Finite Group Databasebeta

← C22×C53×#379 #27241515

Subgroups — C22×C53×#379 #27241515

Order 13568.

33214 conjugacy classes of subgroups; only the 3614 normal subgroups are shown.

normal subgroups only

This group has 33214 conjugacy classes of subgroups — too many to draw. Only its 3614 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
1C27×C53C211yesmaximal, characteristic
2C23×C22⋊C4×C53C216yesmaximal, complemented
3C2×C53×#379C2124yesmaximal, direct product factor
4C26×C53—248no—
5C26×C53C4116yescomplemented
6C26×C53C22112yes—
7C26×C53C2213yes—
8C24×C4×C53C2214yescomplemented
9C22×C22⋊C4×C53—248no—
10C22×C22⋊C4×C53C22172yescomplemented
11C53×#379C22164yesdirect product factor
12C25×C53—2512no—
13C25×C53—2384no—
14C25×C53—2288no—
15C25×C53—248no—
16C25×C53C2×C4124yes—
17C25×C53C2×C4196yescomplemented
18C25×C53D4148yes—
19C25×C53C23116yes—
20C25×C53C23118yes—
21C25×C53C2311yescharacteristic
22C23×C4×C53—248no—
23C23×C4×C53C23124yescomplemented
24C2×C22⋊C4×C53—2288no—
25C2×C22⋊C4×C53C23196yescomplemented
26C24×C53—21536no—
27C24×C53—22048no—
28C24×C53—2192no—
29C24×C53—21152no—
30C24×C53—2384no—
31C24×C53—2192no—
32C24×C53—2144no—
33C24×C53C22⋊C41192yescomplemented
34C24×C53C22×C4172yes—
35C24×C53C22×C414yes—
36C24×C53C22×C4164yescomplemented
37C24×C53C2×D41144yes—
38C24×C53C2×D4124yes—
39C24×C53C24112yes—
40C24×C53C2413yes—
41C22×C4×C53—2144no—
42C22×C4×C53—464no—
43C22×C4×C53C24116yescomplemented
44C22⋊C4×C53—2192no—
45C23×C53—2512no—
46C23×C53—23072no—
47C23×C53—2288no—
48C23×C53—2384no—
49C23×C53—2576no—
50C23×C53—248no—
51C23×C53—2768no—
52C23×C53C2×C22⋊C4148yes—
53C23×C53C2×C22⋊C41288yescomplemented
54C23×C53#120164yes—
55C23×C53C23×C4124yes—
56C23×C53C23×C416yes—
57C23×C53C22×D4148yes—
58C23×C53C22×D4136yes—
59C23×C53C2511yescharacteristic
60C2×C4×C53—248no—
61C2×C4×C53—496no—
62C22×#379C5311yesmaximal, characteristic, direct product factor, Sylow
63C22×C53—2512no—
64C22×C53—248no—
65C22×C53—296no—
66C22×C53—2576no—
67C22×C53#379164yesdirect product factor
68C22×C53C22×C22⋊C4136yes—
69C22×C53C22×C22⋊C4148yescomplemented
70C22×C53C2×#120148yes—
71C22×C53C24×C411yescharacteristic
72C22×C53C23×D416yes—
73C4×C53—416no—
74C27C2×C5311yescharacteristic
75C23×C22⋊C4C2×C5316yescomplemented
76C2×#379C2×C53124yesdirect product factor
77C2×C53—248no—
78C2×C53C2×#379124yesdirect product factor
79C2×C53C23×C22⋊C413yes—
80C2×C53C22×#12014yes—
81C26—248no—
82C26C4×C53116yescomplemented
83C26C22×C53112yes—
84C26C22×C5313yes—
85C24×C4C22×C5314yescomplemented
86C22×C22⋊C4—248no—
87C22×C22⋊C4C22×C53172yescomplemented
88#379C22×C53164yesdirect product factor
89C53C22×#37911yescharacteristic, direct product factor, Sylow
90C25—2512no—
