Finite Group Databasebeta

← #38744730

Subgroups — #38744730

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#27161254C212yesmaximal, complemented
3#27161251C211yesmaximal, characteristic, complemented
4#27160875C212yesmaximal, complemented
5#355×C197C211yesmaximal, characteristic, complemented
6#20263374C2211yescharacteristic, complemented
7#20263354—22no—
8#20263352—21no—
9#20263315C2212yes—
10#20263314—21no—
11#20263312—22no—
12#20263296—22no—
13#20263294—21no—
14#20263254—21no—
15#20263252—22no—
16#20263245C2211yescharacteristic, complemented
17#124×C197C2211yescharacteristic
18#101×C197C2212yescomplemented
19C23×D197—21no—
20#15691359—41no—
21#15691358—22no—
22#15691358—21no—
23C2×Dic394—21no—
24Q8×D197—41no—
25C2×D788—21no—
26C2×D788D412yes—
27C2×C22⋊D197—22no—
28#15691347—42no—
29#15691346—41no—
30D1576—42no—
31#15691343—41no—
32#15691342—21no—
33C22⋊Dic197—21no—
34#15691335—22no—
35#15691335—22no—
36#15691335—22no—
37C4×Dic197D412yes—
38C2×Q8×C197C2311yescharacteristic
39C2×D4×C197D411yescharacteristic
40M16×C197—22no—
41C22⋊C4×C197—22no—
42C42×C197D411yescharacteristic
43C22×D197—22no—
44C22×D197—82no—
45C22×D197—81no—
46Dic394—21no—
47Dic394—22no—
48Dic394—81no—
49D788—21no—
50D788—22no—
51D788—42no—
52D788—81no—
53C22⋊D197—41no—
54C22⋊D197—82no—
55C2×Dic197—22no—
56C4×D197—42no—
57C4×D197—41no—
58#13096802—41no—
59C23×C197—21no—
60Q8×C197—41no—
61D4×C197—41no—
62C2×C4×C197—21no—
63C2×C4×C197C2×D412yes—
64C2×C4×C197C2×D411yescharacteristic
65C8×C197—42no—
66C22×C197—42no—
67C22×C197—41no—
68C22×C197#12011yescharacteristic
69C2×D197—42no—
70C2×D197—42no—
71C2×D197—41no—
72C2×D197—42no—
73C2×D197—84no—
74C4×C197—22no—
75C4×C197—21no—
76C4×C197—41no—
77Dic197—42no—
78Dic197—41no—
79C2×C197—21no—
80C2×C197—81no—
81C2×C197B(4,𝔽2)11yescharacteristic
82D197—81no—
83D197—82no—
84C197#220211yescharacteristic, complemented, Sylow
85#2202—1971nomaximal, Sylow
86H(3,𝔽4)—1971no—
87#454—1971no—
88#454—1972no—
89#355—1972no—
90#355D19711yescharacteristic, complemented
912−1+4—1971no—
92AGL(1,ℤ/8)—3942no—
93AGL(1,ℤ/8)—3941no—
94#124—3942no—
95#124—3941no—
96#124—1972no—
97#124C2×D19711yescharacteristic
98#120—3942no—
99#120—3941no—
100#104—3942no—
101#104—3941no—
102#101—1971no—
103#101C2×D19712yescomplemented
104C24—3941no—
105#41—7881no—
106#41—3942no—
107#41—3941no—
108C2×Q8—7881no—
109C2×Q8—3941no—
110C2×Q8C22×D19711yescharacteristic
111C2×D4—3941no—
112C2×D4—3942no—
113C2×D4—1972no—
114C2×D4D78811yescharacteristic
115SD16—7882no—
116SD16—7881no—
117D8—7882no—
118D8—7881no—
119M16—22no—
120M16—3941no—
121C22⋊C4—22no—
122C22⋊C4—3942no—
123C22⋊C4—3941no—
124C22⋊C4—3942no—
125C22⋊C4—3942no—
126C42—1972no—
127C42D78811yescharacteristic
128C23—21no—
129C23—15762no—
130C23—15761no—
131C23—3942no—
132Q8—41no—
133Q8—15761no—
134Q8—3942no—
135Q8—3941no—
136D4—41no—
137D4—7882no—
138D4—7881no—
139D4—15761no—
140D4—15762no—
141D4—3942no—
142D4—3941no—
143C2×C4—21no—
144C2×C4—7882no—
145C2×C4—7881no—
146C2×C4—3942no—
147C2×C4D4×D19712yes—
148C2×C4C2×D78811yescharacteristic
149C8—42no—
150C8—7881no—
151C22—42no—
152C22—41no—
153C22—7882no—
154C22—7882no—
155C22—7881no—
156C22—7882no—
157C22—15764no—
158C22#2026329611yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—7882no—
163C4—7881no—
164C2—21no—
165C2—81no—
166C2—15762no—
167C2—15761no—
168C2#2716126711yescharacteristic
subgroupG/H
centerC2#27161267
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting#355×C197C2
socleC2×C197B(4,𝔽2)
hypercenter#355D197
hypocenterC197#2202
LCSG ▷ C2×Q8×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #27161821C2semidirect
2. #27161254C2semidirect
3. #27161251C2semidirect
4. #27160875C2semidirect
5. #355×C197C2semidirect
6. #20263374C22semidirect
16. #20263245C22semidirect
18. #101×C197C22semidirect
84. C197#2202semidirect
90. #355D197semidirect
103. #101C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618212
#271612542
#271612512
#271608752
#355×C1972
#2202197