Finite Group Databasebeta

← #38770873

Subgroups — #38770873

Order 25472.

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#27164768C211yesmaximal, characteristic, complemented
2#27164203C212yesmaximal, complemented
3#27164190C211yesmaximal, characteristic
4#27163816C212yesmaximal, complemented
5#355×C199C211yesmaximal, characteristic, complemented
6#20264003C2211yescharacteristic
7#20263988—22no—
8#20263981—21no—
9#20263953C2212yes—
10#20263948—22no—
11#20263947—21no—
12#20263936—21no—
13#20263932—22no—
14#20263883—21no—
15#20263881—22no—
16#20263874C2211yescharacteristic
17#124×C199C2211yescharacteristic
18#101×C199C2212yes—
19#15691887—41no—
20#15691886—21no—
21#15691886—22no—
22C2×Dic398D412yes—
23Q8×D199—41no—
24C22×Dic199—21no—
25Dic796—42no—
26#15691875—42no—
27#15691874—41no—
28#15691871—41no—
29#15691870—21no—
30C4⋊Dic199—21no—
31C4⋊Dic199—21no—
32Dic199⋊C4—22no—
33Dic199⋊C4—22no—
34Dic199⋊C4—22no—
35C22⋊Dic199—21no—
36C22⋊Dic199—22no—
37C4×Dic199D412yes—
38C2×Q8×C199C2311yescharacteristic
39C2×D4×C199D411yescharacteristic
40M16×C199—22no—
41C22⋊C4×C199—22no—
42C42×C199D411yescharacteristic
43Dic398—21no—
44Dic398—22no—
45Dic398—42no—
46D796—21no—
47D796—22no—
48C22⋊D199—41no—
49C2×Dic199—22no—
50C2×Dic199—22no—
51C2×Dic199—81no—
52C4×D199—42no—
53C4×D199—41no—
54C199⋊C8—41no—
55C23×C199—21no—
56Q8×C199—41no—
57D4×C199—41no—
58C2×C4×C199—21no—
59C2×C4×C199C2×D412yes—
60C2×C4×C199C2×D411yescharacteristic
61C8×C199—42no—
62C22×C199—42no—
63C22×C199—41no—
64C22×C199#12011yescharacteristic
65C2×D199—41no—
66C4×C199—22no—
67C4×C199—21no—
68C4×C199—41no—
69Dic199—42no—
70Dic199—41no—
71Dic199—42no—
72C2×C199—21no—
73C2×C199—81no—
74C2×C199B(4,𝔽2)11yescharacteristic
75D199—81no—
76C199#220311yescharacteristic, complemented, Sylow
77#2203—1991nomaximal, Sylow
78#563—1991no—
79#456—1992no—
80#454—1991no—
81#356—1992no—
82#355D19911yescharacteristic, complemented
832−1+4—1991no—
84#137—3982no—
85AGL(1,ℤ/8)—3981no—
86C4⋊Q8—1992no—
87#126—3982no—
88#125—3981no—
89#124C2×D19911yescharacteristic
90#123—3981no—
91C22⋊Q8—3982no—
92#104—3982no—
93#104—3981no—
94#101—1991no—
95#101C2×D19912yes—
96#41—7961no—
97#41—3982no—
98#41—3981no—
99C2×Q8—7961no—
100C2×Q8—1992no—
101C2×Q8C22×D19911yescharacteristic
102C2×D4D79611yescharacteristic
103C22×C4—3981no—
104Q16—7962no—
105SD16—7962no—
106SD16—7961no—
107D8—7961no—
108M16—22no—
109M16—3981no—
110C4⋊C4—3982no—
111C4⋊C4—3981no—
112C4⋊C4—3981no—
113C4⋊C4—3982no—
114C4⋊C4—3982no—
115C22⋊C4—22no—
116C22⋊C4—3982no—
117C22⋊C4—3981no—
118C42—1992no—
119C42D79611yescharacteristic
120C23—21no—
121Q8—41no—
122Q8—7962no—
123Q8—3981no—
124Q8—3982no—
125D4—41no—
126D4—7961no—
127D4—3981no—
128D4—3982no—
129C2×C4—21no—
130C2×C4—7962no—
131C2×C4—7961no—
132C2×C4—15921no—
133C2×C4—3982no—
134C2×C4—3982no—
135C2×C4D4×D19912yes—
136C2×C4C2×D79611yescharacteristic
137C8—42no—
138C8—7961no—
139C22—42no—
140C22—41no—
141C22—7961no—
142C22#2026392511yescharacteristic
143C4—22no—
144C4—21no—
145C4—41no—
146C4—7962no—
147C4—7961no—
148C4—7962no—
149C2—21no—
150C2—81no—
151C2—15921no—
152C2#2716420611yescharacteristic
subgroupG/H
centerC2#27164206
commutator, cosocleC2×Q8×C199C23
FrattiniC2×Q8C22×D199
Fitting#355×C199C2
socleC2×C199B(4,𝔽2)
prime radical#27164190C2
hypercenter#355D199
hypocenterC199#2203
LCSG ▷ C2×Q8×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
1. #27164768C2semidirect
2. #27164203C2semidirect
4. #27163816C2semidirect
5. #355×C199C2semidirect
76. C199#2203semidirect
82. #355D199semidirect

Maximal subgroups

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

subgroupindex
#271647682
#271642032
#271641902
#271638162
#355×C1992
#2203199