Finite Group Databasebeta

← C53×#3058 #20271995

Subgroups — C53×#3058 #20271995

Order 6784.

524 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 200 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
1C22×D8×C53C211yesmaximal, characteristic, complemented
2C2×#131×C53C211yesmaximal, characteristic, complemented
3C53×#535C212yesmaximal, complemented
4C53×#469C214yesmaximal, complemented
5C53×#466C214yesmaximal, complemented
6C53×#427C211yesmaximal, characteristic, complemented
7C53×#418C212yesmaximal, complemented
8C2×#41×C53C2211yescharacteristic
9C22×D4×C53C2212yes—
10C2×D8×C53—24no—
11C2×D8×C53C2212yes—
12C2×D8×C53C2212yes—
13#131×C53—24no—
14C2×M16×C53C2211yescharacteristic
15C2×M16×C53C2212yescomplemented
16C22×C8×C53C2211yescharacteristic
17C22×C8×C53C2212yescomplemented
18#127×C53—22no—
19#124×C53—22no—
20#121×C53C2218yescomplemented
21#120×C53—24no—
22#117×C53C2212yes—
23#107×C53C2212yescomplemented
24#106×C53C2212yescomplemented
25#102×C53C2218yescomplemented
26C24×C53—22no—
27#41×C53—22no—
28#41×C53—22no—
29C2×Q8×C53D411yescharacteristic
30C2×D4×C53—28no—
31C2×D4×C53—24no—
32C2×D4×C53—22no—
33C2×D4×C53—24no—
34C2×D4×C53D412yes—
35C2×D4×C53D411yescharacteristic
36C2×D4×C53C2314yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39D8×C53—24no—
40D8×C53—24no—
41M16×C53—22no—
42M16×C53—24no—
43C2×C8×C53—24no—
44C2×C8×C53D414yescomplemented
45C2×C8×C53C2312yes—
46C2×C8×C53C2311yescharacteristic
47C2×C8×C53C2311yescharacteristic
48C4⋊C4×C53C2314yes—
49C22⋊C4×C53—22no—
50C22⋊C4×C53—28no—
51C42×C53—22no—
52C23×C53—24no—
53C23×C53—44no—
54C23×C53—84no—
55C23×C53C2×D412yes—
56C23×C53C2×D411yescharacteristic
57Q8×C53—22no—
58D4×C53—24no—
59D4×C53—22no—
60D4×C53—24no—
61D4×C53—24no—
62D4×C53—44no—
63D4×C53—88no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—24no—
67C2×C4×C53C2×D412yes—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D412yes—
70C2×C4×C53#4112yes—
71C2×C4×C53C2411yescharacteristic
72C8×C53—22no—
73C8×C53C2×D412yes—
74C8×C53C2×D412yes—
75C22×C53—22no—
76C22×C53—22no—
77C22×C53—24no—
78C22×C53—44no—
79C22×C53—48no—
80C22×C53—88no—
81C22×C53#12112yes—
82C22×C53C22×D411yescharacteristic
83C4×C53—22no—
84C4×C53—44no—
85C4×C53#12112yes—
86C4×C53C22×D411yescharacteristic
87C4×C53C2×#4111yescharacteristic
88#3058C5311yesmaximal, characteristic, direct product factor, Sylow
89C2×C53—22no—
90C2×C53—42no—
91C2×C53—84no—
92C2×C53C2×#12111yescharacteristic
93C2×C53#57612yes—
94C22×D8C2×C5311yescharacteristic, complemented
95C2×#131C2×C5311yescharacteristic, complemented
96#535C2×C5312yescomplemented
97#469C2×C5314yescomplemented
98#466C2×C5314yescomplemented
99#427C2×C5311yescharacteristic, complemented
100#418C2×C5312yescomplemented
101C53#305811yescharacteristic, direct product factor, Sylow
102C2×#41C22×C5311yescharacteristic
