Finite Group Databasebeta

← #38768149

Subgroups — #38768149

Order 25472.

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

normal subgroups only

This group has 206 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#27164768C211yesmaximal, characteristic, complemented
2#27164408C211yesmaximal, characteristic
3#27164383C212yesmaximal, complemented
4#27163759C212yesmaximal, complemented
5#329×C199C211yesmaximal, characteristic, complemented
6C2×#15691873—21no—
7C2×#15691871—21no—
8#20263953—21no—
9#20263952C2211yescharacteristic
10#20263948—22no—
11#20263947C2212yes—
12#20263936—21no—
13#20263932—22no—
14#20263877—22no—
15#20263877—22no—
16#20263866C2211yescharacteristic
17#124×C199C2211yescharacteristic
18#97×C199C2212yes—
19C2×Dic398—21no—
20C2×D796—21no—
21C2×D796—22no—
22C22×Dic199—21no—
23#15691873—42no—
24#15691871—42no—
25C2×C199⋊C8—21no—
26C4⋊Dic199D412yes—
27C4⋊Dic199D412yes—
28Dic199⋊C4—22no—
29Dic199⋊C4—22no—
30Dic199⋊C4—22no—
31C22⋊Dic199—21no—
32C22⋊Dic199—22no—
33C4×Dic199—21no—
34C2×Q8×C199D411yescharacteristic
35C2×D4×C199D411yescharacteristic
36C2×C8×C199—22no—
37C22⋊C4×C199—22no—
38C42×C199C2311yescharacteristic
39C22×D199—41no—
40Dic398—81no—
41D796—41no—
42D796—41no—
43D796—44no—
44C2×Dic199—22no—
45C2×Dic199—22no—
46C2×Dic199—81no—
47C199⋊C8—22no—
48C23×C199—21no—
49Q8×C199—41no—
50D4×C199—41no—
51C2×C4×C199—21no—
52C2×C4×C199C2×D412yes—
53C2×C4×C199C2×D411yescharacteristic
54C8×C199—42no—
55C22×C199—41no—
56C22×C199—42no—
57C22×C199#12011yescharacteristic
58C2×D199—81no—
59C2×D199—82no—
60C4×C199—21no—
61C4×C199—22no—
62C4×C199—41no—
63Dic199—42no—
64Dic199—42no—
65C2×C199—81no—
66C2×C199#45311yescharacteristic
67C2×C199#45412yes—
68D199—161no—
69C199#165711yescharacteristic, complemented, Sylow
70#1657—1991nomaximal, Sylow
71#563—1991no—
72#496—1991no—
73#490—1992no—
74#330—1992no—
75#329D19911yescharacteristic, complemented
76C2×SD16—3981no—
77C2×D8—3981no—
78C4⋊Q8—3981no—
79#127—1991no—
80#126—3982no—
81#125—1992no—
82#124C2×D19911yescharacteristic
83#123—3981no—
84C22⋊Q8—3982no—
85#102—3982no—
86#102—3982no—
87#97—1991no—
88#97C2×D19912yes—
89C2×Q8—3981no—
90C2×Q8D79611yescharacteristic
91C2×D4—3982no—
92C2×D4—3981no—
93C2×D4D79611yescharacteristic
94C22×C4—3981no—
95SD16—7962no—
96D8—7962no—
97C2×C8—22no—
98C2×C8—3981no—
99C4⋊C4—3982no—
100C4⋊C4—3982no—
101C4⋊C4—3982no—
102C4⋊C4—1992no—
103C4⋊C4—1992no—
104C22⋊C4—22no—
105C22⋊C4—3982no—
106C22⋊C4—3981no—
107C42—3981no—
108C42C22×D19911yescharacteristic
109C23—21no—
110C23—7961no—
111Q8—41no—
112Q8—15921no—
113D4—41no—
114D4—7961no—
115D4—7961no—
116D4—7964no—
117C2×C4—21no—
118C2×C4—15921no—
119C2×C4—3982no—
120C2×C4—3982no—
121C2×C4D4×D19912yes—
122C2×C4C2×D79611yescharacteristic
123C8—42no—
124C8—3982no—
125C22—41no—
126C22—42no—
127C22—15921no—
128C22—15922no—
129C22#2026392511yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—7962no—
134C4—7962no—
135C2—81no—
136C2—31841no—
137C2#2716418811yescharacteristic
138C2#2716419212yes—
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#329×C199C2
socleC22×C199#120
prime radical#27164408C2
hypercenter#329D199
hypocenterC199#1657
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #27164768C2semidirect
3. #27164383C2semidirect
4. #27163759C2semidirect
5. #329×C199C2semidirect
69. C199#1657semidirect
75. #329D199semidirect

Maximal subgroups

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

subgroupindex
#271647682
#271644082
#271643832
#271637592
#329×C1992
#1657199