Finite Group Databasebeta

← #38751938

Subgroups — #38751938

Order 25216.

994 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×D788C211yesmaximal, characteristic, complemented
2C2×#20263351C212yesmaximal, complemented
3#27161229C214yesmaximal, complemented
4#27161212C214yesmaximal, complemented
5#27161083C212yesmaximal, complemented
6#27161043C211yesmaximal, characteristic, complemented
7#525×C197C211yesmaximal, characteristic, complemented
8C24×D197—21no—
9C22×D788—28no—
10C22×D788—22no—
11C22×D788—21no—
12C22×D788C2214yes—
13C22×D788C2212yes—
14#20263351—28no—
15C2×#15691345C2214yescomplemented
16C2×#15691343C2214yescomplemented
17C2×#15691342C2212yescomplemented
18#20263246C2218yescomplemented
19C22⋊#13096802C2214yescomplemented
20C2×#41×C197C2211yescharacteristic, complemented
21C23×C4×C197C2211yescharacteristic
22#123×C197—21no—
23C22⋊Q8×C197C2212yescomplemented
24#121×C197C2212yescomplemented
25#120×C197—21no—
26C4×D4×C197—22no—
27#117×C197C2211yescharacteristic, complemented
28C23×D197—28no—
29C23×D197—22no—
30C23×D197—24no—
31C23×D197—48no—
32C2×D788—28no—
33C2×D788—216no—
34C2×D788—216no—
35C2×D788—24no—
36C2×D788—24no—
37C2×D788—22no—
38C2×D788D418yescomplemented
39C2×D788C2314yes—
40#15691345—28no—
41#15691343—28no—
42#15691342—24no—
43C2×#13096802C2314yescomplemented
44C24×C197D411yescharacteristic
45#41×C197—24no—
46C2×Q8×C197C2311yescharacteristic, complemented
47C2×D4×C197—22no—
48C2×D4×C197—21no—
49C2×D4×C197C2311yescharacteristic, complemented
50C22×C4×C197—24no—
51C22×C4×C197—21no—
52C22×C4×C197D412yes—
53C22×C4×C197D411yescharacteristic
54C22×C4×C197C2312yes—
55C22×C4×C197C2311yescharacteristic
56C4⋊C4×C197—22no—
57C4⋊C4×C197C2312yescomplemented
58C22⋊C4×C197—21no—
59C22⋊C4×C197—22no—
60C22⋊C4×C197—22no—
61C42×C197—21no—
62C22×D197—216no—
63C22×D197—28no—
64C22×D197—24no—
65C22×D197—432no—
66C22×D197—48no—
67C22×D197—48no—
68C22×D197—48no—
69D788—216no—
70D788—28no—
71D788—24no—
72D788C2×D418yes—
73#13096802—24no—
74C23×C197—22no—
75C23×C197—24no—
76C23×C197—21no—
77C23×C197C2×D411yescharacteristic
78C23×C197C2×D412yes—
79Q8×C197—22no—
80D4×C197—22no—
81D4×C197—42no—
82D4×C197—44no—
83C2×C4×C197—22no—
84C2×C4×C197—24no—
85C2×C4×C197—24no—
86C2×C4×C197—22no—
87C2×C4×C197—21no—
88C2×C4×C197—42no—
89C2×C4×C197C2×D414yes—
90C2×C4×C197C2×D412yes—
91C2×C4×C197C2×D411yescharacteristic
92C2×C4×C197C2411yescharacteristic
93C22×C197—28no—
94C22×C197—22no—
95C22×C197—22no—
96C22×C197—22no—
97C22×C197—41no—
98C22×C197—42no—
99C22×C197#12012yes—
100C22×C197AGL(1,ℤ/8)14yescomplemented
101C22×C197C22×D411yescharacteristic
102C2×D197—28no—
103C2×D197—416no—
104C2×D197—432no—
105C2×D197—48no—
106C4×C197—22no—
107C4×C197—41no—
108C4×C197—42no—
109C4×C197#12012yes—
110C4×C197C22×D412yes—
111C2×C197—22no—
112C2×C197—24no—
113C2×C197—81no—
114C2×C197C2×#12011yescharacteristic
115C2×C197C2×AGL(1,ℤ/8)12yes—
116D197—48no—
117C197#300311yescharacteristic, complemented, Sylow
118#3003—1971nomaximal, Sylow
119C23×D4—1971no—
120C2×AGL(1,ℤ/8)—1972no—
121#525D19711yescharacteristic, complemented
122#450—1974no—
123#447—1974no—
124#418—1972no—
125#407—1971no—
126C25—3941no—
127C2×#41C2×D19711yescharacteristic, complemented
128C22×D4—3948no—
