Finite Group Databasebeta

← C53×#2201 #20271138

Subgroups — C53×#2201 #20271138

Order 6784.

226 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 56 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
1C53×#563C211yesmaximal, characteristic, complemented
2C53×#458C212yesmaximal
3C53×#455C211yesmaximal, characteristic
4C53×#354C211yesmaximal, characteristic, complemented
5C53×#352C212yesmaximal, complemented
6H(5,𝔽2)×C53C2211yescharacteristic
7#137×C53—21no—
8C4⋊Q8×C53—21no—
9#126×C53—22no—
10#125×C53—21no—
11#124×C53C2211yescharacteristic
12#123×C53—22no—
13#123×C53C2212yes—
14C22⋊Q8×C53—22no—
15#104×C53—21no—
16#100×C53C2211yescharacteristic
17#99×C53—22no—
18#99×C53C2212yes—
19#41×C53—21no—
20#41×C53—41no—
21C2×Q8×C53—21no—
22C2×Q8×C53D411yescharacteristic
23C2×D4×C53—22no—
24C2×D4×C53—41no—
25C2×D4×C53C2311yescharacteristic
26C22×C4×C53D412yes—
27Q16×C53—41no—
28SD16×C53—41no—
29M16×C53—21no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—22no—
32C4⋊C4×C53—22no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—22no—
35C4⋊C4×C53—42no—
36C22⋊C4×C53—22no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C22⋊C4×C53—42no—
40C22⋊C4×C53D412yes—
41C42×C53—21no—
42C42×C53D411yescharacteristic
43C23×C53—22no—
44C23×C53C2×D412yes—
45Q8×C53—21no—
46Q8×C53—41no—
47Q8×C53—81no—
48D4×C53—21no—
49D4×C53—41no—
50D4×C53—42no—
51D4×C53—41no—
52C2×C4×C53—22no—
53C2×C4×C53—22no—
54C2×C4×C53—21no—
55C2×C4×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53—41no—
58C2×C4×C53—42no—
59C2×C4×C53—42no—
60C2×C4×C53C2×D411yescharacteristic
61C8×C53—41no—
62C22×C53—22no—
63C22×C53—42no—
64C22×C53—41no—
65C22×C53—82no—
66C22×C53#12011yescharacteristic
67C4×C53—21no—
68C4×C53—42no—
69C4×C53—41no—
70C4×C53—41no—
71C4×C53—42no—
72C4×C53—41no—
73C4×C53—82no—
74#2201C5311yesmaximal, characteristic, direct product factor, Sylow
75C2×C53—21no—
76C2×C53—42no—
77C2×C53—81no—
78C2×C53B(4,𝔽2)11yescharacteristic
79#563C2×C5311yescharacteristic, complemented
80#458C2×C5312yes—
81#455C2×C5311yescharacteristic
82#354C2×C5311yescharacteristic, complemented
83#352C2×C5312yescomplemented
84C53#220111yescharacteristic, direct product factor, Sylow
85H(5,𝔽2)C22×C5311yescharacteristic
86#137—21no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C22×C5311yescharacteristic
91#123—22no—
92#123C22×C5312yes—
93C22⋊Q8—22no—
94#104—21no—
95#100C22×C5311yescharacteristic
96#99—22no—
97#99C22×C5312yes—
98#41—21no—
99#41—41no—
100C2×Q8—21no—
101C2×Q8D4×C5311yescharacteristic
102C2×D4—22no—
103C2×D4—41no—
104C2×D4C23×C5311yescharacteristic
105C22×C4D4×C5312yes—
106Q16—41no—
107SD16—41no—
108M16—21no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—42no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—42no—
119C22⋊C4D4×C5312yes—
120C42—21no—
121C42D4×C5311yescharacteristic
122C23—22no—
123C23C2×D4×C5312yes—
124Q8—21no—
125Q8—41no—
126Q8—81no—
127D4—21no—
128D4—41no—
129D4—42no—
130D4—41no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4C2×D4×C5311yescharacteristic
140C8—41no—
141C22—22no—
142C22—42no—
143C22—41no—
144C22—82no—
145C22#120×C5311yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—41no—
150C4—42no—
151C4—41no—
152C4—82no—
153C2—21no—
154C2—42no—
155C2—81no—
156C2C53×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C53B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C53
prime radicalH(5,𝔽2)×C53C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C22×C53 ◁ C2×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#563C2semidirect
4. C53×#354C2semidirect
5. C53×#352C2semidirect
74. #2201C53direct (N × H)
79. #563C2×C53semidirect
82. #354C2×C53semidirect
83. #352C2×C53semidirect
84. C53#2201direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#5632
C53×#4582
C53×#4552
C53×#3542
C53×#3522
#220153