Finite Group Databasebeta

← #38744743

Subgroups — #38744743

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#27161264C212yesmaximal
3#27161261C211yesmaximal, characteristic
4#27160877C212yesmaximal, complemented
5#356×C197C211yesmaximal, characteristic, complemented
6#20263374C2211yescharacteristic
7#20263359—22no—
8#20263357—21no—
9#20263325—22no—
10#20263324—21no—
11#20263324C2212yes—
12C4×Dic394—21no—
13#20263291—22no—
14#20263254—21no—
15#20263252—22no—
16#20263245C2211yescharacteristic
17C4⋊Q8×C197C2211yescharacteristic
18#101×C197C2212yes—
19#15691359—41no—
20#15691358—22no—
21#15691358—21no—
22C2×Dic394—21no—
23C2×Dic394—21no—
24C2×Dic394D412yes—
25Q8×D197—41no—
26Dic788—42no—
27#15691348—41no—
28#15691347—42no—
29#15691345—41no—
30#15691342—21no—
31C4⋊Dic197—21no—
32#15691337—22no—
33#15691337—22no—
34#15691337—22no—
35C4×Dic197—21no—
36C4×Dic197—22no—
37C4×Dic197D412yes—
38C2×Q8×C197D411yescharacteristic
39C2×Q8×C197C2311yescharacteristic
40M16×C197—22no—
41C4⋊C4×C197—22no—
42C42×C197D411yescharacteristic
43Dic394—24no—
44Dic394—21no—
45Dic394—22no—
46Dic394—42no—
47Dic394—81no—
48D788—21no—
49D788—22no—
50C22⋊D197—41no—
51C2×Dic197—22no—
52C2×Dic197—22no—
53C4×D197—42no—
54C4×D197—41no—
55#13096802—41no—
56Q8×C197—41no—
57Q8×C197—41no—
58C2×C4×C197—21no—
59C2×C4×C197—21no—
60C2×C4×C197C2×D412yes—
61C2×C4×C197C2×D411yescharacteristic
62C8×C197—42no—
63C22×C197#12011yescharacteristic
64C2×D197—41no—
65C4×C197—22no—
66C4×C197—22no—
67C4×C197—21no—
68C4×C197—41no—
69Dic197—24no—
70Dic197—41no—
71Dic197—42no—
72C2×C197—21no—
73C2×C197B(4,𝔽2)11yescharacteristic
74D197—81no—
75C197#220511yescharacteristic, complemented, Sylow
76#2205—1971nomaximal, Sylow
77Q82—1971no—
78#456—1971no—
79#456—1972no—
80#356—1972no—
81#356D19711yescharacteristic, complemented
822−1+4—1971no—
83#137—3942no—
84#137—3941no—
85C4⋊Q8—3942no—
86C4⋊Q8—3941no—
87C4⋊Q8—1972no—
88C4⋊Q8C2×D19711yescharacteristic
89C4×Q8—3942no—
90C4×Q8—3941no—
91#104—3942no—
92#104—3941no—
93#101—1971no—
94#101C2×D19712yes—
95#41—7881no—
96#41—3942no—
97#41—3941no—
98C2×Q8—7881no—
99C2×Q8—3941no—
100C2×Q8—3941no—
101C2×Q8—1972no—
102C2×Q8D78811yescharacteristic
103C2×Q8C22×D19711yescharacteristic
104Q16—7882no—
105Q16—7881no—
106SD16—7882no—
107SD16—7881no—
108M16—22no—
109M16—3941no—
110C4⋊C4—22no—
111C4⋊C4—3942no—
112C4⋊C4—3941no—
113C4⋊C4—3942no—
114C4⋊C4—3942no—
115C42—3942no—
116C42—3941no—
117C42—1972no—
118C42D78811yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—7882no—
122Q8—15761no—
123Q8—3942no—
124Q8—3941no—
125Q8—3944no—
126D4—7881no—
127D4—3942no—
128D4—3941no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—7882no—
132C2×C4—7881no—
133C2×C4—3942no—
134C2×C4—3942no—
135C2×C4D4×D19712yes—
136C2×C4C2×D78811yescharacteristic
137C8—42no—
138C8—7881no—
139C22—7881no—
140C22#2026329611yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—7881no—
146C4—7882no—
147C4—3944no—
148C2—21no—
149C2—15761no—
150C2#2716126711yescharacteristic
subgroupG/H
centerC2#27161267
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting#356×C197C2
socleC2×C197B(4,𝔽2)
prime radical#20263374C22
hypercenter#356D197
hypocenterC197#2205
LCSG ▷ C2×Q8×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #27161809C2semidirect
4. #27160877C2semidirect
5. #356×C197C2semidirect
75. C197#2205semidirect
81. #356D197semidirect

Maximal subgroups

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

subgroupindex
#271618092
#271612642
#271612612
#271608772
#356×C1972
#2205197