Finite Group Databasebeta

← #38768154

Subgroups — #38768154

Order 25472.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#27164760C211yesmaximal, characteristic
2#27164380C212yesmaximal, complemented
3#27163758C211yesmaximal, characteristic, complemented
4#27163757C212yesmaximal, complemented
5#497×C199C211yesmaximal, characteristic, complemented
6#20263944—21no—
7#20263943C2212yes—
8#20263941—22no—
9#20263925—22no—
10#20263923—21no—
11#20263878—22no—
12#20263875—22no—
13#20263866C2212yescomplemented
14C2×Q16×C199—21no—
15C2×SD16×C199—21no—
16C4⋊Q8×C199C2211yescharacteristic, complemented
17#124×C199C2211yescharacteristic
18#97×C199C2211yescharacteristic, complemented
19C23×D199—21no—
20C2×Dic398D412yes—
21C2×D796—21no—
22C2×D796D412yes—
23C2×C22⋊D199—22no—
24C2×C199⋊C8—22no—
25C22⋊Dic199—21no—
26(C2×D199)⋊C4—22no—
27(C2×D199)⋊C4—22no—
28(C2×D199)⋊C4—22no—
29C4×Dic199—21no—
30C2×Q8×C199—21no—
31C2×Q8×C199D411yescharacteristic
32C2×D4×C199D411yescharacteristic
33Q16×C199—42no—
34SD16×C199—42no—
35C2×C8×C199—21no—
36C4⋊C4×C199—22no—
37C22⋊C4×C199—22no—
38C42×C199C2311yescharacteristic
39C22×D199—22no—
40C22×D199—81no—
41C22×D199—82no—
42Dic398—42no—
43D796—42no—
44D796—81no—
45C22⋊D199—82no—
46C2×Dic199—22no—
47C199⋊C8—42no—
48C23×C199—21no—
49Q8×C199—41no—
50Q8×C199—41no—
51Q8×C199—41no—
52D4×C199—41no—
53C2×C4×C199—21no—
54C2×C4×C199—41no—
55C2×C4×C199C2×D412yes—
56C2×C4×C199C2×D411yescharacteristic
57C8×C199—22no—
58C22×C199—41no—
59C22×C199—42no—
60C22×C199#12011yescharacteristic
61C2×D199—42no—
62C2×D199—42no—
63C2×D199—42no—
64C2×D199—84no—
65C4×C199—21no—
66C4×C199—22no—
67C4×C199—41no—
68C4×C199—81no—
69Dic199—42no—
70C2×C199—81no—
71C2×C199#45311yescharacteristic
72C2×C199#45512yes—
73D199—82no—
74C199#165911yescharacteristic, complemented, Sylow
75#1659—1991nomaximal, Sylow
76H(3,𝔽4)—1991no—
77#497D19911yescharacteristic, complemented
78#489—1992no—
79#329—1991no—
80#329—1992no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D19911yescharacteristic, complemented
84#124—3982no—
85#124—3981no—
86#124—1992no—
87#124C2×D19911yescharacteristic
88#120—3982no—
89#120—3981no—
90#103—3982no—
91#102—3982no—
92#97—1992no—
93#97C2×D19911yescharacteristic, complemented
94C24—3981no—
95C2×Q8—21no—
96C2×Q8—1992no—
97C2×Q8C22⋊D19911yescharacteristic
98C2×D4—3981no—
99C2×D4—3982no—
100C2×D4—1992no—
101C2×D4C22⋊D19911yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—3982no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—3982no—
109C22⋊C4—3981no—
110C22⋊C4—3982no—
111C22⋊C4—3982no—
112C42—3981no—
113C42C22×D19911yescharacteristic
114C23—21no—
115C23—15921no—
116C23—15922no—
117C23—3982no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—7962no—
122D4—41no—
123D4—7962no—
124D4—15921no—
125D4—15922no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—3982no—
129C2×C4D4×D19912yes—
130C2×C4C2×C22⋊D19911yescharacteristic
131C8—22no—
132C8—7962no—
133C22—41no—
134C22—42no—
135C22—7962no—
136C22—7962no—
137C22—7962no—
138C22—15924no—
139C22#2026392311yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—7962no—
145C2—81no—
146C2—15922no—
147C2#2716418511yescharacteristic
148C2#2716419412yes—
subgroupG/H
centerC22#20263923
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#497×C199C2
socleC22×C199#120
prime radical#27164760C2
hypercenter#497D199
hypocenterC199#1659
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #27164380C2semidirect
3. #27163758C2semidirect
4. #27163757C2semidirect
5. #497×C199C2semidirect
13. #20263866C22semidirect
16. C4⋊Q8×C199C22semidirect
18. #97×C199C22semidirect
74. C199#1659semidirect
77. #497D199semidirect
83. C4⋊Q8C2×D199semidirect
93. #97C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271647602
#271643802
#271637582
#271637572
#497×C1992
#1659199