Finite Group Databasebeta

← #38752950

Subgroups — #38752950

Order 25216.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 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#27161553C211yesmaximal, characteristic, complemented
2#27161400C212yesmaximal, complemented
3#27161376C212yesmaximal, complemented
4#27161271C212yesmaximal, complemented
5#27161212C212yesmaximal, complemented
6#27161189C212yesmaximal, complemented
7C4×#15691343C212yesmaximal, complemented
8#27161140C211yesmaximal, characteristic, complemented
9#540×C197C211yesmaximal, characteristic, complemented
10C22×D788C2211yescharacteristic
11C2×#15691343C2212yes—
12C2×#15691343C2212yes—
13C4×D788C2212yes—
14C4×D788C2212yes—
15#20263288—22no—
16#20263285C2211yescharacteristic
17C22⋊C4×D197—22no—
18#20263259C2212yescomplemented
19#20263257C2212yescomplemented
20C4⋊#13096802C2212yescomplemented
21#20263247C2212yescomplemented
22#20263247C2212yescomplemented
23#20263246C2212yes—
24#20263246C2212yes—
25C22⋊#13096802C2212yes—
26#20263237C2211yescharacteristic, complemented
27C4×#13096802C2211yescharacteristic, complemented
28#127×C197C2211yescharacteristic, complemented
29#125×C197C2211yescharacteristic, complemented
30#123×C197—22no—
31#121×C197—22no—
32#121×C197C2212yes—
33C4×D4×C197C2212yescomplemented
34#117×C197C2211yescharacteristic
35C23×D197—21no—
36C2×D788—24no—
37C2×D788—21no—
38C2×D788C2312yes—
39C2×C4×D197—22no—
40C2×C4×D197—22no—
41#15691343—24no—
42#15691343—24no—
43C2×#13096802C2312yes—
44C2×#13096802C2312yes—
45C4⋊Dic197C2312yes—
46C22⋊Dic197—21no—
47#15691335—22no—
48#15691335—22no—
49C4×Dic197—21no—
50C2×D4×C197—22no—
51C2×D4×C197—22no—
52C2×D4×C197C2312yes—
53C22×C4×C197—22no—
54C22×C4×C197C2311yescharacteristic
55C4⋊C4×C197—22no—
56C4⋊C4×C197D412yes—
57C4⋊C4×C197C2312yes—
58C22⋊C4×C197—22no—
59C22⋊C4×C197—22no—
60C22⋊C4×C197D412yes—
61C42×C197C2311yescharacteristic
62C42×C197C2311yescharacteristic
63C22×D197—22no—
64C22×D197—24no—
65C22×D197—82no—
66D788—24no—
67D788—22no—
68D788#4114yes—
69C2×Dic197—22no—
70C4×D197—82no—
71C4×D197—82no—
72#13096802—22no—
73#13096802—22no—
74C23×C197—22no—
75C23×C197C2×D411yescharacteristic
76D4×C197—22no—
77D4×C197—22no—
78D4×C197—82no—
79D4×C197—82no—
80C2×C4×C197—22no—
81C2×C4×C197—22no—
82C2×C4×C197—82no—
83C2×C4×C197C2×D412yes—
84C2×C4×C197C2×D412yes—
85C2×C4×C197C2×D411yescharacteristic
86C2×C4×C197C2411yescharacteristic
87C22×C197—21no—
88C22×C197—22no—
89C22×C197—42no—
90C22×C197—42no—
91C22×C197—42no—
92C22×C197C22×D411yescharacteristic
93C2×D197—24no—
94C2×D197—42no—
95C2×D197—42no—
96C2×D197—84no—
97C4×C197—21no—
98C4×C197—22no—
99C4×C197—22no—
100C4×C197—42no—
101C4×C197C2×#4112yes—
102Dic197—42no—
103C2×C197—41no—
104C2×C197—82no—
105C2×C197#52511yescharacteristic
106C2×C197#57612yes—
107D197—44no—
108C197#312211yescharacteristic, complemented, Sylow
109#3122—1971nomaximal, Sylow
110#540D19711yescharacteristic, complemented
111#518—1971no—
112#482—1972no—
113#478—1972no—
114#459—1972no—
115#447—1972no—
116#442—1972no—
117C4×D8—1972no—
118#433—1971no—
119C22×D4—1971no—
120C2×D8—1972no—
121C2×D8—1972no—
122#127C2×D19711yescharacteristic, complemented
123#125C2×D19711yescharacteristic, complemented
124#123—22no—
125#121—22no—
126#121C2×D19712yes—
127C4×D4—3942no—
128C4×D4—1972no—
129C4×D4—1972no—
130C4×D4C2×D19712yescomplemented
131#117—1971no—
132#117C2×D19711yescharacteristic
133C2×C22⋊C4—3942no—
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—3941no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—3944no—
148C2×D4—3941no—
149C2×D4—1972no—
150C2×D4C22×D19712yes—
151C22×C4—22no—
152C22×C4—3942no—
153C22×C4—3942no—
154C22×C4C22×D19711yescharacteristic
155D8—3944no—
156D8—3944no—
157C2×C8—1972no—
158C2×C8—1972no—
159C4⋊C4—22no—
160C4⋊C4—1972no—
161C4⋊C4C22⋊D19712yes—
162C4⋊C4C22×D19712yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—3941no—
166C22⋊C4—3942no—
167C22⋊C4—3942no—
168C22⋊C4C22⋊D19712yes—
169C42—3941no—
170C42C22×D19711yescharacteristic
171C42C22×D19711yescharacteristic
172C23—22no—
173C23—15762no—
174C23—3944no—
175C23—3942no—
176C23C2×C22⋊D19711yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—82no—
181D4—3944no—
182D4—3942no—
183D4—1974no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—15762no—
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—42no—
199C22—42no—
200C22—42no—
201C22—7882no—
202C22—7882no—
203C22—15764no—
204C22—3944no—
205C22C22×C22⋊D19711yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—42no—
210C4—7882no—
211C4#41×D19712yes—
212C2—41no—
213C2—82no—
214C2—7884no—
215C2#2716158611yescharacteristic
216C2#2716188212yes—
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#540×C197C2
socleC22×C197C22×D4
hypercenter#540D197
hypocenterC197#3122
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #27161553C2semidirect
2. #27161400C2semidirect
3. #27161376C2semidirect
4. #27161271C2semidirect
5. #27161212C2semidirect
6. #27161189C2semidirect
7. C4×#15691343C2semidirect
8. #27161140C2semidirect
9. #540×C197C2semidirect
18. #20263259C22semidirect
19. #20263257C22semidirect
20. C4⋊#13096802C22semidirect
21. #20263247C22semidirect
22. #20263247C22semidirect
26. #20263237C22semidirect
27. C4×#13096802C22semidirect
28. #127×C197C22semidirect
29. #125×C197C22semidirect
33. C4×D4×C197C22semidirect
108. C197#3122semidirect
110. #540D197semidirect
122. #127C2×D197semidirect
123. #125C2×D197semidirect
130. C4×D4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271615532
#271614002
#271613762
#271612712
#271612122
#271611892
C4×#156913432
#271611402
#540×C1972
#3122197