Finite Group Databasebeta

← #38744741

Subgroups — #38744741

Order 25216.

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

normal subgroups only

This group has 224 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#27161809C211yesmaximal, characteristic, complemented
2#27161262C212yesmaximal
3#27160878C211yesmaximal, characteristic, complemented
4#27160877C212yesmaximal, complemented
5#456×C197C211yesmaximal, characteristic
6#20263356—22no—
7#20263325—22no—
8#20263324—21no—
9#20263324C2212yes—
10C4×Dic394—21no—
11#20263291—22no—
12#20263253—22no—
13#20263245C2212yes—
142−1+4×C197C2211yescharacteristic
15#137×C197—21no—
16C4⋊Q8×C197C2211yescharacteristic
17#104×C197—21no—
18#101×C197C2211yescharacteristic
19C2×Dic394—21no—
20C2×Dic394—21no—
21C2×Dic394D412yes—
22#15691348—42no—
23#15691346—42no—
24#15691342—22no—
25C4⋊Dic197—21no—
26#15691337—22no—
27#15691337—22no—
28#15691337—22no—
29C4×Dic197—21no—
30C4×Dic197—22no—
31C4×Dic197D412yes—
32#41×C197—21no—
33#41×C197—22no—
34#41×C197—41no—
35C2×Q8×C197—41no—
36C2×Q8×C197D411yescharacteristic
37C2×Q8×C197C2311yescharacteristic
38Q16×C197—41no—
39SD16×C197—41no—
40M16×C197—21no—
41C4⋊C4×C197—22no—
42C42×C197D411yescharacteristic
43Dic394—24no—
44Dic394—42no—
45Dic394—81no—
46C2×Dic197—22no—
47C2×Dic197—22no—
48#13096802—42no—
49Q8×C197—21no—
50Q8×C197—22no—
51Q8×C197—41no—
52Q8×C197—41no—
53D4×C197—21no—
54D4×C197—22no—
55D4×C197—41no—
56C2×C4×C197—21no—
57C2×C4×C197—21no—
58C2×C4×C197—41no—
59C2×C4×C197—42no—
60C2×C4×C197C2×D412yes—
61C2×C4×C197C2×D411yescharacteristic
62C8×C197—41no—
63C22×C197—41no—
64C22×C197#12011yescharacteristic
65C4×C197—22no—
66C4×C197—21no—
67C4×C197—22no—
68C4×C197—41no—
69C4×C197—41no—
70Dic197—24no—
71Dic197—42no—
72C2×C197—21no—
73C2×C197—81no—
74C2×C197B(4,𝔽2)11yescharacteristic
75C197#220511yescharacteristic, complemented, Sylow
76#2205—1971nomaximal, Sylow
77Q82—1971no—
78#456—1972no—
79#456D19711yescharacteristic
80#356—1971no—
81#356—1972no—
822−1+4C2×D19711yescharacteristic
83#137—21no—
84#137—3942no—
85C4⋊Q8—3942no—
86C4⋊Q8—3941no—
87C4⋊Q8—1972no—
88C4⋊Q8C2×D19711yescharacteristic
89C4×Q8—3942no—
90C4×Q8—3941no—
91#104—21no—
92#104—3942no—
93#101—1972no—
94#101C2×D19711yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—3941no—
100C2×Q8—3941no—
101C2×Q8—1972no—
102C2×Q8C22⋊D19711yescharacteristic
103C2×Q8C22×D19711yescharacteristic
104Q16—41no—
105Q16—7882no—
106SD16—41no—
107SD16—7882no—
108M16—21no—
109M16—3942no—
110C4⋊C4—22no—
111C4⋊C4—3942no—
112C4⋊C4—3941no—
113C4⋊C4—3942no—
114C4⋊C4—3942no—
115C42—3942no—
116C42—3941no—
117C42—1972no—
118C42C22⋊D19711yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—7882no—
124Q8—15761no—
125Q8—3944no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—3942no—
134C2×C4—3942no—
135C2×C4D4×D19712yes—
136C2×C4C2×C22⋊D19711yescharacteristic
137C8—41no—
138C8—7882no—
139C22—41no—
140C22#2026329411yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—7882no—
147C4—3944no—
148C2—21no—
149C2—81no—
150C2#2716126511yescharacteristic
subgroupG/H
centerC2#27161265
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting#456×C197C2
socleC2×C197B(4,𝔽2)
prime radical2−1+4×C197C22
hypercenter#456D197
hypocenterC197#2205
LCSG ▷ C2×Q8×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #27161809C2semidirect
3. #27160878C2semidirect
4. #27160877C2semidirect
75. C197#2205semidirect

Maximal subgroups

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

subgroupindex
#271618092
#271612622
#271608782
#271608772
#456×C1972
#2205197