Finite Group Databasebeta

← #38744732

Subgroups — #38744732

Order 25216.

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

normal subgroups only

This group has 216 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#27161829C211yesmaximal, characteristic, complemented
2#27161262C212yesmaximal, complemented
3#27160877C212yesmaximal, complemented
4#27160876C211yesmaximal, characteristic, complemented
5#454×C197C211yesmaximal, characteristic
6#20263356—22no—
7#20263324C2212yes—
8#20263319—22no—
9#20263318—21no—
10#20263307—21no—
11#20263303—22no—
12#20263253—22no—
13#20263245C2212yes—
142−1+4×C197C2211yescharacteristic
15AGL(1,ℤ/8)×C197—21no—
16#124×C197C2211yescharacteristic
17#104×C197—21no—
18#101×C197C2211yescharacteristic
19C2×Dic394D412yes—
20C22×Dic197—21no—
21#15691348—42no—
22#15691346—42no—
23#15691342—22no—
24C4⋊Dic197—21no—
25C4⋊Dic197—21no—
26#15691337—22no—
27#15691337—22no—
28#15691337—22no—
29C22⋊Dic197—22no—
30C22⋊Dic197—21no—
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
43Dic394—42no—
44C2×Dic197—22no—
45C2×Dic197—22no—
46C2×Dic197—81no—
47#13096802—42no—
48C23×C197—21no—
49Q8×C197—21no—
50Q8×C197—22no—
51Q8×C197—41no—
52D4×C197—21no—
53D4×C197—22no—
54D4×C197—41no—
55D4×C197—41no—
56C2×C4×C197—21no—
57C2×C4×C197—41no—
58C2×C4×C197—42no—
59C2×C4×C197C2×D412yes—
60C2×C4×C197C2×D411yescharacteristic
61C8×C197—41no—
62C22×C197—42no—
63C22×C197—41no—
64C22×C197—41no—
65C22×C197#12011yescharacteristic
66C4×C197—21no—
67C4×C197—22no—
68C4×C197—41no—
69C4×C197—41no—
70Dic197—42no—
71Dic197—42no—
72C2×C197—21no—
73C2×C197—81no—
74C2×C197—81no—
75C2×C197B(4,𝔽2)11yescharacteristic
76C197#220311yescharacteristic, complemented, Sylow
77#2203—1971nomaximal, Sylow
78#563—1971no—
79#456—1972no—
80#454D19711yescharacteristic
81#356—1972no—
82#355—1971no—
832−1+4C2×D19711yescharacteristic
84#137—3942no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—1972no—
87#126—3942no—
88#125—3941no—
89#124C2×D19711yescharacteristic
90#123—3941no—
91C22⋊Q8—3942no—
92#104—21no—
93#104—3942no—
94#101—1972no—
95#101C2×D19711yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—1972no—
101C2×Q8C22×D19711yescharacteristic
102C2×D4C22⋊D19711yescharacteristic
103C22×C4—3941no—
104Q16—7882no—
105SD16—41no—
106SD16—7882no—
107D8—41no—
108M16—21no—
109M16—3942no—
110C4⋊C4—3942no—
111C4⋊C4—3941no—
112C4⋊C4—3941no—
113C4⋊C4—3942no—
114C4⋊C4—3942no—
115C22⋊C4—22no—
116C22⋊C4—3942no—
117C22⋊C4—3941no—
118C42—1972no—
119C42C22⋊D19711yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—7882no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—15761no—
133C2×C4—3942no—
134C2×C4—3942no—
135C2×C4D4×D19712yes—
136C2×C4C2×C22⋊D19711yescharacteristic
137C8—41no—
138C8—7882no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#2026329411yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—7882no—
148C4—7882no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#2716126511yescharacteristic
subgroupG/H
centerC2#27161265
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting, prime radical#454×C197C2
socleC2×C197B(4,𝔽2)
hypercenter#454D197
hypocenterC197#2203
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. #27161829C2semidirect
2. #27161262C2semidirect
3. #27160877C2semidirect
4. #27160876C2semidirect
76. C197#2203semidirect

Maximal subgroups

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

subgroupindex
#271618292
#271612622
#271608772
#271608762
#454×C1972
#2203197