103C22×D4C22×C5312yes—
104C2×D8—24no—
105C2×D8C22×C5312yes—
106C2×D8C22×C5312yes—
107#131—24no—
108C2×M16C22×C5311yescharacteristic
109C2×M16C22×C5312yescomplemented
110C22×C8C22×C5311yescharacteristic
111C22×C8C22×C5312yescomplemented
112#127—22no—
113#124—22no—
114#121C22×C5318yescomplemented
115#120—24no—
116#117C22×C5312yes—
117#107C22×C5312yescomplemented
118#106C22×C5312yescomplemented
119#102C22×C5318yescomplemented
120C24—22no—
121#41—22no—
122#41—22no—
123C2×Q8D4×C5311yescharacteristic
124C2×D4—28no—
125C2×D4—24no—
126C2×D4—22no—
127C2×D4—24no—
128C2×D4D4×C5312yes—
129C2×D4D4×C5311yescharacteristic
130C2×D4C23×C5314yes—
131C22×C4C23×C5312yes—
132C22×C4C23×C5311yescharacteristic
133D8—24no—
134D8—24no—
135M16—22no—
136M16—24no—
137C2×C8—24no—
138C2×C8D4×C5314yescomplemented
139C2×C8C23×C5312yes—
140C2×C8C23×C5311yescharacteristic
141C2×C8C23×C5311yescharacteristic
142C4⋊C4C23×C5314yes—
143C22⋊C4—22no—
144C22⋊C4—28no—
145C42—22no—
146C23—24no—
147C23—44no—
148C23—84no—
149C23C2×D4×C5312yes—
150C23C2×D4×C5311yescharacteristic
151Q8—22no—
152D4—24no—
153D4—22no—
154D4—24no—
155D4—24no—
156D4—44no—
157D4—88no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4C2×D4×C5312yes—
162C2×C4C2×D4×C5311yescharacteristic
163C2×C4C2×D4×C5312yes—
164C2×C4#41×C5312yes—
165C2×C4C24×C5311yescharacteristic
166C8—22no—
167C8C2×D4×C5312yes—
168C8C2×D4×C5312yes—
169C22—22no—
170C22—22no—
171C22—24no—
172C22—44no—
173C22—48no—
174C22—88no—
175C22#121×C5312yes—
176C22C22×D4×C5311yescharacteristic
177C4—22no—
178C4—44no—
179C4#121×C5312yes—
180C4C22×D4×C5311yescharacteristic
181C4C2×#41×C5311yescharacteristic
182C2—22no—
183C2—42no—
184C2—84no—
185C2C2×#121×C5311yescharacteristic
186C2C53×#57612yes—
subgroupG/H
center, socleC22×C53C22×D4
commutator, Frattini, cosocleC2×C4C24×C53
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×D8×C53C2semidirect
2. C2×#131×C53C2semidirect
3. C53×#535C2semidirect
4. C53×#469C2semidirect
5. C53×#466C2semidirect
6. C53×#427C2semidirect
7. C53×#418C2semidirect
15. C2×M16×C53C22semidirect
17. C22×C8×C53C22semidirect
20. #121×C53C22semidirect
23. #107×C53C22semidirect
24. #106×C53C22semidirect
25. #102×C53C22semidirect
44. C2×C8×C53D4semidirect
88. #3058C53direct (N × H)
94. C22×D8C2×C53semidirect
95. C2×#131C2×C53semidirect
96. #535C2×C53semidirect
97. #469C2×C53semidirect
98. #466C2×C53semidirect
99. #427C2×C53semidirect
100. #418C2×C53semidirect
101. C53#3058direct (N × H)
109. C2×M16C22×C53semidirect
111. C22×C8C22×C53semidirect
114. #121C22×C53semidirect
117. #107C22×C53semidirect
118. #106C22×C53semidirect
119. #102C22×C53semidirect
138. C2×C8D4×C53semidirect

Maximal subgroups

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

subgroupindex
C22×D8×C532
C2×#131×C532
C53×#5352
C53×#4692
C53×#4662
C53×#4272
C53×#4182
#305853