Finite Group Databasebeta

← #38768140

Subgroups — #38768140

Order 25472.

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

normal subgroups only

This group has 262 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, complemented
2#27164408C211yesmaximal, characteristic, complemented
3#27164407C212yesmaximal, complemented
4#27163757C212yesmaximal, complemented
5#329×C199C211yesmaximal, characteristic, complemented
6C2×#15691875—22no—
7C2×#15691873—21no—
8C2×D1592—22no—
9C2×#15691871—21no—
10#20263952C2211yescharacteristic, complemented
11#20263944—21no—
12#20263943C2212yes—
13#20263941—22no—
14#20263925—22no—
15#20263923—21no—
16#20263866C2211yescharacteristic, complemented
17#124×C199C2211yescharacteristic
18#97×C199C2212yescomplemented
19C23×D199—21no—
20C2×Dic398D412yes—
21C2×D796—21no—
22C2×D796—21no—
23C2×D796—22no—
24C2×D796D412yes—
25C2×C22⋊D199—22no—
26#15691875—44no—
27#15691873—42no—
28D1592—44no—
29#15691871—42no—
30C2×C199⋊C8—21no—
31C22⋊Dic199—21no—
32(C2×D199)⋊C4—22no—
33(C2×D199)⋊C4—22no—
34(C2×D199)⋊C4—22no—
35C4×Dic199—21no—
36C2×Q8×C199D411yescharacteristic
37C2×D4×C199D411yescharacteristic
38C2×C8×C199—22no—
39C22⋊C4×C199—22no—
40C42×C199C2311yescharacteristic
41C22×D199—22no—
42C22×D199—41no—
43C22×D199—82no—
44C22×D199—81no—
45Dic398—42no—
46D796—42no—
47D796—41no—
48D796—42no—
49D796—41no—
50D796—42no—
51D796—81no—
52C22⋊D199—82no—
53C2×Dic199—22no—
54C199⋊C8—22no—
55C23×C199—21no—
56Q8×C199—41no—
57D4×C199—41no—
58C2×C4×C199—21no—
59C2×C4×C199C2×D412yes—
60C2×C4×C199C2×D411yescharacteristic
61C8×C199—24no—
62C22×C199—41no—
63C22×C199—42no—
64C22×C199#12011yescharacteristic
65C2×D199—42no—
66C2×D199—42no—
67C2×D199—42no—
68C2×D199—84no—
69C2×D199—82no—
70C2×D199—81no—
71C4×C199—22no—
72C4×C199—21no—
73C4×C199—41no—
74Dic199—42no—
75C2×C199—81no—
76C2×C199#45312yes—
77C2×C199#45311yescharacteristic
78D199—82no—
79D199—161no—
80C199#165511yescharacteristic, complemented, Sylow
81#1655—1991nomaximal, Sylow
82H(3,𝔽4)—1991no—
83#496—1991no—
84#496—1992no—
85#329—1992no—
86#329D19911yescharacteristic, complemented
87C2×SD16—3982no—
88C2×SD16—3981no—
89C2×D8—3982no—
90C2×D8—3981no—
91#127—1991no—
92#124—3982no—
93#124—3981no—
94#124—1992no—
95#124C2×D19911yescharacteristic
96#120—3982no—
97#120—3981no—
98#97—1991no—
99#97C2×D19912yescomplemented
100C24—3981no—
101C2×Q8—1992no—
102C2×Q8D79611yescharacteristic
103C2×D4—3981no—
104C2×D4—3982no—
105C2×D4—3982no—
106C2×D4—3981no—
107C2×D4—1992no—
108C2×D4D79611yescharacteristic
109SD16—7964no—
110SD16—7962no—
111D8—7964no—
112D8—7962no—
113C2×C8—22no—
114C2×C8—3981no—
115C22⋊C4—22no—
116C22⋊C4—3982no—
117C22⋊C4—3981no—
118C22⋊C4—3982no—
119C22⋊C4—3982no—
120C42—3981no—
121C42C22×D19911yescharacteristic
122C23—21no—
123C23—7961no—
124C23—15922no—
125C23—15921no—
126C23—3982no—
127Q8—41no—
128Q8—7962no—
129D4—41no—
130D4—7962no—
131D4—7962no—
132D4—7961no—
133D4—7962no—
134D4—7961no—
135D4—15921no—
136D4—15922no—
137C2×C4—21no—
138C2×C4—3982no—
139C2×C4D4×D19912yes—
140C2×C4C2×D79611yescharacteristic
141C8—24no—
142C8—3982no—
143C22—42no—
144C22—41no—
145C22—7962no—
146C22—7962no—
147C22—7962no—
148C22—15924no—
149C22—15922no—
150C22—15921no—
151C22#2026392511yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—7962no—
156C2—81no—
157C2—15922no—
158C2—31841no—
159C2#2716418712yes—
160C2#2716418811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#329×C199C2
socleC22×C199#120
hypercenter#329D199
hypocenterC199#1655
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. #27164760C2semidirect
2. #27164408C2semidirect
3. #27164407C2semidirect
4. #27163757C2semidirect
5. #329×C199C2semidirect
10. #20263952C22semidirect
16. #20263866C22semidirect
18. #97×C199C22semidirect
80. C199#1655semidirect
86. #329D199semidirect
99. #97C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271647602
#271644082
#271644072
#271637572
#329×C1992
#1655199