Finite Group Databasebeta

← C23×C3×C7×C8×C23 #72207668

Subgroups — C23×C3×C7×C8×C23 #72207668

Order 30912.

1352 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×C7×C23C211yesmaximal, characteristic
2C22×C3×C7×C8×C23C2114yesmaximal, direct product factor
3C23×C7×C8×C23C311yesmaximal, characteristic, direct product factor, Hall
4C24×C3×C7×C23C411yescharacteristic
5C22×C3×C4×C7×C23C417yes—
6C22×C3×C4×C7×C23C2217yes—
7C2×C3×C7×C8×C23C22128yesdirect product factor
8C23×C4×C7×C23C2×C311yescharacteristic
9C22×C7×C8×C23C2×C3114yesdirect product factor
10C23×C3×C8×C23C711yesmaximal, characteristic, direct product factor, Hall
11C23×C3×C7×C23C818yesdirect product factor
12C23×C3×C7×C23C2×C417yes—
13C2×C3×C4×C7×C23C2×C4121yes—
14C2×C3×C4×C7×C23C2317yes—
15C3×C7×C8×C23C2318yesdirect product factor
16C24×C7×C23C3×C411yescharacteristic
17C22×C4×C7×C23C3×C417yes—
18C22×C4×C7×C23C22×C317yes—
19C2×C7×C8×C23C22×C3128yesdirect product factor
20C23×C3×C4×C23C2×C711yescharacteristic
21C22×C3×C8×C23C2×C7114yesdirect product factor
22C22×C3×C7×C23C2×C8128yesdirect product factor
23C22×C3×C7×C23C22×C417yes—
24C3×C4×C7×C23C22×C417yes—
25C3×C4×C7×C23C2411yescharacteristic
26C23×C8×C23C3×C711yescharacteristic, direct product factor, Hall
27C23×C3×C7×C8C2311yesmaximal, characteristic, direct product factor, Hall
28C23×C7×C23C3×C818yesdirect product factor
29C23×C7×C23C2×C3×C417yes—
30C2×C4×C7×C23C2×C3×C4121yes—
31C2×C4×C7×C23C23×C317yes—
32C7×C8×C23C23×C318yesdirect product factor
33C24×C3×C23C4×C711yescharacteristic
34C22×C3×C4×C23C4×C717yes—
35C22×C3×C4×C23C22×C717yes—
36C2×C3×C8×C23C22×C7128yesdirect product factor
37C2×C3×C7×C23C22×C8114yesdirect product factor
38C2×C3×C7×C23C23×C411yescharacteristic
39C23×C4×C23C2×C3×C711yescharacteristic
40C22×C8×C23C2×C3×C7114yesdirect product factor
41C23×C3×C4×C7C2×C2311yescharacteristic
42C22×C3×C7×C8C2×C23114yesdirect product factor
43C22×C7×C23C2×C3×C8128yesdirect product factor
44C22×C7×C23C22×C3×C417yes—
45C4×C7×C23C22×C3×C417yes—
46C4×C7×C23C24×C311yescharacteristic
47C23×C3×C23C7×C818yesdirect product factor
48C23×C3×C23C2×C4×C717yes—
49C2×C3×C4×C23C2×C4×C7121yes—
50C2×C3×C4×C23C23×C717yes—
51C3×C8×C23C23×C718yesdirect product factor
52C3×C7×C23C23×C811yescharacteristic, direct product factor, Hall
53C23×C7×C8C3×C2311yescharacteristic, direct product factor, Hall
54C24×C23C3×C4×C711yescharacteristic
55C22×C4×C23C3×C4×C717yes—
56C22×C4×C23C22×C3×C717yes—
57C2×C8×C23C22×C3×C7128yesdirect product factor
58C24×C3×C7C4×C2311yescharacteristic
59C22×C3×C4×C7C4×C2317yes—
60C22×C3×C4×C7C22×C2317yes—
61C2×C3×C7×C8C22×C23128yesdirect product factor
62C2×C7×C23C22×C3×C8114yesdirect product factor
63C2×C7×C23C23×C3×C411yescharacteristic
64C22×C3×C23C2×C7×C8128yesdirect product factor
65C22×C3×C23C22×C4×C717yes—
66C3×C4×C23C22×C4×C717yes—
67C3×C4×C23C24×C711yescharacteristic
68C23×C4×C7C2×C3×C2311yescharacteristic
