Finite Group Databasebeta

← #38767889

Subgroups — #38767889

Order 25472.

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

normal subgroups only

This group has 246 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#27164337C212yesmaximal, complemented
3#27163957C211yesmaximal, characteristic, complemented
4#27163753C212yesmaximal, complemented
5#324×C199C211yesmaximal, characteristic, complemented
6C2×#15691886—21no—
7#20263935—22no—
8#20263930—22no—
9#20263929—21no—
10#20263929C2212yes—
11C2×C4⋊Dic199C2211yescharacteristic
12#20263887—22no—
13#20263877—22no—
14C199⋊#95—21no—
15C199⋊#95—21no—
16C199⋊#95C2211yescharacteristic
17C2×C4⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19#15691886—42no—
20#15691886—42no—
21C2×Dic398—21no—
22C2×D796—21no—
23C2×D796—21no—
24C2×D796D412yes—
25C2×C22⋊D199—22no—
26C22×Dic199—21no—
27C22×Dic199—21no—
28C2×C4×D199—22no—
29C4⋊Dic199—21no—
30C4⋊Dic199—22no—
31C4⋊Dic199D412yes—
32(C2×D199)⋊C4—22no—
33(C2×D199)⋊C4—22no—
34C22×C4×C199D411yescharacteristic
35C22×C4×C199D411yescharacteristic
36C22×C4×C199C2311yescharacteristic
37C2×C8×C199—22no—
38C4⋊C4×C199—22no—
39C22×D199—22no—
40Dic398—42no—
41D796—42no—
42D796—42no—
43D796—81no—
44C22⋊D199—42no—
45C22⋊D199—42no—
46C2×Dic199—22no—
47C2×Dic199—41no—
48C2×Dic199—41no—
49C2×Dic199—42no—
50C2×Dic199—41no—
51C2×Dic199—41no—
52C4×D199—44no—
53C23×C199C2×D411yescharacteristic
54C2×C4×C199—21no—
55C2×C4×C199—21no—
56C2×C4×C199—21no—
57C2×C4×C199—41no—
58C2×C4×C199—41no—
59C2×C4×C199SD1612yes—
60C2×C4×C199C2×D412yes—
61C8×C199—42no—
62C22×C199—21no—
63C22×C199—21no—
64C22×C199#12011yescharacteristic
65C22×C199C2×SD1611yescharacteristic
66C22×C199AGL(1,ℤ/8)11yescharacteristic
67C2×D199—42no—
68C2×D199—42no—
69C2×D199—42no—
70C4×C199—22no—
71C4×C199—22no—
72C4×C199—41no—
73Dic199—42no—
74Dic199—81no—
75Dic199—81no—
76C2×C199—21no—
77C2×C199—21no—
78C2×C199#45011yescharacteristic
79C2×C199#45411yescharacteristic
80C2×C199#45811yescharacteristic
81D199—82no—
82C199#161611yescharacteristic, complemented, Sylow
83#1616—1991nomaximal, Sylow
84#537—1991no—
85#481—1992no—
86#400—1991no—
87#327—1992no—
88#324D19911yescharacteristic, complemented
89C2×#41—3981no—
90C22⋊Q8—3982no—
91#121—3982no—
92#121—3981no—
93#121—1992no—
94C2×C4⋊C4—1991no—
95C2×C4⋊C4C2×D19911yescharacteristic
96#106—3982no—
97#102—3982no—
98C22⋊C8C2×D19912yes—
99#95—3981no—
100#95—3981no—
101#95—1991no—
102#41—7962no—
103#41—7962no—
104C2×Q8—3981no—
105C2×D4—3981no—
106C2×D4—3981no—
107C2×D4—3982no—
108C2×D4—1992no—
109C22×C4—3982no—
110C22×C4—3981no—
111C22×C4—3981no—
112C22×C4D79611yescharacteristic
113C22×C4D79611yescharacteristic
114C22×C4C22×D19911yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—3981no—
118C4⋊C4—3982no—
119C4⋊C4—1992no—
120C22⋊C4—3982no—
121C22⋊C4—3982no—
122C23—3982no—
123C23C2×D79611yescharacteristic
124Q8—7962no—
125D4—7962no—
126D4—7962no—
127D4—7962no—
128D4—7962no—
129D4—15921no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—7964no—
136C2×C4—7961no—
137C2×C4—7961no—
138C2×C4—7962no—
139C2×C4—7961no—
140C2×C4—7961no—
141C2×C4—3982no—
142C2×C4#1569187512yes—
143C2×C4D4×D19912yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—7962no—
148C22—7962no—
149C22—7962no—
150C22#2026392511yescharacteristic
151C22C2×#1569187511yescharacteristic
152C22#2026398311yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—7962no—
157C4—15921no—
158C4—15921no—
159C2—21no—
160C2—21no—
161C2—15922no—
162C2#2716417111yescharacteristic
163C2#2716419311yescharacteristic
164C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199#120
prime radical#27164599C2
hypercenter#324D199
hypocenterC199#1616
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. #27164337C2semidirect
3. #27163957C2semidirect
4. #27163753C2semidirect
5. #324×C199C2semidirect
82. C199#1616semidirect
88. #324D199semidirect

Maximal subgroups

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

subgroupindex
#271645992
#271643372
#271639572
#271637532
#324×C1992
#1616199