Finite Group Databasebeta

← #38767852

Subgroups — #38767852

Order 25472.

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

normal subgroups only

This group has 254 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#27164331C212yesmaximal, complemented
3#27163751C212yesmaximal, complemented
4#27163744C211yesmaximal, characteristic, complemented
5#398×C199C211yesmaximal, characteristic, complemented
6C2×#15691886—21no—
7#20263935—22no—
8#20263930—22no—
9#20263929—21no—
10#20263929C2212yes—
11#20263888—22no—
12#20263875—22no—
13C22⋊(C199⋊C8)C2212yes—
14C2×C4⋊C4×C199—21no—
15C2×C4⋊C4×C199C2211yescharacteristic
16C2×C4⋊C4×C199C2211yescharacteristic
17#95×C199—21no—
18#95×C199C2211yescharacteristic
19#15691886—42no—
20#15691886—42no—
21C2×Dic398—21no—
22C2×D796—21no—
23C2×D796—21no—
24C2×D796D412yes—
25C2×C22⋊D199—22no—
26C2×C4×D199—22no—
27C2×C199⋊C8—22no—
28C4⋊Dic199—21no—
29C4⋊Dic199D412yes—
30(C2×D199)⋊C4—22no—
31(C2×D199)⋊C4—22no—
32C22×C4×C199—21no—
33C22×C4×C199—21no—
34C22×C4×C199D411yescharacteristic
35C22×C4×C199D411yescharacteristic
36C22×C4×C199C2311yescharacteristic
37C4⋊C4×C199—22no—
38C4⋊C4×C199—22no—
39C4⋊C4×C199—42no—
40C22×D199—22no—
41Dic398—42no—
42D796—42no—
43D796—42no—
44D796—81no—
45C22⋊D199—42no—
46C22⋊D199—42no—
47C2×Dic199—22no—
48C4×D199—44no—
49C199⋊C8—42no—
50C23×C199C2×D411yescharacteristic
51C2×C4×C199—21no—
52C2×C4×C199—21no—
53C2×C4×C199—22no—
54C2×C4×C199—21no—
55C2×C4×C199—41no—
56C2×C4×C199—41no—
57C2×C4×C199—41no—
58C2×C4×C199—41no—
59C2×C4×C199—41no—
60C2×C4×C199—41no—
61C2×C4×C199—41no—
62C2×C4×C199D812yes—
63C2×C4×C199C2×D412yes—
64C22×C199—21no—
65C22×C199—21no—
66C22×C199#12011yescharacteristic
67C22×C199C2×D811yescharacteristic
68C22×C199AGL(1,ℤ/8)11yescharacteristic
69C2×D199—42no—
70C2×D199—42no—
71C2×D199—42no—
72C4×C199—22no—
73C4×C199—22no—
74C4×C199—42no—
75C4×C199—41no—
76C4×C199—81no—
77Dic199—42no—
78C2×C199—21no—
79C2×C199—21no—
80C2×C199#44711yescharacteristic
81C2×C199#45611yescharacteristic
82C2×C199#45811yescharacteristic
83D199—82no—
84C199#161011yescharacteristic, complemented, Sylow
85#1610—1991nomaximal, Sylow
86#537—1991no—
87#480—1992no—
88#398D19911yescharacteristic, complemented
89#327—1992no—
90#324—1991no—
91C2×#41—3981no—
92C22⋊Q8—3982no—
93#121—3982no—
94#121—3981no—
95#121—1992no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C2×D19911yescharacteristic
98C2×C4⋊C4C2×D19911yescharacteristic
99#107—3982no—
100#102—3982no—
101C22⋊C8—1992no—
102#95—21no—
103#95C2×D19911yescharacteristic
104#41—7962no—
105#41—7962no—
106C2×Q8—3981no—
107C2×D4—3981no—
108C2×D4—3981no—
109C2×D4—3982no—
110C2×D4—1992no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—3982no—
114C22×C4C22⋊D19911yescharacteristic
115C22×C4C22⋊D19911yescharacteristic
116C22×C4C22×D19911yescharacteristic
117C2×C8—3982no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—3981no—
122C4⋊C4—1992no—
123C22⋊C4—3982no—
124C22⋊C4—3982no—
125C23—3982no—
126C23C2×C22⋊D19911yescharacteristic
127Q8—7962no—
128D4—7962no—
129D4—7962no—
130D4—7962no—
131D4—7962no—
132D4—15921no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—7964no—
145C2×C4—3982no—
146C2×C4#1569187112yes—
147C2×C4D4×D19912yes—
148C8—7962no—
149C22—21no—
150C22—21no—
151C22—7962no—
152C22—7962no—
153C22—7962no—
154C22#2026392311yescharacteristic
155C22C2×#1569187111yescharacteristic
156C22#2026398011yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—81no—
162C4—7962no—
163C2—21no—
164C2—21no—
165C2—15922no—
166C2#2716415111yescharacteristic
167C2#2716420011yescharacteristic
168C2#2716420711yescharacteristic
subgroupG/H
centerC22#20263923
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#398×C199C2
socleC22×C199#120
prime radical#27164599C2
hypercenter#398D199
hypocenterC199#1610
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #27164331C2semidirect
3. #27163751C2semidirect
4. #27163744C2semidirect
5. #398×C199C2semidirect
84. C199#1610semidirect
88. #398D199semidirect

Maximal subgroups

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

subgroupindex
#271645992
#271643312
#271637512
#271637442
#398×C1992
#1610199