69C22×C7×C8C2×C3×C23114yesdirect product factor
70C23×C3×C8C7×C2311yescharacteristic, direct product factor, Hall
71C23×C23C3×C7×C818yesdirect product factor
72C23×C23C2×C3×C4×C717yes—
73C2×C4×C23C2×C3×C4×C7121yes—
74C2×C4×C23C23×C3×C717yes—
75C8×C23C23×C3×C718yesdirect product factor
76C23×C3×C7C8×C2318yesdirect product factor
77C23×C3×C7C2×C4×C2317yes—
78C2×C3×C4×C7C2×C4×C23121yes—
79C2×C3×C4×C7C23×C2317yes—
80C3×C7×C8C23×C2318yesdirect product factor
81C7×C23C23×C3×C811yescharacteristic, direct product factor, Hall
82C2×C3×C23C22×C7×C8114yesdirect product factor
83C2×C3×C23C23×C4×C711yescharacteristic
84C24×C7C3×C4×C2311yescharacteristic
85C22×C4×C7C3×C4×C2317yes—
86C22×C4×C7C22×C3×C2317yes—
87C2×C7×C8C22×C3×C23128yesdirect product factor
88C23×C3×C4C2×C7×C2311yescharacteristic
89C22×C3×C8C2×C7×C23114yesdirect product factor
90C22×C23C2×C3×C7×C8128yesdirect product factor
91C22×C23C22×C3×C4×C717yes—
92C4×C23C22×C3×C4×C717yes—
93C4×C23C24×C3×C711yescharacteristic
94C22×C3×C7C2×C8×C23128yesdirect product factor
95C22×C3×C7C22×C4×C2317yes—
96C3×C4×C7C22×C4×C2317yes—
97C3×C4×C7C24×C2311yescharacteristic
98C3×C23C23×C7×C811yescharacteristic, direct product factor, Hall
99C23×C8C3×C7×C2311yescharacteristic, direct product factor, Sylow
100C23×C7C3×C8×C2318yesdirect product factor
101C23×C7C2×C3×C4×C2317yes—
102C2×C4×C7C2×C3×C4×C23121yes—
103C2×C4×C7C23×C3×C2317yes—
104C7×C8C23×C3×C2318yesdirect product factor
105C24×C3C4×C7×C2311yescharacteristic
106C22×C3×C4C4×C7×C2317yes—
107C22×C3×C4C22×C7×C2317yes—
108C2×C3×C8C22×C7×C23128yesdirect product factor
109C2×C23C22×C3×C7×C8114yesdirect product factor
110C2×C23C23×C3×C4×C711yescharacteristic
111C2×C3×C7C22×C8×C23114yesdirect product factor
112C2×C3×C7C23×C4×C2311yescharacteristic
113C23×C4C2×C3×C7×C2311yescharacteristic
114C22×C8C2×C3×C7×C23114yesdirect product factor
115C22×C7C2×C3×C8×C23128yesdirect product factor
116C22×C7C22×C3×C4×C2317yes—
117C4×C7C22×C3×C4×C2317yes—
118C4×C7C24×C3×C2311yescharacteristic
119C23×C3C7×C8×C2318yesdirect product factor
120C23×C3C2×C4×C7×C2317yes—
121C2×C3×C4C2×C4×C7×C23121yes—
122C2×C3×C4C23×C7×C2317yes—
123C3×C8C23×C7×C2318yesdirect product factor
124C23C23×C3×C7×C811yescharacteristic, direct product factor, Sylow
125C3×C7C23×C8×C2311yescharacteristic, direct product factor, Hall
126C24C3×C4×C7×C2311yescharacteristic
127C22×C4C3×C4×C7×C2317yes—
128C22×C4C22×C3×C7×C2317yes—
129C2×C8C22×C3×C7×C23128yesdirect product factor
130C2×C7C22×C3×C8×C23114yesdirect product factor
131C2×C7C23×C3×C4×C2311yescharacteristic
132C22×C3C2×C7×C8×C23128yesdirect product factor
133C22×C3C22×C4×C7×C2317yes—
134C3×C4C22×C4×C7×C2317yes—
135C3×C4C24×C7×C2311yescharacteristic
136C23C3×C7×C8×C2318yesdirect product factor
137C23C2×C3×C4×C7×C2317yes—
138C2×C4C2×C3×C4×C7×C23121yes—
139C2×C4C23×C3×C7×C2317yes—
140C8C23×C3×C7×C2318yesdirect product factor
141C7C23×C3×C8×C2311yescharacteristic, direct product factor, Sylow
