Finite Group Databasebeta

← #38770871

Subgroups — #38770871

Order 25472.

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#27164200C212yesmaximal, complemented
2#27164190C211yesmaximal, characteristic
3#27163817C212yesmaximal, complemented
4#27163815C211yesmaximal, characteristic, complemented
5#563×C199C211yesmaximal, characteristic, complemented
6#20264003C2211yescharacteristic
7#20263986—22no—
8#20263981—21no—
9#20263883—21no—
10#20263883—22no—
11#20263874C2211yescharacteristic
12#20263874C2212yes—
13C4⋊Q8×C199C2212yes—
14#126×C199—22no—
15#125×C199—21no—
16#124×C199C2211yescharacteristic
17#123×C199—21no—
18C22⋊Q8×C199—22no—
19#15691887—41no—
20#15691886—21no—
21#15691886—22no—
22Q8×D199—41no—
23#15691876—42no—
24#15691874—41no—
25#15691873—42no—
26#15691871—41no—
27#15691870—21no—
28#15691870—22no—
29C2×Q8×C199D412yes—
30C2×Q8×C199C2311yescharacteristic
31C2×D4×C199D411yescharacteristic
32C22×C4×C199—21no—
33C4⋊C4×C199—21no—
34C4⋊C4×C199—22no—
35C4⋊C4×C199—21no—
36C4⋊C4×C199—22no—
37C4⋊C4×C199—22no—
38C22⋊C4×C199—22no—
39C22⋊C4×C199—21no—
40C22⋊C4×C199—22no—
41C42×C199D411yescharacteristic
42C42×C199D412yes—
43Dic398—21no—
44Dic398—22no—
45D796—21no—
46D796—22no—
47C22⋊D199—41no—
48C4×D199—41no—
49C4×D199—42no—
50C199⋊C8—41no—
51C199⋊C8—42no—
52C23×C199—21no—
53Q8×C199—41no—
54Q8×C199—42no—
55D4×C199—41no—
56C2×C4×C199—21no—
57C2×C4×C199—22no—
58C2×C4×C199—22no—
59C2×C4×C199—81no—
60C2×C4×C199C2×D412yes—
61C2×C4×C199C2×D411yescharacteristic
62C22×C199—42no—
63C22×C199—41no—
64C22×C199#12011yescharacteristic
65C2×D199—41no—
66C4×C199—21no—
67C4×C199—22no—
68C4×C199—41no—
69C4×C199—42no—
70C4×C199—42no—
71Dic199—41no—
72C2×C199—21no—
73C2×C199—81no—
74C2×C199B(4,𝔽2)11yescharacteristic
75D199—81no—
76C199#220311yescharacteristic, complemented, Sylow
77#2203—1991nomaximal, Sylow
78#563D19911yescharacteristic, complemented
79#456—1992no—
80#454—1991no—
81#356—1992no—
82#355—1991no—
832−1+4—1991no—
84#137—3982no—
85AGL(1,ℤ/8)—3981no—
86C4⋊Q8C2×D19912yes—
87#126—22no—
88#125—21no—
89#124C2×D19911yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—3981no—
93#104—3982no—
94#101—1992no—
95#101—1991no—
96#41—7961no—
97#41—3981no—
98#41—3982no—
99C2×Q8—7961no—
100C2×Q8C22⋊D19912yes—
101C2×Q8C22×D19911yescharacteristic
102C2×D4C22⋊D19911yescharacteristic
103C22×C4—21no—
104Q16—7962no—
105SD16—7961no—
106SD16—7962no—
107D8—7961no—
108M16—3981no—
109M16—3982no—
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⋊D19911yescharacteristic
119C42C22⋊D19912yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—3981no—
124Q8—3982no—
125D4—41no—
126D4—7961no—
127D4—3981no—
128D4—3982no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—7961no—
134C2×C4—7962no—
135C2×C4C2×C22⋊D19912yes—
136C2×C4C2×C22⋊D19911yescharacteristic
137C8—7961no—
138C8—7962no—
139C22—42no—
140C22—41no—
141C22—7961no—
142C22#2026392411yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—7961no—
149C2—21no—
150C2—81no—
151C2—15921no—
152C2#2716420511yescharacteristic
subgroupG/H
centerC2#27164205
commutator, cosocleC2×Q8×C199C23
FrattiniC2×Q8C22×D199
Fitting#563×C199C2
socleC2×C199B(4,𝔽2)
prime radical#27164190C2
hypercenter#563D199
hypocenterC199#2203
LCSG ▷ C2×Q8×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #27164200C2semidirect
3. #27163817C2semidirect
4. #27163815C2semidirect
5. #563×C199C2semidirect
76. C199#2203semidirect
78. #563D199semidirect

Maximal subgroups

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

subgroupindex
#271642002
#271641902
#271638172
#271638152
#563×C1992
#2203199