Finite Group Databasebeta

← #38767357

Subgroups — #38767357

Order 25472.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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#27164568C211yesmaximal, characteristic
2#27164075C211yesmaximal, characteristic, complemented
3#27163751C214yesmaximal, complemented
4#376×C199C211yesmaximal, characteristic, complemented
5C2×#15691886—21no—
6#20263955C411yescharacteristic, complemented
7#20263952C411yescharacteristic, complemented
8#20263943—21no—
9#20263929—21no—
10#20263929C2212yes—
11C4×D796—21no—
12C4×D796C412yescomplemented
13C4⋊(C199⋊C8)—21no—
14C22⋊(C199⋊C8)C2212yes—
15#20263866—21no—
16C2×C42×C199—21no—
17C2×C42×C199C2211yescharacteristic
18#95×C199—21no—
19#95×C199C2212yes—
20#15691886—44no—
21C2×Dic398—21no—
22C2×D796—21no—
23C2×D796—21no—
24C2×D796C2×C412yes—
25C2×C22⋊D199—22no—
26C2×C4×D199—22no—
27C2×C199⋊C8—22no—
28C4⋊Dic199—21no—
29C4⋊Dic199C2×C412yes—
30(C2×D199)⋊C4—22no—
31(C2×D199)⋊C4—22no—
32C22×C4×C199—22no—
33C22×C4×C199D411yescharacteristic
34C22×C4×C199D411yescharacteristic
35C22×C4×C199C2311yescharacteristic
36C42×C199—24no—
37C42×C199C2×C411yescharacteristic
38C42×C199C2×C411yescharacteristic
39C42×C199D411yescharacteristic
40C42×C199D411yescharacteristic
41C22×D199—22no—
42Dic398—42no—
43D796—42no—
44D796—42no—
45D796—42no—
46C22⋊D199—44no—
47C2×Dic199—22no—
48C4×D199—44no—
49C199⋊C8—42no—
50C23×C199C2×D411yescharacteristic
51C2×C4×C199—22no—
52C2×C4×C199—22no—
53C2×C4×C199—24no—
54C2×C4×C199—41no—
55C2×C4×C199—42no—
56C2×C4×C199—42no—
57C2×C4×C199C22⋊C411yescharacteristic
58C2×C4×C199C22⋊C411yescharacteristic
59C2×C4×C199C22⋊C411yescharacteristic
60C2×C4×C199C22⋊C411yescharacteristic
61C2×C4×C199C22×C411yescharacteristic
62C2×C4×C199C2×D411yescharacteristic
63C22×C199—21no—
64C22×C199—21no—
65C22×C199C2×C22⋊C411yescharacteristic
66C22×C199AGL(1,ℤ/8)12yes—
67C2×D199—42no—
68C2×D199—42no—
69C2×D199—42no—
70C4×C199—21no—
71C4×C199—21no—
72C4×C199—21no—
73C4×C199—21no—
74C4×C199—21no—
75C4×C199—44no—
76C4×C199#10412yes—
77Dic199—42no—
78C2×C199—22no—
79C2×C199#41011yescharacteristic
80C2×C199#41811yescharacteristic
81C2×C199C2×#10411yescharacteristic
82D199—82no—
83C199#150511yescharacteristic, complemented, Sylow
84#1505—1991nomaximal, Sylow
85#532—1991no—
86#432—1991no—
87#376D19911yescharacteristic, complemented
88#327—1994no—
89C2×#41—3981no—
90C4⋊Q8—1991no—
91#127—1991no—
92#124—3981no—
93#121—3981no—
94#121—1992no—
95C4×D4—3981no—
96C4×D4—1992no—
97C2×C42—21no—
98C2×C42C2×D19911yescharacteristic
99C4⋊C8—3981no—
100C22⋊C8—1992no—
101#97—3981no—
102#95—21no—
103#95C2×D19912yes—
104#41—7964no—
105C2×Q8—3981no—
106C2×D4—3982no—
107C2×D4—3981no—
108C2×D4—3981no—
109C2×D4—1992no—
110C22×C4—22no—
111C22×C4—3982no—
112C22×C4C22⋊D19911yescharacteristic
113C22×C4C22⋊D19911yescharacteristic
114C22×C4C22×D19911yescharacteristic
115C2×C8—3982no—
116C4⋊C4—3981no—
117C4⋊C4—1992no—
118C22⋊C4—3982no—
119C22⋊C4—3982no—
120C42—24no—
121C42C4×D19911yescharacteristic
122C42C4×D19911yescharacteristic
123C42D79611yescharacteristic
124C42D79611yescharacteristic
125C23—3982no—
126C23C2×C22⋊D19911yescharacteristic
127Q8—7962no—
128D4—7962no—
129D4—7962no—
130D4—7964no—
131D4—7962no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—24no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—7964no—
139C2×C4—3982no—
140C2×C4(C2×D199)⋊C411yescharacteristic
141C2×C4(C2×D199)⋊C411yescharacteristic
142C2×C4(C2×D199)⋊C411yescharacteristic
143C2×C4(C2×D199)⋊C411yescharacteristic
144C2×C4C2×C4×D19911yescharacteristic
145C2×C4C2×D79611yescharacteristic
146C8—7962no—
147C22—21no—
148C22—21no—
149C22—7962no—
150C22—7962no—
151C22—7962no—
152C22C2×(C2×D199)⋊C411yescharacteristic
153C22#2026398012yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—44no—
160C4—7962no—
161C4#2026388312yes—
162C2—22no—
163C2—15922no—
164C2#2716399111yescharacteristic
165C2#2716402211yescharacteristic
166C2C2×#2026388311yescharacteristic
subgroupG/H
centerC22C2×(C2×D199)⋊C4
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#376×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164568C2
cosocleC22×C4×C199C23
hypercenter#376D199
hypocenterC199#1505
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
2. #27164075C2semidirect
3. #27163751C2semidirect
4. #376×C199C2semidirect
6. #20263955C4semidirect
7. #20263952C4semidirect
12. C4×D796C4semidirect
83. C199#1505semidirect
87. #376D199semidirect

Maximal subgroups

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

subgroupindex
#271645682
#271640752
#271637512
#376×C1992
#1505199