Finite Group Databasebeta

← C53×#1605 #20270542

Subgroups — C53×#1605 #20270542

Order 6784.

238 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 68 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×#544C211yesmaximal, characteristic
2C53×#451C212yesmaximal
3C53×#395C211yesmaximal, characteristic
4C53×#328C212yesmaximal
5C53×#324C211yesmaximal, characteristic
6C22×Q8×C53C2211yescharacteristic
7C2×Q16×C53—22no—
8C4⋊Q8×C53—22no—
9#125×C53—22no—
10C22⋊Q8×C53—21no—
11C22⋊Q8×C53C2212yes—
12#117×C53—21no—
13C2×C4⋊C4×C53—21no—
14C2×C4⋊C4×C53C2211yescharacteristic
15#103×C53—22no—
16C22⋊C8×C53C2212yes—
17#95×C53—21no—
18#95×C53C2211yescharacteristic
19C2×Q8×C53—21no—
20C2×Q8×C53—22no—
21C2×Q8×C53—42no—
22C2×Q8×C53D412yes—
23C22×C4×C53—21no—
24C22×C4×C53—21no—
25C22×C4×C53D411yescharacteristic
26C22×C4×C53D411yescharacteristic
27C22×C4×C53C2311yescharacteristic
28Q16×C53—44no—
29C2×C8×C53—22no—
30C4⋊C4×C53—21no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—22no—
34C4⋊C4×C53—22no—
35C4⋊C4×C53—22no—
36C4⋊C4×C53—42no—
37C4⋊C4×C53D412yes—
38C22⋊C4×C53—22no—
39C42×C53—22no—
40C23×C53C2×D411yescharacteristic
41Q8×C53—24no—
42Q8×C53—42no—
43Q8×C53—42no—
44Q8×C53—81no—
45C2×C4×C53—21no—
46C2×C4×C53—21no—
47C2×C4×C53—22no—
48C2×C4×C53—22no—
49C2×C4×C53—21no—
50C2×C4×C53—22no—
51C2×C4×C53—22no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53Q1612yescomplemented
59C2×C4×C53C2×D412yes—
60C8×C53—42no—
61C22×C53—21no—
62C22×C53—21no—
63C22×C53#12011yescharacteristic
64C22×C53C2×Q1611yescharacteristic
65C22×C53#13711yescharacteristic
66C4×C53—22no—
67C4×C53—22no—
68C4×C53—42no—
69C4×C53—42no—
70C4×C53—42no—
71C4×C53—41no—
72C4×C53—42no—
73#1605C5311yesmaximal, characteristic, direct product factor, Sylow
74C2×C53—21no—
75C2×C53—21no—
76C2×C53#45111yescharacteristic
77C2×C53#45611yescharacteristic
78C2×C53B(4,𝔽2)11yescharacteristic
79#544C2×C5311yescharacteristic
80#451C2×C5312yes—
81#395C2×C5311yescharacteristic
82#328C2×C5312yes—
83#324C2×C5311yescharacteristic
84C53#160511yescharacteristic, direct product factor, Sylow
85C22×Q8C22×C5311yescharacteristic
86C2×Q16—22no—
87C4⋊Q8—22no—
88#125—22no—
89C22⋊Q8—21no—
90C22⋊Q8C22×C5312yes—
91#117—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C22×C5311yescharacteristic
94#103—22no—
95C22⋊C8C22×C5312yes—
96#95—21no—
97#95C22×C5311yescharacteristic
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—42no—
101C2×Q8D4×C5312yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4D4×C5311yescharacteristic
105C22×C4D4×C5311yescharacteristic
106C22×C4C23×C5311yescharacteristic
107Q16—44no—
108C2×C8—22no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—42no—
116C4⋊C4D4×C5312yes—
117C22⋊C4—22no—
118C42—22no—
119C23C2×D4×C5311yescharacteristic
120Q8—24no—
121Q8—42no—
122Q8—42no—
123Q8—81no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4Q16×C5312yescomplemented
138C2×C4C2×D4×C5312yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#120×C5311yescharacteristic
143C22C2×Q16×C5311yescharacteristic
144C22#137×C5311yescharacteristic
145C4—22no—
146C4—22no—
147C4—42no—
148C4—42no—
149C4—42no—
150C4—41no—
151C4—42no—
152C2—21no—
153C2—21no—
154C2C53×#45111yescharacteristic
155C2C53×#45611yescharacteristic
156C2C53×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C53#120
commutator, Frattini, cosocleC22×C4C23×C53
prime radicalC23×C53C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C22×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
58. C2×C4×C53Q16semidirect
73. #1605C53direct (N × H)
84. C53#1605direct (N × H)
137. C2×C4Q16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#5442
C53×#4512
C53×#3952
C53×#3282
C53×#3242
#160553