Finite Group Databasebeta

← #38767821

Subgroups — #38767821

Order 25472.

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

normal subgroups only

This group has 282 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#27164599C211yesmaximal, characteristic
2#27164159C212yesmaximal, complemented
3#27163938C211yesmaximal, characteristic, complemented
4#27163753C212yesmaximal, complemented
5#324×C199C211yesmaximal, characteristic, complemented
6C2×#15691886—21no—
7C22×Dic398C2211yescharacteristic
8C2×#15691875—22no—
9#20263935—22no—
10#20263930—22no—
11#20263929—21no—
12#20263929C2212yes—
13C2×Dic199⋊C4—21no—
14#20263880—22no—
15C199⋊#95—21no—
16C199⋊#95C2211yescharacteristic
17C2×C4⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19#15691886—42no—
20#15691886—42no—
21C2×Dic398—21no—
22C2×Dic398—22no—
23C2×Dic398—42no—
24C2×D796—21no—
25C2×D796—21no—
26C2×D796D412yes—
27C2×C22⋊D199—22no—
28C22×Dic199—21no—
29C22×Dic199—21no—
30C2×C4×D199—22no—
31#15691875—44no—
32C4⋊Dic199—21no—
33C4⋊Dic199D412yes—
34Dic199⋊C4—42no—
35(C2×D199)⋊C4—22no—
36(C2×D199)⋊C4—22no—
37C22×C4×C199D411yescharacteristic
38C22×C4×C199D411yescharacteristic
39C22×C4×C199C2311yescharacteristic
40C2×C8×C199—22no—
41C4⋊C4×C199—22no—
42C22×D199—22no—
43Dic398—24no—
44Dic398—42no—
45Dic398—42no—
46D796—42no—
47D796—42no—
48D796—81no—
49C22⋊D199—42no—
50C22⋊D199—42no—
51C2×Dic199—22no—
52C2×Dic199—22no—
53C2×Dic199—22no—
54C2×Dic199—41no—
55C2×Dic199—41no—
56C2×Dic199—41no—
57C2×Dic199—41no—
58C4×D199—44no—
59C23×C199C2×D411yescharacteristic
60C2×C4×C199—21no—
61C2×C4×C199—21no—
62C2×C4×C199—21no—
63C2×C4×C199—41no—
64C2×C4×C199—41no—
65C2×C4×C199SD1612yescomplemented
66C2×C4×C199C2×D412yes—
67C8×C199—42no—
68C22×C199—21no—
69C22×C199—21no—
70C22×C199#12011yescharacteristic
71C22×C199C2×SD1611yescharacteristic
72C22×C199AGL(1,ℤ/8)11yescharacteristic
73C2×D199—42no—
74C2×D199—42no—
75C2×D199—42no—
76C4×C199—22no—
77C4×C199—22no—
78C4×C199—41no—
79Dic199—42no—
80Dic199—42no—
81Dic199—42no—
82C2×C199—21no—
83C2×C199—21no—
84C2×C199#45011yescharacteristic
85C2×C199#45611yescharacteristic
86C2×C199B(4,𝔽2)11yescharacteristic
87D199—82no—
88C199#160411yescharacteristic, complemented, Sylow
89#1604—1991nomaximal, Sylow
90#537—1991no—
91#448—1992no—
92#395—1991no—
93#327—1992no—
94#324D19911yescharacteristic, complemented
95C2×#41—3981no—
96C22×Q8—1991no—
97C2×SD16—3982no—
98C22⋊Q8—3982no—
99#121—3982no—
100#121—3981no—
101#121—1992no—
102C2×C4⋊C4—3981no—
103C2×C4⋊C4C2×D19911yescharacteristic
104#103—3982no—
105C22⋊C8C2×D19912yes—
106#95—3981no—
107#95—1991no—
108#41—7962no—
109#41—7962no—
110C2×Q8—7962no—
111C2×Q8—3981no—
112C2×Q8—3982no—
113C2×D4—3981no—
114C2×D4—3981no—
115C2×D4—3982no—
116C2×D4—1992no—
117C22×C4—3982no—
118C22×C4—3981no—
119C22×C4—3981no—
120C22×C4D79611yescharacteristic
121C22×C4D79611yescharacteristic
122C22×C4C22×D19911yescharacteristic
123SD16—7964no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—7962no—
127C4⋊C4—3981no—
128C4⋊C4—1992no—
129C22⋊C4—3982no—
130C22⋊C4—3982no—
131C23—3982no—
132C23C2×D79611yescharacteristic
133Q8—7962no—
134Q8—7962no—
135Q8—3984no—
136D4—7962no—
137D4—7962no—
138D4—7962no—
139D4—7962no—
140D4—15921no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—7964no—
147C2×C4—7961no—
148C2×C4—7961no—
149C2×C4—7961no—
150C2×C4—7961no—
151C2×C4—3982no—
152C2×C4—3982no—
153C2×C4—3982no—
154C2×C4#1569187512yescomplemented
155C2×C4D4×D19912yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—7962no—
160C22—7962no—
161C22—7962no—
162C22#2026392511yescharacteristic
163C22C2×#1569187511yescharacteristic
164C22#2026398311yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—7962no—
169C4—7962no—
170C4—7962no—
171C2—21no—
172C2—21no—
173C2—15922no—
174C2#2716417111yescharacteristic
175C2#2716420311yescharacteristic
176C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199#120
prime radical#27164599C2
hypercenter#324D199
hypocenterC199#1604
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27164159C2semidirect
3. #27163938C2semidirect
4. #27163753C2semidirect
5. #324×C199C2semidirect
65. C2×C4×C199SD16semidirect
88. C199#1604semidirect
94. #324D199semidirect
154. C2×C4#15691875semidirect

Maximal subgroups

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

subgroupindex
#271645992
#271641592
#271639382
#271637532
#324×C1992
#1604199