Finite Group Databasebeta

← #38767857

Subgroups — #38767857

Order 25472.

210 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164649C211yesmaximal, characteristic
2#27164348C212yesmaximal
3#27163754C212yesmaximal
4#27163744C211yesmaximal, characteristic
5#398×C199C211yesmaximal, characteristic
6#20263954—22no—
7#20263946—22no—
8#20263934—21no—
9#20263934C2212yes—
10#20263914—21no—
11#20263886—22no—
12#20263878—22no—
13C22⋊(C199⋊C8)C2212yes—
14C2×C4⋊C4×C199—21no—
15C2×C4⋊C4×C199C2211yescharacteristic
16C2×C4⋊C4×C199C2211yescharacteristic
17#95×C199—21no—
18#95×C199C2211yescharacteristic
19C2×Dic398—21no—
20C2×Dic398D412yes—
21C2×C199⋊C8—22no—
22C4⋊Dic199—21no—
23C4⋊Dic199—21no—
24C4⋊Dic199—21no—
25C4⋊Dic199D412yes—
26Dic199⋊C4—22no—
27Dic199⋊C4—22no—
28C22⋊Dic199—22no—
29C4×Dic199—22no—
30C22×C4×C199—21no—
31C22×C4×C199—21no—
32C22×C4×C199D411yescharacteristic
33C22×C4×C199D411yescharacteristic
34C22×C4×C199C2311yescharacteristic
35C4⋊C4×C199—22no—
36C4⋊C4×C199—22no—
37C4⋊C4×C199—42no—
38Dic398—42no—
39Dic398—81no—
40C2×Dic199—22no—
41C2×Dic199—22no—
42C199⋊C8—42no—
43C23×C199C2×D411yescharacteristic
44C2×C4×C199—21no—
45C2×C4×C199—21no—
46C2×C4×C199—22no—
47C2×C4×C199—21no—
48C2×C4×C199—41no—
49C2×C4×C199—41no—
50C2×C4×C199—41no—
51C2×C4×C199—41no—
52C2×C4×C199—41no—
53C2×C4×C199—41no—
54C2×C4×C199—41no—
55C2×C4×C199SD1612yes—
56C2×C4×C199C2×D412yes—
57C22×C199—21no—
58C22×C199—21no—
59C22×C199#12011yescharacteristic
60C22×C199C2×SD1611yescharacteristic
61C22×C199#13711yescharacteristic
62C4×C199—22no—
63C4×C199—22no—
64C4×C199—42no—
65C4×C199—41no—
66C4×C199—81no—
67Dic199—42no—
68Dic199—42no—
69C2×C199—21no—
70C2×C199—21no—
71C2×C199#44811yescharacteristic
72C2×C199#45611yescharacteristic
73C2×C199#45811yescharacteristic
74C199#161111yescharacteristic, complemented, Sylow
75#1611—1991nomaximal, Sylow
76#544—1991no—
77#483—1992no—
78#398D19911yescharacteristic
79#328—1992no—
80#324—1991no—
81C4⋊Q8—3982no—
82#125—3982no—
83C22⋊Q8—3981no—
84C22⋊Q8—1992no—
85#117—3981no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2×D19911yescharacteristic
88C2×C4⋊C4C2×D19911yescharacteristic
89#106—3982no—
90#103—3982no—
91C22⋊C8—1992no—
92#95—21no—
93#95C2×D19911yescharacteristic
94C2×Q8—3981no—
95C2×Q8—1992no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D19911yescharacteristic
99C22×C4C22⋊D19911yescharacteristic
100C22×C4C22×D19911yescharacteristic
101C2×C8—3982no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4—3982no—
106C4⋊C4—3981no—
107C4⋊C4—3981no—
108C4⋊C4—3981no—
109C4⋊C4—3982no—
110C4⋊C4—1992no—
111C22⋊C4—3982no—
112C42—3982no—
113C23C2×C22⋊D19911yescharacteristic
114Q8—7962no—
115Q8—15921no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—3982no—
128C2×C4—3982no—
129C2×C4#1569187412yes—
130C2×C4D4×D19912yes—
131C8—7962no—
132C22—21no—
133C22—21no—
134C22#2026392311yescharacteristic
135C22C2×#1569187411yescharacteristic
136C22#2026398511yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—81no—
142C4—7962no—
143C4—7962no—
144C2—21no—
145C2—21no—
146C2#2716415611yescharacteristic
147C2#2716420011yescharacteristic
148C2#2716420711yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#398×C199C2
socleC22×C199#120
prime radicalC23×C199C2×D4
hypercenter#398D199
hypocenterC199#1611
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
74. C199#1611semidirect

Maximal subgroups

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

subgroupindex
#271646492
#271643482
#271637542
#271637442
#398×C1992
#1611199