91C25—2288no—
92C25—248no—
93C25—2384no—
94C25C2×C4×C53124yes—
95C25C2×C4×C53196yescomplemented
96C25D4×C53148yes—
97C25C23×C53116yes—
98C25C23×C53118yes—
99C25C23×C5311yescharacteristic
100C23×C4—248no—
101C23×C4C23×C53124yescomplemented
102C2×C22⋊C4—2288no—
103C2×C22⋊C4C23×C53196yescomplemented
104C24—21536no—
105C24—22048no—
106C24—2192no—
107C24—21152no—
108C24—2384no—
109C24—2192no—
110C24—2144no—
111C24C22⋊C4×C531192yescomplemented
112C24C22×C4×C53172yes—
113C24C22×C4×C5314yes—
114C24C22×C4×C53164yescomplemented
115C24C2×D4×C531144yes—
116C24C2×D4×C53124yes—
117C24C24×C53112yes—
118C24C24×C5313yes—
119C22×C4—2144no—
120C22×C4—464no—
121C22×C4C24×C53116yescomplemented
122C22⋊C4—2192no—
123C23—23072no—
124C23—2512no—
125C23—2288no—
126C23—2384no—
127C23—2576no—
128C23—248no—
129C23—2768no—
130C23C2×C22⋊C4×C53148yes—
131C23C2×C22⋊C4×C531288yescomplemented
132C23#120×C53164yes—
133C23C23×C4×C53124yes—
134C23C23×C4×C5316yes—
135C23C22×D4×C53148yes—
136C23C22×D4×C53136yes—
137C23C25×C5311yescharacteristic
138C2×C4—248no—
139C2×C4—496no—
140C22—2512no—
141C22—2576no—
142C22—248no—
143C22—296no—
144C22C53×#379164yesdirect product factor
145C22C22×C22⋊C4×C53136yes—
146C22C22×C22⋊C4×C53148yescomplemented
147C22C2×#120×C53148yes—
148C22C24×C4×C5311yescharacteristic
149C22C23×D4×C5316yes—
150C4—416no—
151C2—248no—
152C2C2×C53×#379124yesdirect product factor
153C2C23×C22⋊C4×C5313yes—
154C2C22×#120×C5314yes—
subgroupG/H
center, socleC25×C53C23
commutatorC22C24×C4×C53
Frattini, cosocleC23C25×C53
prime radicalC27×C53C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C22⋊C4×C53C2semidirect
3. C2×C53×#379C2direct (N × H)
5. C26×C53C4semidirect
8. C24×C4×C53C22semidirect
10. C22×C22⋊C4×C53C22semidirect
11. C53×#379C22direct (N × H)
17. C25×C53C2×C4semidirect
23. C23×C4×C53C23semidirect
25. C2×C22⋊C4×C53C23semidirect
33. C24×C53C22⋊C4semidirect
36. C24×C53C22×C4semidirect
43. C22×C4×C53C24semidirect
53. C23×C53C2×C22⋊C4semidirect
62. C22×#379C53direct (N × H)
67. C22×C53#379direct (N × H)
69. C22×C53C22×C22⋊C4semidirect
75. C23×C22⋊C4C2×C53semidirect
76. C2×#379C2×C53direct (N × H)
78. C2×C53C2×#379direct (N × H)
82. C26C4×C53semidirect
85. C24×C4C22×C53semidirect
87. C22×C22⋊C4C22×C53semidirect
88. #379C22×C53direct (N × H)
89. C53C22×#379direct (N × H)
95. C25C2×C4×C53semidirect
101. C23×C4C23×C53semidirect
103. C2×C22⋊C4C23×C53semidirect
111. C24C22⋊C4×C53semidirect
114. C24C22×C4×C53semidirect
121. C22×C4C24×C53semidirect
131. C23C2×C22⋊C4×C53semidirect
144. C22C53×#379direct (N × H)
146. C22C22×C22⋊C4×C53semidirect
152. C2C2×C53×#379direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C532
C23×C22⋊C4×C532
C2×C53×#3792
C22×#37953