Finite Group Databasebeta

← #38744733

Subgroups — #38744733

Order 25216.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#27161261C212yesmaximal, complemented
2#27161251C211yesmaximal, characteristic
3#27160878C212yesmaximal, complemented
4#27160876C211yesmaximal, characteristic, complemented
5#563×C197C211yesmaximal, characteristic, complemented
6#20263374C2211yescharacteristic
7#20263357—22no—
8#20263352—21no—
9#20263254—21no—
10#20263254—22no—
11#20263245C2212yes—
12#20263245C2211yescharacteristic
13C4⋊Q8×C197C2212yes—
14#126×C197—22no—
15#125×C197—21no—
16#124×C197C2211yescharacteristic
17#123×C197—21no—
18C22⋊Q8×C197—22no—
19#15691359—41no—
20#15691358—21no—
21#15691358—22no—
22Q8×D197—41no—
23#15691348—42no—
24#15691346—41no—
25#15691345—42no—
26#15691343—41no—
27#15691342—21no—
28#15691342—22no—
29C2×Q8×C197D412yes—
30C2×Q8×C197C2311yescharacteristic
31C2×D4×C197D411yescharacteristic
32C22×C4×C197—21no—
33C4⋊C4×C197—21no—
34C4⋊C4×C197—22no—
35C4⋊C4×C197—21no—
36C4⋊C4×C197—22no—
37C4⋊C4×C197—22no—
38C22⋊C4×C197—22no—
39C22⋊C4×C197—21no—
40C22⋊C4×C197—22no—
41C42×C197D411yescharacteristic
42C42×C197D412yes—
43Dic394—21no—
44Dic394—22no—
45D788—21no—
46D788—22no—
47C22⋊D197—41no—
48C4×D197—41no—
49C4×D197—42no—
50#13096802—41no—
51#13096802—42no—
52C23×C197—21no—
53Q8×C197—41no—
54Q8×C197—42no—
55D4×C197—41no—
56C2×C4×C197—21no—
57C2×C4×C197—22no—
58C2×C4×C197—22no—
59C2×C4×C197—81no—
60C2×C4×C197C2×D412yes—
61C2×C4×C197C2×D411yescharacteristic
62C22×C197—42no—
63C22×C197—41no—
64C22×C197#12011yescharacteristic
65C2×D197—41no—
66C4×C197—21no—
67C4×C197—22no—
68C4×C197—41no—
69C4×C197—42no—
70C4×C197—42no—
71Dic197—41no—
72C2×C197—21no—
73C2×C197—81no—
74C2×C197B(4,𝔽2)11yescharacteristic
75D197—81no—
76C197#220311yescharacteristic, complemented, Sylow
77#2203—1971nomaximal, Sylow
78#563D19711yescharacteristic, complemented
79#456—1972no—
80#454—1971no—
81#356—1972no—
82#355—1971no—
832−1+4—1971no—
84#137—3942no—
85AGL(1,ℤ/8)—3941no—
86C4⋊Q8C2×D19712yes—
87#126—22no—
88#125—21no—
89#124C2×D19711yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—3941no—
93#104—3942no—
94#101—1972no—
95#101—1971no—
96#41—7881no—
97#41—3941no—
98#41—3942no—
99C2×Q8—7881no—
100C2×Q8C22⋊D19712yes—
101C2×Q8C22×D19711yescharacteristic
102C2×D4C22⋊D19711yescharacteristic
103C22×C4—21no—
104Q16—7882no—
105SD16—7881no—
106SD16—7882no—
107D8—7881no—
108M16—3941no—
109M16—3942no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D19711yescharacteristic
119C42C22⋊D19712yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—3941no—
124Q8—3942no—
125D4—41no—
126D4—7881no—
127D4—3941no—
128D4—3942no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—7881no—
134C2×C4—7882no—
135C2×C4C2×C22⋊D19712yes—
136C2×C4C2×C22⋊D19711yescharacteristic
137C8—7881no—
138C8—7882no—
139C22—42no—
140C22—41no—
141C22—7881no—
142C22#2026329511yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—7881no—
149C2—21no—
150C2—81no—
151C2—15761no—
152C2#2716126611yescharacteristic
subgroupG/H
centerC2#27161266
commutator, cosocleC2×Q8×C197C23
FrattiniC2×Q8C22×D197
Fitting#563×C197C2
socleC2×C197B(4,𝔽2)
prime radical#27161251C2
hypercenter#563D197
hypocenterC197#2203
LCSG ▷ C2×Q8×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #27161261C2semidirect
3. #27160878C2semidirect
4. #27160876C2semidirect
5. #563×C197C2semidirect
76. C197#2203semidirect
78. #563D197semidirect

Maximal subgroups

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

subgroupindex
#271612612
#271612512
#271608782
#271608762
#563×C1972
#2203197