Finite Group Databasebeta

← #38770870

Subgroups — #38770870

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#27164201C212yesmaximal, complemented
3#27163816C212yesmaximal, complemented
4#27163815C211yesmaximal, characteristic, complemented
5#454×C199C211yesmaximal, characteristic
6#20263985—22no—
7#20263953C2212yes—
8#20263948—22no—
9#20263947—21no—
10#20263936—21no—
11#20263932—22no—
12#20263882—22no—
13#20263874C2212yes—
142−1+4×C199C2211yescharacteristic
15AGL(1,ℤ/8)×C199—21no—
16#124×C199C2211yescharacteristic
17#104×C199—21no—
18#101×C199C2211yescharacteristic
19C2×Dic398D412yes—
20C22×Dic199—21no—
21#15691876—42no—
22#15691874—42no—
23#15691870—22no—
24C4⋊Dic199—21no—
25C4⋊Dic199—21no—
26Dic199⋊C4—22no—
27Dic199⋊C4—22no—
28Dic199⋊C4—22no—
29C22⋊Dic199—21no—
30C22⋊Dic199—22no—
31C4×Dic199D412yes—
32#41×C199—22no—
33#41×C199—21no—
34#41×C199—41no—
35C2×Q8×C199—41no—
36C2×Q8×C199C2311yescharacteristic
37C2×D4×C199D411yescharacteristic
38SD16×C199—41no—
39D8×C199—41no—
40M16×C199—21no—
41C22⋊C4×C199—22no—
42C42×C199D411yescharacteristic
43Dic398—42no—
44C2×Dic199—22no—
45C2×Dic199—22no—
46C2×Dic199—81no—
47C199⋊C8—42no—
48C23×C199—21no—
49Q8×C199—21no—
50Q8×C199—22no—
51Q8×C199—41no—
52D4×C199—21no—
53D4×C199—22no—
54D4×C199—41no—
55D4×C199—41no—
56C2×C4×C199—21no—
57C2×C4×C199—41no—
58C2×C4×C199—42no—
59C2×C4×C199C2×D412yes—
60C2×C4×C199C2×D411yescharacteristic
61C8×C199—41no—
62C22×C199—42no—
63C22×C199—41no—
64C22×C199—41no—
65C22×C199#12011yescharacteristic
66C4×C199—21no—
67C4×C199—22no—
68C4×C199—41no—
69C4×C199—41no—
70Dic199—42no—
71Dic199—42no—
72C2×C199—21no—
73C2×C199—81no—
74C2×C199—81no—
75C2×C199B(4,𝔽2)11yescharacteristic
76C199#220311yescharacteristic, complemented, Sylow
77#2203—1991nomaximal, Sylow
78#563—1991no—
79#456—1992no—
80#454D19911yescharacteristic
81#356—1992no—
82#355—1991no—
832−1+4C2×D19911yescharacteristic
84#137—3982no—
85AGL(1,ℤ/8)—21no—
86C4⋊Q8—1992no—
87#126—3982no—
88#125—3981no—
89#124C2×D19911yescharacteristic
90#123—3981no—
91C22⋊Q8—3982no—
92#104—21no—
93#104—3982no—
94#101—1992no—
95#101C2×D19911yescharacteristic
96#41—22no—
97#41—21no—
98#41—41no—
99C2×Q8—41no—
100C2×Q8—1992no—
101C2×Q8C22×D19911yescharacteristic
102C2×D4C22⋊D19911yescharacteristic
103C22×C4—3981no—
104Q16—7962no—
105SD16—41no—
106SD16—7962no—
107D8—41no—
108M16—21no—
109M16—3982no—
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—
119C42C22⋊D19911yescharacteristic
120C23—21no—
121Q8—21no—
122Q8—22no—
123Q8—41no—
124Q8—7962no—
125D4—21no—
126D4—22no—
127D4—41no—
128D4—41no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—15921no—
133C2×C4—3982no—
134C2×C4—3982no—
135C2×C4D4×D19912yes—
136C2×C4C2×C22⋊D19911yescharacteristic
137C8—41no—
138C8—7962no—
139C22—42no—
140C22—41no—
141C22—41no—
142C22#2026392311yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—41no—
147C4—7962no—
148C4—7962no—
149C2—21no—
150C2—81no—
151C2—81no—
152C2#2716420411yescharacteristic
subgroupG/H
centerC2#27164204
commutator, cosocleC2×Q8×C199C23
FrattiniC2×Q8C22×D199
Fitting, prime radical#454×C199C2
socleC2×C199B(4,𝔽2)
hypercenter#454D199
hypocenterC199#2203
LCSG ▷ C2×Q8×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #27164768C2semidirect
2. #27164201C2semidirect
3. #27163816C2semidirect
4. #27163815C2semidirect
76. C199#2203semidirect

Maximal subgroups

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

subgroupindex
#271647682
#271642012
#271638162
#271638152
#454×C1992
#2203199