Finite Group Databasebeta

← C22×C23×#379 #19181629

Subgroups — C22×C23×#379 #19181629

Order 5888.

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×C23C211yesmaximal, characteristic
2C23×C22⋊C4×C23C216yesmaximal, complemented
3C2×C23×#379C2124yesmaximal, direct product factor
4C26×C23—248no—
5C26×C23C4116yescomplemented
6C26×C23C22112yes—
7C26×C23C2213yes—
8C24×C4×C23C2214yescomplemented
9C22×C22⋊C4×C23—248no—
10C22×C22⋊C4×C23C22172yescomplemented
11C23×#379C22164yesdirect product factor
12C25×C23—2512no—
13C25×C23—2384no—
14C25×C23—2288no—
15C25×C23—248no—
16C25×C23C2×C4124yes—
17C25×C23C2×C4196yescomplemented
18C25×C23D4148yes—
19C25×C23C23116yes—
20C25×C23C23118yes—
21C25×C23C2311yescharacteristic
22C23×C4×C23—248no—
23C23×C4×C23C23124yescomplemented
24C2×C22⋊C4×C23—2288no—
25C2×C22⋊C4×C23C23196yescomplemented
26C24×C23—21536no—
27C24×C23—22048no—
28C24×C23—2192no—
29C24×C23—21152no—
30C24×C23—2384no—
31C24×C23—2192no—
32C24×C23—2144no—
33C24×C23C22⋊C41192yescomplemented
34C24×C23C22×C4172yes—
35C24×C23C22×C414yes—
36C24×C23C22×C4164yescomplemented
37C24×C23C2×D41144yes—
38C24×C23C2×D4124yes—
39C24×C23C24112yes—
40C24×C23C2413yes—
41C22×C4×C23—2144no—
42C22×C4×C23—464no—
43C22×C4×C23C24116yescomplemented
44C22⋊C4×C23—2192no—
45C22×#379C2311yesmaximal, characteristic, direct product factor, Sylow
46C23×C23—2512no—
47C23×C23—23072no—
48C23×C23—2288no—
49C23×C23—2384no—
50C23×C23—2576no—
51C23×C23—248no—
52C23×C23—2768no—
53C23×C23C2×C22⋊C4148yes—
54C23×C23C2×C22⋊C41288yescomplemented
55C23×C23#120164yes—
56C23×C23C23×C4124yes—
57C23×C23C23×C416yes—
58C23×C23C22×D4148yes—
59C23×C23C22×D4136yes—
60C23×C23C2511yescharacteristic
61C2×C4×C23—248no—
62C2×C4×C23—496no—
63C27C2×C2311yescharacteristic
64C23×C22⋊C4C2×C2316yescomplemented
65C2×#379C2×C23124yesdirect product factor
66C22×C23—2512no—
67C22×C23—248no—
68C22×C23—296no—
69C22×C23—2576no—
70C22×C23#379164yesdirect product factor
71C22×C23C22×C22⋊C4136yes—
72C22×C23C22×C22⋊C4148yescomplemented
73C22×C23C2×#120148yes—
74C22×C23C24×C411yescharacteristic
75C22×C23C23×D416yes—
76C4×C23—416no—
77C26—248no—
78C26C4×C23116yescomplemented
79C26C22×C23112yes—
80C26C22×C2313yes—
81C24×C4C22×C2314yescomplemented
82C22×C22⋊C4—248no—
83C22×C22⋊C4C22×C23172yescomplemented
84#379C22×C23164yesdirect product factor
85C2×C23—248no—
86C2×C23C2×#379124yesdirect product factor
87C2×C23C23×C22⋊C413yes—
88C2×C23C22×#12014yes—
89C25—2512no—
90C25—2384no—
91C25—2288no—
92C25—248no—
93C25C2×C4×C23124yes—
94C25C2×C4×C23196yescomplemented
95C25D4×C23148yes—
96C25C23×C23116yes—
97C25C23×C23118yes—
