Finite Group Databasebeta

← #38744728

Subgroups — #38744728

Order 25216.

248 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 41 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#27161821C211yesmaximal, characteristic, complemented
2#27161252C212yesmaximal, complemented
3#27160876C211yesmaximal, characteristic, complemented
4#27160875C212yesmaximal, complemented
5#454×C197C211yesmaximal, characteristic, complemented
6#20263351—22no—
7#20263315C2212yes—
8#20263314—21no—
9#20263312—22no—
10#20263296—22no—
11#20263294—21no—
12#20263253—22no—
13#20263245C2212yescomplemented
142−1+4×C197C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C197—21no—
16#124×C197C2211yescharacteristic
17#104×C197—21no—
18#101×C197C2211yescharacteristic, complemented
19C23×D197—21no—
20C2×Dic394—21no—
21C2×D788—21no—
22C2×D788D412yes—
23C2×C22⋊D197—22no—
24#15691345—42no—
25#15691343—42no—
26#15691342—22no—
27C22⋊Dic197—21no—
28#15691335—22no—
29#15691335—22no—
30#15691335—22no—
31C4×Dic197D412yes—
32#41×C197—21no—
33#41×C197—22no—
34#41×C197—41no—
35C2×Q8×C197—41no—
36C2×Q8×C197C2311yescharacteristic
37C2×D4×C197D411yescharacteristic
38SD16×C197—41no—
39D8×C197—41no—
40M16×C197—21no—
41C22⋊C4×C197—22no—
42C42×C197D411yescharacteristic
43C22×D197—22no—
44C22×D197—82no—
45C22×D197—81no—
46Dic394—81no—
47D788—42no—
48D788—81no—
49C22⋊D197—82no—
50C2×Dic197—22no—
51#13096802—42no—
52C23×C197—21no—
53Q8×C197—21no—
54Q8×C197—22no—
55Q8×C197—41no—
56D4×C197—21no—
57D4×C197—22no—
58D4×C197—41no—
59D4×C197—41no—
60C2×C4×C197—21no—
61C2×C4×C197—41no—
62C2×C4×C197—42no—
63C2×C4×C197C2×D412yes—
64C2×C4×C197C2×D411yescharacteristic
65C8×C197—41no—
66C22×C197—42no—
67C22×C197—41no—
68C22×C197—41no—
69C22×C197#12011yescharacteristic
70C2×D197—42no—
71C2×D197—42no—
72C2×D197—42no—
73C2×D197—84no—
74C4×C197—21no—
75C4×C197—22no—
76C4×C197—41no—
77C4×C197—41no—
78Dic197—42no—
79C2×C197—21no—
80C2×C197—81no—
81C2×C197—81no—
82C2×C197B(4,𝔽2)11yescharacteristic
83D197—82no—
84C197#220211yescharacteristic, complemented, Sylow
85#2202—1971nomaximal, Sylow
86H(3,𝔽4)—1971no—
87#454—1972no—
88#454D19711yescharacteristic, complemented
89#355—1971no—
90#355—1972no—
912−1+4C2×D19711yescharacteristic, complemented
92AGL(1,ℤ/8)—21no—
93AGL(1,ℤ/8)—3942no—
94#124—3942no—
95#124—3941no—
96#124—1972no—
97#124C2×D19711yescharacteristic
98#120—3942no—
99#120—3941no—
100#104—21no—
101#104—3942no—
102#101—1972no—
103#101C2×D19711yescharacteristic, complemented
104C24—3941no—
105#41—22no—
106#41—21no—
107#41—41no—
108C2×Q8—41no—
109C2×Q8—3941no—
110C2×Q8C22×D19711yescharacteristic
111C2×D4—3941no—
112C2×D4—3942no—
113C2×D4—1972no—
114C2×D4C22⋊D19711yescharacteristic
115SD16—41no—
116SD16—7882no—
117D8—41no—
118D8—7882no—
119M16—21no—
120M16—3942no—
121C22⋊C4—22no—
122C22⋊C4—3942no—
123C22⋊C4—3941no—
124C22⋊C4—3942no—
125C22⋊C4—3942no—
126C42—1972no—
127C42C22⋊D19711yescharacteristic
128C23—21no—
129C23—15762no—
130C23—15761no—
131C23—3942no—
132Q8—21no—
133Q8—22no—
134Q8—41no—
135Q8—15761no—
136D4—21no—
137D4—22no—
138D4—41no—
139D4—41no—
140D4—7882no—
141D4—15761no—
142D4—15762no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—3942no—
147C2×C4D4×D19712yes—
148C2×C4C2×C22⋊D19711yescharacteristic
149C8—41no—
150C8—7882no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—7882no—
155C22—7882no—
156C22—7882no—
157C22—15764no—
158C22#2026329411yescharacteristic
159C4—21no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—7882no—
164C2—21no—
165C2—81no—
166C2—81no—
167C2—15762no—
168C2#2716126511yescharacteristic
subgroupG/H
centerC2#27161265
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting#454×C197C2
socleC2×C197B(4,𝔽2)
hypercenter#454D197
hypocenterC197#2202
LCSG ▷ C2×Q8×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #27161821C2semidirect
2. #27161252C2semidirect
3. #27160876C2semidirect
4. #27160875C2semidirect
5. #454×C197C2semidirect
13. #20263245C22semidirect
14. 2−1+4×C197C22semidirect
18. #101×C197C22semidirect
84. C197#2202semidirect
88. #454D197semidirect
91. 2−1+4C2×D197semidirect
103. #101C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618212
#271612522
#271608762
#271608752
#454×C1972
#2202197