129C22×D4—3942no—
130C22×D4—3941no—
131C22×D4—1974no—
132C22×D4—1972no—
133C23×C4C2×D19711yescharacteristic
134AGL(1,ℤ/8)—3948no—
135C2×SD16—1974no—
136C2×D8—1974no—
137C2×M16—1972no—
138#123—21no—
139C22⋊Q8C2×D19712yescomplemented
140#121C2×D19712yescomplemented
141#120—21no—
142C4×D4—22no—
143#117C2×D19711yescharacteristic, complemented
144#102—1978no—
145C22⋊C8—1974no—
146C24—7888no—
147C24—3948no—
148C24—3942no—
149C24—3944no—
150C24C22⋊D19711yescharacteristic
151#41—24no—
152C2×Q8C22×D19711yescharacteristic, complemented
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—3948no—
156C2×D4—39416no—
157C2×D4—39416no—
158C2×D4—3944no—
159C2×D4—3944no—
160C2×D4—3942no—
161C2×D4—1978no—
162C2×D4—1974no—
163C2×D4C22×D19711yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4C22⋊D19712yes—
167C22×C4C22⋊D19711yescharacteristic
168C22×C4C22×D19712yes—
169C22×C4C22×D19711yescharacteristic
170SD16—3948no—
171D8—3948no—
172M16—3944no—
173C2×C8—1974no—
174C4⋊C4—22no—
175C4⋊C4C22×D19712yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—22no—
181C23—24no—
182C23—21no—
183C23—78832no—
184C23—7888no—
185C23—7888no—
186C23—7888no—
187C23—39416no—
188C23—3948no—
189C23—3944no—
190C23C2×C22⋊D19711yescharacteristic
191C23C2×C22⋊D19712yes—
192Q8—22no—
193D4—22no—
194D4—42no—
195D4—44no—
196D4—39416no—
197D4—3948no—
198D4—3944no—
199D4—1978no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4D4×D19714yes—
207C2×C4C2×C22⋊D19712yes—
208C2×C4C2×C22⋊D19711yescharacteristic
209C2×C4C23×D19711yescharacteristic
210C8—3944no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—42no—
216C22—41no—
217C22—78816no—
218C22—78832no—
219C22—7888no—
220C22—3948no—
221C22#2026329412yes—
222C22#2026335114yescomplemented
223C22C22×C22⋊D19711yescharacteristic
224C4—22no—
225C4—42no—
226C4—41no—
227C4#2026329412yes—
228C4C2×D4×D19712yes—
229C2—22no—
230C2—24no—
231C2—81no—
232C2—7888no—
233C2C2×#2026329411yescharacteristic
234C2C2×#2026335112yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#525×C197C2
socleC22×C197C22×D4
hypercenter#525D197
hypocenterC197#3003
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C23×D788C2semidirect
2. C2×#20263351C2semidirect
3. #27161229C2semidirect
4. #27161212C2semidirect
5. #27161083C2semidirect
6. #27161043C2semidirect
7. #525×C197C2semidirect
15. C2×#15691345C22semidirect
16. C2×#15691343C22semidirect
17. C2×#15691342C22semidirect
18. #20263246C22semidirect
19. C22⋊#13096802C22semidirect
20. C2×#41×C197C22semidirect
23. C22⋊Q8×C197C22semidirect
24. #121×C197C22semidirect
27. #117×C197C22semidirect
38. C2×D788D4semidirect
43. C2×#13096802C23semidirect
46. C2×Q8×C197C23semidirect
49. C2×D4×C197C23semidirect
57. C4⋊C4×C197C23semidirect
100. C22×C197AGL(1,ℤ/8)semidirect
117. C197#3003semidirect
121. #525D197semidirect
127. C2×#41C2×D197semidirect
139. C22⋊Q8C2×D197semidirect
140. #121C2×D197semidirect
143. #117C2×D197semidirect
152. C2×Q8C22×D197semidirect
163. C2×D4C22×D197semidirect
175. C4⋊C4C22×D197semidirect
222. C22#20263351semidirect

Maximal subgroups

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

subgroupindex
C23×D7882
C2×#202633512
#271612292
#271612122
#271610832
#271610432
#525×C1972
#3003197