Finite Group Databasebeta

← #38770881

Subgroups — #38770881

Order 25472.

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#27164748C211yesmaximal, characteristic, complemented
2#27164203C212yesmaximal
3#27164200C211yesmaximal, characteristic
4#27163816C212yesmaximal, complemented
5#356×C199C211yesmaximal, characteristic, complemented
6#20264003C2211yescharacteristic
7#20263988—22no—
8#20263986—21no—
9#20263954—22no—
10#20263953—21no—
11#20263953C2212yes—
12C4×Dic398—21no—
13#20263920—22no—
14#20263883—21no—
15#20263881—22no—
16#20263874C2211yescharacteristic
17C4⋊Q8×C199C2211yescharacteristic
18#101×C199C2212yes—
19#15691887—41no—
20#15691886—21no—
21#15691886—22no—
22C2×Dic398—21no—
23C2×Dic398—21no—
24C2×Dic398D412yes—
25Q8×D199—41no—
26Dic796—42no—
27#15691876—41no—
28#15691875—42no—
29#15691873—41no—
30#15691870—21no—
31C4⋊Dic199—21no—
32Dic199⋊C4—22no—
33Dic199⋊C4—22no—
34Dic199⋊C4—22no—
35C4×Dic199—21no—
36C4×Dic199—22no—
37C4×Dic199D412yes—
38C2×Q8×C199D411yescharacteristic
39C2×Q8×C199C2311yescharacteristic
40M16×C199—22no—
41C4⋊C4×C199—22no—
42C42×C199D411yescharacteristic
43Dic398—24no—
44Dic398—21no—
45Dic398—22no—
46Dic398—42no—
47Dic398—81no—
48D796—21no—
49D796—22no—
50C22⋊D199—41no—
51C2×Dic199—22no—
52C2×Dic199—22no—
53C4×D199—42no—
54C4×D199—41no—
55C199⋊C8—41no—
56Q8×C199—41no—
57Q8×C199—41no—
58C2×C4×C199—21no—
59C2×C4×C199—21no—
60C2×C4×C199C2×D412yes—
61C2×C4×C199C2×D411yescharacteristic
62C8×C199—42no—
63C22×C199#12011yescharacteristic
64C2×D199—41no—
65C4×C199—22no—
66C4×C199—22no—
67C4×C199—21no—
68C4×C199—41no—
69Dic199—24no—
70Dic199—41no—
71Dic199—42no—
72C2×C199—21no—
73C2×C199B(4,𝔽2)11yescharacteristic
74D199—81no—
75C199#220511yescharacteristic, complemented, Sylow
76#2205—1991nomaximal, Sylow
77Q82—1991no—
78#456—1991no—
79#456—1992no—
80#356—1992no—
81#356D19911yescharacteristic, complemented
822−1+4—1991no—
83#137—3982no—
84#137—3981no—
85C4⋊Q8—3982no—
86C4⋊Q8—3981no—
87C4⋊Q8—1992no—
88C4⋊Q8C2×D19911yescharacteristic
89C4×Q8—3982no—
90C4×Q8—3981no—
91#104—3982no—
92#104—3981no—
93#101—1991no—
94#101C2×D19912yes—
95#41—7961no—
96#41—3982no—
97#41—3981no—
98C2×Q8—7961no—
99C2×Q8—3981no—
100C2×Q8—3981no—
101C2×Q8—1992no—
102C2×Q8D79611yescharacteristic
103C2×Q8C22×D19911yescharacteristic
104Q16—7962no—
105Q16—7961no—
106SD16—7962no—
107SD16—7961no—
108M16—22no—
109M16—3981no—
110C4⋊C4—22no—
111C4⋊C4—3982no—
112C4⋊C4—3981no—
113C4⋊C4—3982no—
114C4⋊C4—3982no—
115C42—3982no—
116C42—3981no—
117C42—1992no—
118C42D79611yescharacteristic
119Q8—41no—
120Q8—41no—
121Q8—7962no—
122Q8—15921no—
123Q8—3982no—
124Q8—3981no—
125Q8—3984no—
126D4—7961no—
127D4—3982no—
128D4—3981no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—7962no—
132C2×C4—7961no—
133C2×C4—3982no—
134C2×C4—3982no—
135C2×C4D4×D19912yes—
136C2×C4C2×D79611yescharacteristic
137C8—42no—
138C8—7961no—
139C22—7961no—
140C22#2026392511yescharacteristic
141C4—22no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—7961no—
146C4—7962no—
147C4—3984no—
148C2—21no—
149C2—15921no—
150C2#2716420611yescharacteristic
subgroupG/H
centerC2#27164206
commutator, cosocleC2×Q8×C199C23
FrattiniC2×Q8C22×D199
Fitting#356×C199C2
socleC2×C199B(4,𝔽2)
prime radical#20264003C22
hypercenter#356D199
hypocenterC199#2205
LCSG ▷ C2×Q8×C199 ▷ C22×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
1. #27164748C2semidirect
4. #27163816C2semidirect
5. #356×C199C2semidirect
75. C199#2205semidirect
81. #356D199semidirect

Maximal subgroups

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

subgroupindex
#271647482
#271642032
#271642002
#271638162
#356×C1992
#2205199