
#271614692
#271614682
#271614642
#271614602
#271610312
#497×C1972
#3153197