Finite Group Databasebeta

← #38752955

Subgroups — #38752955

Order 25216.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#27161684C211yesmaximal, characteristic, complemented
2#27161403C212yesmaximal, complemented
3#27161379C212yesmaximal, complemented
4#27161274C212yesmaximal, complemented
5#27161213C212yesmaximal, complemented
6#27161189C212yesmaximal, complemented
7C4×#15691343C212yesmaximal, complemented
8#27161140C211yesmaximal, characteristic, complemented
9#518×C197C211yesmaximal, characteristic, complemented
10C2×#15691343C2212yes—
11C2×#15691343C2212yes—
12#20263323C2211yescharacteristic, complemented
13#20263318C2211yescharacteristic, complemented
14#20263311—22no—
15#20263300C2212yes—
16#20263299—22no—
17C4×D788C2212yescomplemented
18#20263259C2212yescomplemented
19#20263257C2212yescomplemented
20C4⋊#13096802C2212yescomplemented
21#20263247C2212yes—
22#20263247C2212yes—
23#20263246C2212yescomplemented
24#20263246C2212yescomplemented
25C22⋊#13096802C2212yes—
26#20263237C2211yescharacteristic, complemented
27C4×#13096802C2211yescharacteristic, complemented
28C22×D4×C197C2211yescharacteristic
29C4×D4×C197—22no—
30C4×D4×C197C2212yes—
31C4×D4×C197C2212yes—
32#117×C197C2211yescharacteristic
33#117×C197C2211yescharacteristic
34C2×C22⋊C4×C197—22no—
35C2×D788—22no—
36C2×D788C2312yes—
37C2×C22⋊D197—22no—
38C2×C4×D197—22no—
39#15691343—24no—
40#15691343—24no—
41C2×#13096802C2312yes—
42C2×#13096802C2312yes—
43C4⋊Dic197C2312yes—
44#15691337—22no—
45#15691335—22no—
46#15691335—22no—
47C24×C197—21no—
48C2×D4×C197—24no—
49C2×D4×C197—21no—
50C2×D4×C197C2312yes—
51C22×C4×C197—22no—
52C22×C4×C197—22no—
53C22×C4×C197C2311yescharacteristic
54C4⋊C4×C197D412yes—
55C4⋊C4×C197C2312yes—
56C22⋊C4×C197—21no—
57C22⋊C4×C197—22no—
58C22⋊C4×C197—22no—
59C22⋊C4×C197D412yes—
60C42×C197—21no—
61C42×C197C2311yescharacteristic
62C42×C197C2311yescharacteristic
63C22×D197—22no—
64D788—22no—
65D788—22no—
66D788—82no—
67C22⋊D197—82no—
68C2×Dic197—22no—
69C4×D197—82no—
70#13096802—22no—
71#13096802—22no—
72C23×C197—24no—
73C23×C197—22no—
74C23×C197—82no—
75C23×C197C2×D411yescharacteristic
76D4×C197—24no—
77D4×C197—22no—
78D4×C197#4114yes—
79C2×C4×C197—22no—
80C2×C4×C197—22no—
81C2×C4×C197—82no—
82C2×C4×C197—82no—
83C2×C4×C197C2×D411yescharacteristic
84C2×C4×C197C2×D412yes—
85C2×C4×C197C2×D412yes—
86C2×C4×C197C2411yescharacteristic
87C22×C197—21no—
88C22×C197—24no—
89C22×C197—22no—
90C22×C197—42no—
91C22×C197—42no—
92C22×C197—84no—
93C22×C197C22×D411yescharacteristic
94C2×D197—42no—
95C2×D197—42no—
96C2×D197—42no—
97C4×C197—21no—
98C4×C197—22no—
99C4×C197—22no—
100C4×C197—42no—
101C4×C197C2×#4112yes—
102Dic197—42no—
103C2×C197—41no—
104C2×C197—44no—
105C2×C197#52511yescharacteristic
106C2×C197#57612yes—
107D197—82no—
108C197#312211yescharacteristic, complemented, Sylow
109#3122—1971nomaximal, Sylow
110#540—1971no—
111#518D19711yescharacteristic, complemented
112#482—1972no—
113#478—1972no—
114#459—1972no—
115#447—1972no—
116#442—1972no—
117C4×D8—1972no—
118#433—1971no—
119C22×D4C2×D19711yescharacteristic
120C2×D8—1972no—
121C2×D8—1972no—
122#127—1971no—
123#125—1971no—
124#123—3942no—
125#121—3942no—
126#121—1972no—
127C4×D4—22no—
128C4×D4—1972no—
129C4×D4C2×D19712yes—
130C4×D4C2×D19712yes—
131#117C2×D19711yescharacteristic
132#117C2×D19711yescharacteristic
133C2×C22⋊C4—22no—
134#107—1972no—
135#106—1972no—
136C4⋊C8—1972no—
137#102—1972no—
138#102—1972no—
139#102—1972no—
140#102—1972no—
141C22⋊C8—1972no—
142#97—1971no—
143C4×C8—1971no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—3942no—
148C2×D4—3942no—
149C2×D4—1972no—
150C2×D4C22×D19712yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—3942no—
154C22×C4C22×D19711yescharacteristic
155D8—3944no—
156D8—3944no—
157C2×C8—1972no—
158C2×C8—1972no—
159C4⋊C4—3942no—
160C4⋊C4—1972no—
161C4⋊C4C22⋊D19712yes—
162C4⋊C4C22×D19712yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—3942no—
167C22⋊C4—3942no—
168C22⋊C4C22⋊D19712yes—
169C42—21no—
170C42C22×D19711yescharacteristic
171C42C22×D19711yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—3942no—
176C23C2×C22⋊D19711yescharacteristic
177D4—24no—
178D4—22no—
179D4—15762no—
180D4—15762no—
181D4—3942no—
182D4—3942no—
183D4#1569135814yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—15762no—
189C2×C4—3942no—
190C2×C4C2×C22⋊D19711yescharacteristic
191C2×C4C2×C22⋊D19712yes—
192C2×C4C2×C22⋊D19712yes—
193C2×C4C23×D19711yescharacteristic
194C8—3942no—
195C8—3942no—
196C22—22no—
197C22—21no—
198C22—24no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—7882no—
203C22—7882no—
204C22—7882no—
205C22C22×C22⋊D19711yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—42no—
210C4—7882no—
211C4C2×#1569135812yes—
212C2—41no—
213C2—44no—
214C2—15762no—
215C2#2716159011yescharacteristic
216C2#2716188212yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#518×C197C2
socleC22×C197C22×D4
hypercenter#518D197
hypocenterC197#3122
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #27161684C2semidirect
2. #27161403C2semidirect
3. #27161379C2semidirect
4. #27161274C2semidirect
5. #27161213C2semidirect
6. #27161189C2semidirect
7. C4×#15691343C2semidirect
8. #27161140C2semidirect
9. #518×C197C2semidirect
12. #20263323C22semidirect
13. #20263318C22semidirect
17. C4×D788C22semidirect
18. #20263259C22semidirect
19. #20263257C22semidirect
20. C4⋊#13096802C22semidirect
23. #20263246C22semidirect
24. #20263246C22semidirect
26. #20263237C22semidirect
27. C4×#13096802C22semidirect
108. C197#3122semidirect
111. #518D197semidirect

Maximal subgroups

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

subgroupindex
#271616842
#271614032
#271613792
#271612742
#271612132
#271611892
C4×#156913432
#271611402
#518×C1972
#3122197