98C25C23×C2311yescharacteristic
99C23×C4—248no—
100C23×C4C23×C23124yescomplemented
101C2×C22⋊C4—2288no—
102C2×C22⋊C4C23×C23196yescomplemented
103C23C22×#37911yescharacteristic, direct product factor, Sylow
104C24—22048no—
105C24—21536no—
106C24—2192no—
107C24—21152no—
108C24—2384no—
109C24—2192no—
110C24—2144no—
111C24C22⋊C4×C231192yescomplemented
112C24C22×C4×C23172yes—
113C24C22×C4×C2314yes—
114C24C22×C4×C23164yescomplemented
115C24C2×D4×C231144yes—
116C24C2×D4×C23124yes—
117C24C24×C23112yes—
118C24C24×C2313yes—
119C22×C4—2144no—
120C22×C4—464no—
121C22×C4C24×C23116yescomplemented
122C22⋊C4—2192no—
123C23—2512no—
124C23—23072no—
125C23—248no—
126C23—2768no—
127C23—2288no—
128C23—2384no—
129C23—2576no—
130C23C2×C22⋊C4×C23148yes—
131C23C2×C22⋊C4×C231288yescomplemented
132C23C23×#120164yes—
133C23C23×C4×C23124yes—
134C23C23×C4×C2316yes—
135C23C22×D4×C23148yes—
136C23C22×D4×C23136yes—
137C23C25×C2311yescharacteristic
138C2×C4—248no—
139C2×C4—496no—
140C22—2512no—
141C22—296no—
142C22—2576no—
143C22—248no—
144C22C23×#379164yesdirect product factor
145C22C22×C22⋊C4×C23136yes—
146C22C22×C22⋊C4×C23148yescomplemented
147C22C2×C23×#120148yes—
148C22C24×C4×C2311yescharacteristic
149C22C23×D4×C2316yes—
150C4—416no—
151C2—248no—
152C2C2×C23×#379124yesdirect product factor
153C2C23×C22⋊C4×C2313yes—
154C2C22×C23×#12014yes—
subgroupG/H
center, socleC25×C23C23
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
prime radicalC27×C23C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×C22⋊C4×C23C2semidirect
3. C2×C23×#379C2direct (N × H)
5. C26×C23C4semidirect
8. C24×C4×C23C22semidirect
10. C22×C22⋊C4×C23C22semidirect
11. C23×#379C22direct (N × H)
17. C25×C23C2×C4semidirect
23. C23×C4×C23C23semidirect
25. C2×C22⋊C4×C23C23semidirect
33. C24×C23C22⋊C4semidirect
36. C24×C23C22×C4semidirect
43. C22×C4×C23C24semidirect
45. C22×#379C23direct (N × H)
54. C23×C23C2×C22⋊C4semidirect
64. C23×C22⋊C4C2×C23semidirect
65. C2×#379C2×C23direct (N × H)
70. C22×C23#379direct (N × H)
72. C22×C23C22×C22⋊C4semidirect
78. C26C4×C23semidirect
81. C24×C4C22×C23semidirect
83. C22×C22⋊C4C22×C23semidirect
84. #379C22×C23direct (N × H)
86. C2×C23C2×#379direct (N × H)
94. C25C2×C4×C23semidirect
100. C23×C4C23×C23semidirect
102. C2×C22⋊C4C23×C23semidirect
103. C23C22×#379direct (N × H)
111. C24C22⋊C4×C23semidirect
114. C24C22×C4×C23semidirect
121. C22×C4C24×C23semidirect
131. C23C2×C22⋊C4×C23semidirect
144. C22C23×#379direct (N × H)
146. C22C22×C22⋊C4×C23semidirect
152. C2C2×C23×#379direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C232
C23×C22⋊C4×C232
C2×C23×#3792
C22×#37923