Finite Group Databasebeta

← #38768143

Subgroups — #38768143

Order 25472.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#27164408C212yesmaximal, complemented
2#27164384C211yesmaximal, characteristic, complemented
3#27163760C211yesmaximal, characteristic, complemented
4#27163758C212yesmaximal, complemented
5#562×C199C211yesmaximal, characteristic, complemented
6C2×#15691873—22no—
7C2×#15691871—22no—
8#20263952C2211yescharacteristic
9#20263875—22no—
10#20263866C2211yescharacteristic
11#20263866C2212yes—
12#125×C199C2211yescharacteristic
13#124×C199C2212yes—
14#123×C199—22no—
15C22⋊Q8×C199—22no—
16#121×C199—22no—
17C2×D796—22no—
18C2×D796—21no—
19#15691873—44no—
20#15691871—44no—
21C2×C199⋊C8—22no—
22C2×C199⋊C8—21no—
23C2×Q8×C199D412yes—
24C2×D4×C199—22no—
25C2×D4×C199D412yes—
26C22×C4×C199—22no—
27C4⋊C4×C199—22no—
28C4⋊C4×C199—22no—
29C4⋊C4×C199D412yes—
30C22⋊C4×C199—22no—
31C22⋊C4×C199—22no—
32C22⋊C4×C199—22no—
33C42×C199C2311yescharacteristic
34C22×D199—41no—
35D796—42no—
36D796—42no—
37D796—42no—
38C199⋊C8—24no—
39C199⋊C8—41no—
40C23×C199—22no—
41Q8×C199—42no—
42D4×C199—42no—
43D4×C199—82no—
44C2×C4×C199—22no—
45C2×C4×C199—22no—
46C2×C4×C199—82no—
47C2×C4×C199C2×D411yescharacteristic
48C2×C4×C199C2×D412yes—
49C22×C199—42no—
50C22×C199—42no—
51C22×C199—42no—
52C22×C199#12011yescharacteristic
53C2×D199—82no—
54C2×D199—81no—
55C4×C199—22no—
56C4×C199—21no—
57C4×C199—42no—
58C4×C199—42no—
59C2×C199—82no—
60C2×C199#45312yes—
61C2×C199#45411yescharacteristic
62D199—161no—
63C199#165611yescharacteristic, complemented, Sylow
64#1656—1991nomaximal, Sylow
65#562D19911yescharacteristic, complemented
66#496—1992no—
67#490—1991no—
68#330—1991no—
69#329—1992no—
70C2×SD16—3982no—
71C2×D8—3982no—
72#127—1991no—
73#125C2×D19911yescharacteristic
74#124C2×D19912yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—3982no—
79#97—1991no—
80#97—1992no—
81C2×Q8C22⋊D19912yes—
82C2×D4—22no—
83C2×D4—3981no—
84C2×D4—3982no—
85C2×D4C22⋊D19912yes—
86C22×C4—22no—
87SD16—7964no—
88D8—7964no—
89C2×C8—3981no—
90C2×C8—3982no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D19912yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D19911yescharacteristic
98C23—22no—
99C23—7961no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—7962no—
104D4—7962no—
105D4—7962no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D19911yescharacteristic
110C2×C4C2×C22⋊D19912yes—
111C8—7961no—
112C8—3984no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—15922no—
117C22—15921no—
118C22#2026392411yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—31841no—
125C2#2716418612yes—
126C2#2716419111yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#562×C199C2
socleC22×C199#120
hypercenter#562D199
hypocenterC199#1656
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #27164408C2semidirect
2. #27164384C2semidirect
3. #27163760C2semidirect
4. #27163758C2semidirect
5. #562×C199C2semidirect
63. C199#1656semidirect
65. #562D199semidirect

Maximal subgroups

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

subgroupindex
#271644082
#271643842
#271637602
#271637582
#562×C1992
#1656199