142C2×C3C22×C7×C8×C23114yesdirect product factor
143C2×C3C23×C4×C7×C2311yescharacteristic
144C22C2×C3×C7×C8×C23128yesdirect product factor
145C22C22×C3×C4×C7×C2317yes—
146C4C22×C3×C4×C7×C2317yes—
147C4C24×C3×C7×C2311yescharacteristic
148C3C23×C7×C8×C2311yescharacteristic, direct product factor, Sylow
149C2C22×C3×C7×C8×C23114yesdirect product factor
150C2C23×C3×C4×C7×C2311yescharacteristic
subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC4C24×C3×C7×C23
socle, prime radicalC24×C3×C7×C23C4
LCSG ▷ C1
UCSC1 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×C7×C8×C23C2direct (N × H)
3. C23×C7×C8×C23C3direct (N × H)
7. C2×C3×C7×C8×C23C22direct (N × H)
9. C22×C7×C8×C23C2×C3direct (N × H)
10. C23×C3×C8×C23C7direct (N × H)
11. C23×C3×C7×C23C8direct (N × H)
15. C3×C7×C8×C23C23direct (N × H)
19. C2×C7×C8×C23C22×C3direct (N × H)
21. C22×C3×C8×C23C2×C7direct (N × H)
22. C22×C3×C7×C23C2×C8direct (N × H)
26. C23×C8×C23C3×C7direct (N × H)
27. C23×C3×C7×C8C23direct (N × H)
28. C23×C7×C23C3×C8direct (N × H)
32. C7×C8×C23C23×C3direct (N × H)
36. C2×C3×C8×C23C22×C7direct (N × H)
37. C2×C3×C7×C23C22×C8direct (N × H)
40. C22×C8×C23C2×C3×C7direct (N × H)
42. C22×C3×C7×C8C2×C23direct (N × H)
43. C22×C7×C23C2×C3×C8direct (N × H)
47. C23×C3×C23C7×C8direct (N × H)
51. C3×C8×C23C23×C7direct (N × H)
52. C3×C7×C23C23×C8direct (N × H)
53. C23×C7×C8C3×C23direct (N × H)
57. C2×C8×C23C22×C3×C7direct (N × H)
61. C2×C3×C7×C8C22×C23direct (N × H)
62. C2×C7×C23C22×C3×C8direct (N × H)
64. C22×C3×C23C2×C7×C8direct (N × H)
69. C22×C7×C8C2×C3×C23direct (N × H)
70. C23×C3×C8C7×C23direct (N × H)
71. C23×C23C3×C7×C8direct (N × H)
75. C8×C23C23×C3×C7direct (N × H)
76. C23×C3×C7C8×C23direct (N × H)
80. C3×C7×C8C23×C23direct (N × H)
81. C7×C23C23×C3×C8direct (N × H)
82. C2×C3×C23C22×C7×C8direct (N × H)
87. C2×C7×C8C22×C3×C23direct (N × H)
89. C22×C3×C8C2×C7×C23direct (N × H)
90. C22×C23C2×C3×C7×C8direct (N × H)
94. C22×C3×C7C2×C8×C23direct (N × H)
98. C3×C23C23×C7×C8direct (N × H)
99. C23×C8C3×C7×C23direct (N × H)
100. C23×C7C3×C8×C23direct (N × H)
104. C7×C8C23×C3×C23direct (N × H)
108. C2×C3×C8C22×C7×C23direct (N × H)
109. C2×C23C22×C3×C7×C8direct (N × H)
111. C2×C3×C7C22×C8×C23direct (N × H)
114. C22×C8C2×C3×C7×C23direct (N × H)
115. C22×C7C2×C3×C8×C23direct (N × H)
119. C23×C3C7×C8×C23direct (N × H)
123. C3×C8C23×C7×C23direct (N × H)
124. C23C23×C3×C7×C8direct (N × H)
125. C3×C7C23×C8×C23direct (N × H)
129. C2×C8C22×C3×C7×C23direct (N × H)
130. C2×C7C22×C3×C8×C23direct (N × H)
132. C22×C3C2×C7×C8×C23direct (N × H)
136. C23C3×C7×C8×C23direct (N × H)
140. C8C23×C3×C7×C23direct (N × H)
141. C7C23×C3×C8×C23direct (N × H)
142. C2×C3C22×C7×C8×C23direct (N × H)
144. C22C2×C3×C7×C8×C23direct (N × H)
148. C3C23×C7×C8×C23direct (N × H)
149. C2C22×C3×C7×C8×C23direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×C7×C232
C22×C3×C7×C8×C232
C23×C7×C8×C233
C23×C3×C8×C237
C23×C3×C7×C823